The Atlas of Rules

Build each stage of a platform as a sloping talud surmounted by a vertical tablero: a rectangular panel framed by a projecting border and cantilevered slightly over the slope below; repeat the pair at every stage.

Teotihuacan: tablero taller than talud (often ≈ 1½–2 × its height)

Treat the stepped pyramid as a platform whose purpose is to lift a small temple towards the sky, reached by a single steep stair on its principal face; enlarge it by building a new shell over the old.

To make a small temple dominate the skyline, build a tall crest of masonry on its roof, set over the heavy rear wall and pierced with openings to lighten it; model it in painted stucco with the figure of the ruler or god.

Align the principal avenue and every building of the city to a single bearing (at Teotihuacan about 15½° east of north) so that the whole city shares one orientation with its great pyramids.

In a corbelled vault each course may project only a little beyond the one below, and each overhanging half must have its centre of gravity behind the inner face of its wall; load the tails of the stones with heavy hearting until the capstone is set.

stable if Σ W·x (behind the springing edge) > Σ W·x (in front)

Keep the span of a corbelled room to about three metres or less and make it long instead of wide; to enlarge a building, add parallel galleries rather than widening them.

Maya room spans ≈ 1.5–3 m; vault rise ≈ 1–1½ × span

Shape each block of the finest walls to the irregular outline of the stones already laid, grinding the joints until they close without mortar; let the stones be as large and as many-sided as they come.

Give doors, windows and niches sides that lean inward, so that each opening is narrower at the top than at the bottom, and span it with a single stone lintel.

top width ≈ ⅘ of bottom width

Batter every wall a few degrees, thicker at the foot than at the top, and round its outer corners.

batter ≈ 3–5°

For the most sacred buildings lay rectangular blocks in level courses that diminish in height towards the top of the wall, with pillowed faces and joints closed without mortar.

Frame the doorway of a temple, palace or royal precinct with a second, larger recessed trapezoid, so that the opening steps back in two planes; keep the single jamb for ordinary buildings.

A beam is squeezed along its top and stretched along its bottom; stone resists squeezing well but stretching badly, so a stone lintel must be deep in proportion to its span and cannot be long. Doubling the span quadruples the stress from its own weight.

σ ≈ ¾ ρ g L² ⁄ d (self-weight) · + ≈ 0.43 ρ g L³ ⁄ d² (a wall above)

In a bonded masonry wall the courses above an opening corbel over it of their own accord, so the lintel carries roughly the triangle of wall with sides rising at 60° from its ends; everything above that triangle arches onto the jambs.

load ≈ equilateral triangle on the span: W ≈ 0.43 · L² · t · ρ g

Weight for weight, timber is many times stronger in bending than stone; long spans in trabeated buildings are therefore roofed with timber beams resting on stone walls and columns.

bending strength ÷ density: oak ≈ 70 · limestone ≈ 3 (kPa per kg ⁄ m³)

At Stonehenge each lintel is pegged to its uprights by a mortise-and-tenon joint and locked to its neighbours by tongue-and-groove joints, and each is curved to follow the circle: the joints of a timber frame reproduced in dressed sarsen.

Orient the principal axis of the monument on the rising or setting of the sun at a solstice, so that the architecture frames it.

In a megalithic chamber the room can be no wider than the largest slab the builders can raise over it; the upright stones are set to suit the capstone, not the other way round.

Specify the slope of a pyramid face by its seked: the number of palms (and fingers) by which the face runs inward for every cubit of rise. The height follows from half the base divided by the seked, times seven.

seked = 7 × (½ base ⁄ height) palms · angle = arctan(7 ⁄ seked)

The Great Pyramid of Khufu was laid out 440 royal cubits square and 280 cubits high, a seked of exactly 5½ palms and a slope of about 51°50′.

½ × 440 ÷ 280 × 7 = 5½ palms → arctan(7 ⁄ 5.5) ≈ 51.84°

Lay out Egyptian buildings in royal cubits (≈ 52.4 cm) of seven palms of four fingers, so that every dimension and every slope can be stated in whole or simple fractional units.

1 royal cubit = 7 palms = 28 fingers ≈ 0.524 m

The royal tomb grew in about a century from a low battered mastaba, through a stack of diminishing mastabas (the step pyramid), to a smooth-faced true pyramid; a failure of stability in between forced the change of slope of the Bent Pyramid.

When a steep pyramid shows signs of distress, flatten the slope of the upper part: Sneferu’s pyramid at Dahshur rises at about 54° for its lower part and about 43° above.

lower seked 5 palms (≈ 54.5°) → upper ≈ 7½ palms (≈ 43°)

Make the outer faces of pylons, enclosure walls and mastabas lean inward as they rise, so that the wall is thickest at its foot; mark the corners with a torus roll.

batter of pylons ≈ 1 : 8 (about 83°)

Finish every Egyptian wall, gate and shrine with a torus roll and, above it, a deep cavetto (the gorge) curving outward under a flat band.

Arrange the temple on a single axis (pylon, open court, hypostyle hall, sanctuary) so that at each threshold the floor rises, the roof descends and the light diminishes, until the god’s shrine is reached in darkness.

In a great columned hall make the columns of the central aisle taller than the rest, and fill the difference in height with stone grilles, so that light enters high up over the side aisles.

Model the column on a plant of the Nile (a papyrus stem or bundle, a bundle of lotus, a palm trunk), bound at the neck with five bands, standing on a round base disc and carrying a small square abacus.

Give the columns of the tall, clerestory-lit central aisle open papyrus capitals and those of the lower aisles closed buds.

Cut a shaft with sixteen shallow flutes or facets, give it a square abacus and a low round base: the “proto-Doric” column of the Middle Kingdom rock tombs at Beni Hasan.

Cut the obelisk from a single block of granite, square in section, tapering to about two-thirds of its base width, capped by a pyramidion, and about ten times as tall as it is wide; set them in pairs before the pylon.

height ≈ 9–11 × base width

Build the temple tower as a solid stepped mass of brick, each stage set back on the one below, with a shrine on its summit and stairs climbing its principal face.

Build the core of sun-dried mud brick; case it with baked brick laid in bitumen, with layers of reed matting through the mass and weep holes to let water out.

Break the long faces of brick walls with regularly spaced shallow buttresses (or recesses), so that light and shadow give scale and rhythm to a material without mouldings.

Where the roof is of timber, make the columns tall, slender (about twelve diameters) and widely spaced, and fill the hall with them on a square grid.

height ≈ 12 D · spacing ≈ 5 D (Apadana: ≈ 19 m, D ≈ 1.6 m, axes ≈ 8.7 m)

Crown the Persian column with two kneeling bulls (or lions, or griffins) set back to back; lay the main beam in the saddle between their necks, and the cross-beams on their heads.

Set the royal buildings on a raised terrace; climb it by broad double stairs with very low, deep steps, and line their parapets with reliefs of the procession that climbs them.

steps of the Great Stairway ≈ 10 cm high

Build a round chamber of horizontal courses, each ring projecting a little beyond the one below, dress the inner face to a smooth curve, close the top with a capstone, and heap earth over the whole.

Over a large lintel leave a triangular opening by corbelling the courses on either side, so that the wall’s weight passes to the jambs; close the triangle with a thin slab.

Build fortification walls of huge, roughly shaped limestone blocks, with small stones packed into the gaps between them; reserve cut ashlar for gates and façades.

Plan the principal hall as a long rectangle entered on its short side through a porch with two columns between the side walls and a vestibule, and put a great round hearth in the centre of the hall, ringed by four columns carrying a raised roof.

Make the timber column narrower at the foot than at the top, set it on a stone disc, and crown it with a fat cushion capital and a square abacus.

Build an arch of wedge-shaped stones (voussoirs) whose joints all point towards the centre of the curve: each stone can then fall only by forcing its neighbours apart, and so the ring holds itself up once the last stone, the keystone, is in place.

joints ⟂ intrados (radial to the centre)

An arch pushes outward on its supports as well as down. For the same load and span the horizontal push grows as the rise falls: halve the rise and you roughly double the thrust.

H ≈ w·s² ⁄ 8f (uniform load w per metre, span s, rise f)

A chain hanging freely takes the one shape in which it is in pure tension; turn that shape upside down and it is the shape of an arch in pure compression for the same loads.

free-hanging chain (catenary) ↔ ideal arch under its own weight

Trace the path of the resultant force through the stones from crown to foundations. If a line can be found that stays everywhere within the masonry, the arch will stand; where it touches an edge, a hinge will open, and four hinges make it collapse.

Draw a pointed arch from two centres on the springing line. With the centres at the springings, the arch is equilateral; with centres beyond them, a lancet; with centres between them, a drop arch. Late Gothic added arches of four centres and the reversed curve of the ogee.

r = s (equilateral) · r > s (lancet) · s⁄2 < r < s (drop)

Classify arches by the geometry of the intrados: the number and position of the centres (one for round and segmental, two for pointed, three or four for basket and Tudor arches), or a single continuous curve (ellipse, parabola, catenary), and distinguish true arches, with voussoirs, from corbelled “arches” of oversailing courses.

A round arch must rise half its span, so arches of different spans on one springing line have crowns at different heights. A pointed arch can be given any rise at least half its span, so arches of different widths can all be made to reach the same height.

round: f = s⁄2 (fixed) · pointed: r = (s²⁄4 + f²) ⁄ s for any f ≥ s⁄2

To find how thick the pier of an arch must be, divide the intrados into three equal parts; draw a line from the springing through the first division, and produce it beyond the springing by its own length. A vertical through the end of that line marks the outer face of the pier.

pier ≈ ¼ span (semicircle) · ≈ 0.23 span (equilateral) · less for a lancet

An arch cannot support itself until it is complete. Build it on a timber centring shaped to the intrados, strike the centring only when the keystone is set and the mortar has taken, and reuse it from bay to bay, which is one reason to keep the spans of an arcade equal.

In a row of arches the thrusts of neighbouring arches cancel over each intermediate pier, which may therefore be slender; at the ends of the row nothing balances them, so the outermost piers must be made wider or buttressed.

Over a doorway or window spanned by a lintel, turn an arch in the wall above so that the weight of the masonry is carried round the opening onto the jambs, leaving the lintel to carry only the small triangle of wall beneath the arch.

Front a basilican church with two storeys of the order: the lower as wide as nave and chapels, the upper as wide as the nave alone, joined by scrolls and crowned by a pediment.

Il Gesù (reconstructed): lower ≈ 34 m wide, upper ≈ 20 m; scrolls fill the difference

Build up a façade in layers: let the wall advance in stages towards the central bay, double the pilasters or columns as it advances, and break the entablature forward over every support.

Mark the main door by setting a segmental pediment inside a larger triangular one, or by breaking the pediment open to hold a cartouche or statue.

To unify a façade of two or more storeys, run a single order of pilasters or columns through all of them, from the base to the main cornice.

To combine the unity of a central plan with the direction a church needs, plan the space as an oval, set across the axis to widen the view of the altar, or along it to lead the eye towards it.

Divide the inside of a dome into coffers that shrink in step with the meridians as they rise, so that their perspective deepens the dome and draws the eye to the light of the lantern.

Let the wall of a façade curve in plan (concave, convex, concave) so that the front answers the street, the light and the space within, and the order bends with it.

Light the climax of a Baroque interior (the altar or the dome) from windows the visitor cannot see, so that it seems to glow of itself.

Proportion every step so that twice its height plus its depth equals the length of a natural pace on the level: two feet, or 24 pouces.

2 r + g = 24 pouces ≈ 65 cm (modern practice: 60–66 cm)

Hold that the beauty of the orders comes from fixed proportions that please by nature, as consonances do in music; where the ancients differ, the differences answer to circumstances such as distance and height.

Make the stair of a palace a processional space: one broad flight rising to a landing, then turning back in two flights either side to arrive at the principal floor beneath a single great vault.

Give the flight of steps before a temple an odd number, so that one who starts with the right foot will also set the right foot first on the top.

Raise the central nave above the lean-to roofs of the aisles and pierce the upper walls with a row of windows, the clerestory, so that the middle of a wide hall is lit from high on both sides.

nave height ≈ 1½–2 × nave width; one window over each bay of the arcade

Cover a wide nave with triangulated timber trusses rather than a vault: their tie-beams take the outward thrust, so the walls need only carry weight and may stand on slender columns.

nave span ≈ 13–24 m on trusses (Santa Sabina ≈ 13.5 m; Old St Peter’s ≈ 24 m)

Order the church as a single path: a colonnaded atrium, a narthex, the long nave drawn toward the altar, and the apse framed by a triumphal arch at the end of the axis.

To set a round dome on a square bay, imagine a sphere passing through the four corners of the square; cut it by the four vertical planes of the sides and by the horizontal plane of their crowns. The four spherical triangles left in the corners are the pendentives; the circle at their top carries the dome.

sphere radius R = S⁄√2; arches = sphere ∩ side planes (radius S⁄2); ring at h₀ + S⁄2 with radius S⁄2

Throw an arch diagonally across each upper corner of the square, backed by a small conch or cone; the four squinches and the four walls between them make an octagon, on which a round dome can sit.

octagon side = S (√2 − 1) ≈ 0.414 S; squinch span = octagon side

If the sphere that forms the pendentives is simply continued above the ring, the bay is covered by a single shallow dome, the sail vault or pendentive dome, which springs directly from the four corners.

crown = h₀ + S⁄√2 (lower than a hemisphere on the ring, h₀ + S)

Lift the dome on a cylindrical or polygonal drum pierced with windows between the ring of the pendentives and the springing of the dome, to light the centre of the church and to make the dome visible from afar.

drum height ≈ ⅓–1 × its radius (Byzantine); much taller in Russian churches

Pierce the base of the dome with a close ring of windows between buttressing ribs, so that the shell seems to float on light rather than to rest on the arches.

Hagia Sophia: 40 windows and 40 ribs; dome ≈ 31 m across, crown 55.6 m above the floor

To turn a domed square into a long church, open two of its great arches into semidomes of the same radius, and open the semidomes in turn into smaller exedrae, so that the space expands by stages from the central dome.

Hagia Sophia: central square 31 m; two semidomes of radius ≈ 15.5 m; four exedrae; interior ≈ 31 × 70 m

Where no semidome takes the thrust of the dome, on the north and south, carry it across the aisles to massive buttress piers that rise above the roofs, and fill the great arches with thin walls pierced by windows.

A shallow dome pushes harder outward than a steep one of the same span: when the first dome of Hagia Sophia fell, it was rebuilt with a higher rise so that its thrust would be more nearly vertical.

rise raised by ≈ 6 m (about 20 Byzantine feet), 558–562

Divide a square naos into nine bays: dome the central one on a drum over four columns, barrel-vault the four arms of the cross, and cover the four corner bays lower; add three apses to the east and a narthex to the west.

arms = central bay width; corners lower; dome on pendentives over four columns

Crown the four corner bays of a cross-in-square with smaller domes on drums around the great central dome, making a five-domed quincunx.

Lift each dome on a tall drum and give it a bulbous outer shell that swells beyond the drum and draws up to a point under the cross; build the shell as a light timber frame sheathed in metal over the structural vault below.

onion Ø ≈ 1.2–1.3 × drum Ø; height ≈ 2 × its base radius

Divide each wall into bays with pilaster strips and end each bay in a semicircular gable, the zakomara, that follows the curve of the barrel vault behind it; shape the roof to the gables.

usually 3 zakomary per façade, one per bay

Mount from the walls to the drum in stepped tiers of decorative gables with pointed ogee heads, the kokoshniki, each tier set back and narrower than the one below, so that the body of the church rises into its domes in a pyramid of arches.

Instead of a dome, crown the church with a tall octagonal pyramid of masonry (the shatyor, or tent) set on an octagon over the square body, with kokoshniki at the transitions and a small cupola at the top.

Ascension, Kolomenskoye (1532): ≈ 62 m high; cube → octagon → tent → cupola

Take the section of the standard timber, the cai, as the module of the building: it is 15 fen high and 10 fen wide, and every member is dimensioned in fen.

1 cai = 15 fen (height) × 10 fen (width)

There are eight grades of cai, from 9 × 6 cun down to 4½ × 3 cun; use the first for great halls of nine or eleven bays and the smallest for pavilions and ceiling coffers.

grades 1–8: 9×6, 8¼×5½, 7½×5, 7.2×4.8, 6.6×4.4, 6×4, 5¼×3½, 4½×3 cun

Between two stacked arms leave a spacer of 6 fen, the zhi; a cai plus a zhi, 21 fen, is a “full timber” (zucai) and the vertical step of each tier of a bracket set.

zucai = 15 + 6 = 21 fen

Build the bracket set (puzuo) of blocks (dou) and arms (gong) crossing at right angles, each tier projecting no more than 30 fen beyond the one below; a set of n projecting tiers is called an (n + 3)-puzuo, from four to eight.

projection per tier ≤ 30 fen · rise per tier = 21 fen · puzuo = tiers + 3

Where the eaves must project far, replace the outer arms by slanting levers (ang) that run down and out beyond the bracket set and rise inward to be held down under an interior purlin.

Shape the underside of every arm’s end in four or five facets, and reduce the upper third of a spindle column by the same method, so that no member ends in a hard corner.

For a great hall, measure the distance between the front and back eave purlins and raise the ridge purlin above them by one third of it; for lesser halls, by about a quarter.

rise = ⅓ × depth between eave purlins (halls); ≈ ¼ (+ a small addition) for ting-tang halls

Draw a straight line from the ridge purlin to the eave purlin and lower the first purlin below the ridge from it by one tenth of the rise; from that point draw a new line to the eave purlin and lower the next purlin by one twentieth; continue, halving each time.

drop₁ = H⁄10 · drop₂ = H⁄20 · drop₃ = H⁄40 · …

Give each rafter span a fixed slope, increasing from the eaves to the ridge: half-pitch (5 jǔ) for the eave span, then about 6½ or 7, 7½, and 9 tenths for the span below the ridge.

rise of span k = run × (0.5, 0.65, 0.75, 0.9) for four spans

From the intermediate bracket set of the second bay the rafter ends advance progressively outward, so that at the corner the eave projects further (by 4 cun for a hall of one bay, 5 for three bays, 7 for five bays) and rises with the corner beam.

Give the hipped roof (wudian) to the highest halls, the hip-and-gable (xieshan) to the next, and the gabled roofs, overhanging (xuanshan) and flush (yingshan), to lesser buildings and houses; double the eaves for the very highest.

double wudian > double xieshan > wudian > xieshan > xuanshan > yingshan

Make the central bay (mingjian) the widest and let the bays diminish toward the ends; keep the eave columns no taller than the width of their bay.

Give a hall an odd number of bays across the front (three, five, seven, nine or eleven) so that the central axis passes through a bay and a doorway, never through a column.

Make each column a little taller than the one nearer the centre, so that the corner columns stand highest: by 2 cun per column outward from the central bay, giving 6 cun at the corners of a seven-bay hall and 1 chi 2 cun for thirteen bays.

corner rise = (bays − 1) cun · each step outward = 2 cun

Set the heads of the outer columns inward: on the front and back by 1 fen for every chi of height (1/100), on the ends by 8 li per chi (8/1000); the corner columns lean both ways.

lean = 1/100 of height (front) · 8/1000 (flanks)

Lay out the royal city as a square of nine li on each side, with three gates in each wall and, within, nine roads running north–south and nine east–west, each wide enough for nine chariots abreast.

side = 9 li · 3 gates × 4 sides · 9 jing + 9 wei · road width = 9 gui (tracks)

With the palace at the centre facing south, place the ancestral temple on its left (east) and the altar of soil and grain on its right (west); put the court in front and the market behind, each one fu (a hundred paces) square.

Find true north and south with a plumb-lined post and the shadows of sunrise and sunset, checked by the pole star; then turn the principal hall of every compound to face south.

Project the eaves from the centre of the eave purlin by 3½ chi for rafters of 3 cun diameter and 4 to 4½ chi for rafters of 5 cun; beyond them add flying rafters projecting six-tenths as far again.

eave projection ≈ 3½–4½ chi · flying rafters = 0.6 × eave projection

Prefer richness of meaning to clarity of meaning: accept the complex and contradictory demands of program, structure and context, and resolve them in a “difficult whole” rather than excluding them for the sake of simplicity.

Let an element do more than one thing and belong to more than one order at once: a wall that is both inside and outside, a column that is both structure and sign.

A building can carry meaning in two ways: as a duck, whose whole form is shaped into a symbol, or as a decorated shed, a plain, efficient shelter with symbols applied to it. Prefer the shed.

Where a building is seen from a moving car across a car park, communication depends on large signs placed at the road, and the building behind them can be a simple shed.

Speak two languages at once: the technical language of modern construction for the profession, and a language of familiar, historical or popular signs for everyone else.

Give every room in which people spend time daylight from at least two sides; plan the building in narrow wings so that this is possible.

Shape the space between buildings as carefully as the rooms: make outdoor spaces partly enclosed and convex, like outdoor rooms, rather than leftover gaps around freestanding objects.

Make the roof visible, and bring its eaves down low enough, and far enough out, that the building reads as a shelter from the street and the garden.

Frame a sphere by subdividing the triangular faces of an icosahedron and projecting the points onto the sphere; the triangulated net of short struts spreads loads through the whole surface and encloses the greatest volume for the least material.

full sphere of frequency ν: 30ν² struts, 20ν² triangles, 10ν² + 2 hubs

A saddle surface in which two opposite corners are raised and two lowered is a hyperbolic paraboloid; it contains two families of straight lines, so it can be formed with straight boards and yet has double curvature.

y = k·x·z (or any bilinear patch through four points not in one plane)

A thin curved shell carries load by the membrane forces in its surface, not by bending; so shape it to its loads and supports, and it can span great distances with a thickness of a few centimetres.

span ÷ thickness of a good shell ≈ 400–1000 (an egg: ≈ 100)

A fabric or cable-net roof is stable only if it is prestressed into an anticlastic (saddle) shape, curved upward in one direction and downward in the other, so that wind from either side meets a surface already in tension.

Hang a chain (or a model of strings loaded like the building) from two points; the curve it takes, turned upside down, is the ideal shape of an arch carrying the same loads, in pure compression.

A masonry arch stands if a line of thrust (the path of the compressive force through it) can be drawn entirely within its thickness; the catenary arch contains its own line of thrust exactly, the semicircle only if it is thick enough.

catenary: thrust line = centre line · semicircle: needs t ⁄ R ≳ 0.11

Move ducts, pipes, lifts and escalators out of the building’s core onto its exterior, colour-code them, and let them become the architecture, leaving the interior free and making the plant easy to reach and replace.

Span the whole depth of the building with the structure so that each floor is a single open plate, adaptable to uses that cannot be foreseen.

Pompidou: 48 m clear span between column lines, 13 bays of 12.8 m

Define a form by a few parameters and a rule, and let every element vary continuously with them, instead of repeating identical elements.

floor i: rotation = i × θ, scale = 1 − (1 − τ)·(i ⁄ n)^1.3

Place a point (or curve) in the design and let a property of every element (size, rotation, opening) be a function of its distance from that point.

r(panel) = r_min + (r_max − r_min) · d ⁄ d_max

Tilt, twist, fragment and collide the volumes of a composition, so that no floor is level, no wall upright and no angle square except where the ground forces it.

On an equator-facing window, make the overhang deep enough to shade the whole window from the high noon sun of summer, and no deeper, so that the low winter sun passes beneath it into the room.

noon sun altitude = 90° − latitude ± 23.44° · overhang D = window height ⁄ tan(summer altitude)

Place heavy materials (concrete, stone, brick, earth) where the winter sun falls on them inside the insulated envelope, so that they absorb heat by day and give it back at night.

Make the envelope so well insulated and airtight, and recover so much heat from the exhaust air, that the building needs almost no active heating.

heating ≤ 15 kWh/m²·a · airtightness n₅₀ ≤ 0.6 h⁻¹ · heat recovery ≥ 75%

Edge the beds before the portico with box, clip box into figures of animals set facing one another, and in the garden beyond cut it into many shapes, even into the letters of the owner’s name.

Give the villa garden porticoes for sun and for shade, grottoes made in imitation of nature with tufa and shells, evergreens planted in straight rows at equal distances, and hedges of box; set vines on columns over the walks.

Lay the ideal garden out on a circular plan in concentric rings, woods outside, then meadows and orchards, then gardens of knots and clipped evergreens divided by walks, with water and the fountain of the goddess at the centre.

On a hillside, cut the garden into level terraces held by retaining walls, join them by stairs and ramps on a single axis that runs from the house, and lead water down the axis from level to level.

Bring water down a slope in a stepped stone channel whose scrolled sides break it into ripples and small falls, on the axis of the garden, so that the water itself becomes an ornament as it descends.

Beside the house, enclose a small garden with walls or high hedges for the owner’s private use, with flowers, herbs, fruit trees and a fountain, apart from the public garden on the axis.

Lay the parterres where they are seen from above, from the windows of the house or from a raised terrace, since their patterns are lost to someone standing among them; keep everything near them low.

Approach the house by a wide avenue of double or quadruple rows of trees; behind it lay the parterres of embroidery, then parterres of grass, then the groves with their walks and fountains, and close the walks with perspectives that deceive the eye.

Make the parterre oblong rather than square, since figures somewhat oblong look better on the ground than square ones; later practice made it as broad as the front of the house and about twice as long.

parterre oblong; later practice: width ≈ house front, length ≈ 2 × width

Proportion the width of an allée to its length: a walk of about 100 toises (some 200 m) should be 5 or 6 toises wide, one of 200 toises 7 or 8, and so on, so that a long walk does not seem to narrow to a slot in the distance.

100 toises long → 5–6 wide · 200 → 7–8 · 300 → 9–10 · 400 → 10–12 (1 toise ≈ 1.95 m)

Organise the whole garden on one axis from the centre of the house to the horizon; lay out the parts symmetrically about it, and use water, slopes and hidden changes of level so that the view along it reveals the garden by stages.

In place of a wall, bound the garden with a ditch whose side towards the garden is a vertical retaining wall and whose other side slopes up to the field, so that cattle are kept out and the view from the house runs on uninterrupted into the park.

ditch c. 1.5–2 m deep: wall on the garden side, turf slope on the park side

Let the character of the site direct the design: whether the ground should be raised or lowered, a view opened or closed, water led or stilled. Prefer variety, surprise and concealed bounds to symmetry.

Compose the garden as a painter composes a landscape: in groups of trees for light and shade, with water winding through the hollows and a temple or bridge placed where it will be seen across it, so that each turn of the walk opens a new picture.

Bring grass up to the walls of the house and keep the cattle off it with a ha-ha; plant trees in clumps on the rising ground and as single specimens on the lawn; and enclose the park with a belt of woodland, with a drive inside it from which the park is seen.

Make the lake by damming a stream in its valley, give it a winding, irregular outline with promontories, and hide both its ends behind trees, a bend or a bridge, so that it seems a river flowing through the park.

Show the natural beauties of a place and hide its defects; give it an appearance of extent and freedom by disguising the boundary; conceal every interference of art, so that the whole seems natural; and remove or hide whatever serves only convenience and cannot be made ornamental.

Let the approach be the natural road to the house, without needless turns; let it show the house first at an angle, so that two sides are seen, and not head-on down an avenue; and make it plain that the road leads to this house and nowhere else.

Do not bring the park up to the walls of a house: give the house a terrace, steps and a flower garden in formal lines between it and the park, because the house is a work of art and must be joined to the landscape by art.

Value in a landscape the qualities painters seek, roughness, sudden variation and irregularity, which are neither beautiful nor sublime but picturesque; keep old trees, broken banks and intricate views, and do not smooth everything into lawn and clump.

Plant each tree and shrub apart, with room to grow to its full natural form, so that it can be seen and admired as a specimen; prefer foreign kinds to native ones, so that the place is recognised at once as a garden, a work of art.

Raise tender plants in glasshouses through the winter and plant them out in early summer, massed in beds cut in the lawn or on a terrace, in blocks, ribbons or patterns of a few strong colours; lift them in autumn and replant for the next season.

Glaze a roof in a series of small ridges and valleys, the panes set at an angle on light sash bars, so that the low sun of morning and evening strikes the glass more squarely, the roof is stiffened by its folds, and the rain runs into gutters in the furrows.

Build on a single grid of 24 feet, set by the 8-foot spacing of the roof ridges and the size of the largest pane of glass, so that every column, girder, sash bar and pane is one of a few standard parts that can be made by machine, delivered and bolted together in any number.

pane 49 × 10 in · ridge 8 ft · bay 24 ft · spans 24, 48, 72 ft · 1,848 × 408 ft

Lay out the public park with a drive round its edge, lakes dug in the hollows and the spoil raised into mounds, winding paths through groves and lawns inside; sell plots for villas round the boundary to pay for it, their gardens facing the park.

Carry the city’s cross traffic through the park on roads sunk below its surface, out of sight; inside it, give carriages, riders and walkers each their own system of drives, rides and paths, crossing one another by bridges and arches so that none need meet.

Place perfectly hardy plants, foreign as well as native, in woods, meadows, banks and the edges of water where they will thrive and spread without further care, instead of renewing tender bedding every year.

Design the garden with the house and as part of it: carry the lines of the house into the garden in terraces, walls, walks and hedges, divide it into enclosures, and let the architect who designs the house design these as well.

Arrange a long border in a sequence of colours: at the ends, cool blues, whites, pale yellows and pinks with grey foliage; towards the middle, stronger yellows and oranges rising to the deepest reds at the centre; then back again. Plant in long, thin, overlapping drifts rather than blocks.

Munstead Wood main border: 200 × 14 ft · cool → warm → hot → warm → cool

Divide the garden into a sequence of enclosures by hedges and walls, each with its own character, colour or season, joined by openings on axes, so that the garden is discovered room by room and never seen at once.

Where the ground is flat and the water lies deep, raise it from wells by wheels into tanks and channels, and lay the garden out in regular, symmetrical plots round them, with walks, beds of flowers and trees in order.

Draw into the garden the views beyond its walls: a distant hill, a neighbour’s trees, a temple roof, the sky above and the water below, and the changes of the seasons. Adapt the design to the site and borrow from what lies round it.

Make mountains of piled rock, and choose single rocks for their forms: the limestones of Lake Tai, eroded by water into holes, hollows and folds, are the finest; set them up like peaks, singly or in groups, against white walls or by water.

Set stones by first placing a principal stone and then others that answer it, following the “request” each makes; recall famous landscapes; and obey the taboos: a stone that lay flat in nature should not be set upright, nor one that stood upright laid flat.

Where there is no pond or stream, make the landscape of stones alone: set rocks as mountains, cliffs and islands, and let gravel or sand stand for water.

Gather the weight and thrust of the vaults onto a few points by ribs, carry them down piers and out across the aisles by flying buttresses to external buttress piers, and the wall between the supports carries almost nothing, so it can be opened into windows.

Divide the elevation of the nave into three storeys: a tall arcade opening into the aisles, a low triforium passage in the thickness of the wall against the aisle roof, and a clerestory of large windows reaching up into the vault.

Chartres: arcade ≈ clerestory, triforium a narrow band between

Carry the outward thrust of a high vault across the aisle roof by a half-arch of stone (the flyer) that leans against the clerestory wall at the level of the vault’s haunches and springs from a massive buttress pier outside the aisle, which takes the thrust down to the ground.

Load the top of a buttress pier with a heavy pinnacle: the added weight steepens the resultant of the flyer’s thrust and the pier’s own weight, keeping it well inside the base of the pier.

Draw a square the width of the pinnacle; inscribe a second square on the mid-points of its sides, and a third on the mid-points of the second. Turned back into line, the squares give the widths of the body, its panels and the spire; every height is a multiple of the first side.

widths: s, s⁄√2, s⁄2, s⁄(2√2) … · heights: n × s

To make a square of half the area of another, join the mid-points of its sides: the rotated square so drawn has exactly half the area. Repeat to halve again.

side of inscribed square = s ⁄ √2 · area = ½ s²

Proportion the cross-section of a church from the square: let the total height to the crown of the nave vault equal the total width, and take the heights of the aisles and the springing of the vaults at whole divisions of that square.

H = W · levels at k × (W ⁄ n)

Alternatively, fit the section to an equilateral triangle standing on the width, so that the height is √3⁄2 of the width, and take the levels at equal divisions of that height.

H = W · √3⁄2 ≈ 0.866 W · Milan: 96 braccia wide → 83.1, rounded to 84 = 6 × 14

A building’s form must follow the geometrical rules of the craft (scientia) as well as the experience of the builder (ars); neither is sufficient alone.

Proportion every member of a vaulted choir from its clear width: the wall a fixed fraction of the width (a tenth, as the rule is usually read), and buttresses, window jambs, mullions and ribs as simple multiples or fractions of the wall.

wall ≈ W ⁄ 10 · buttress, jambs, ribs = multiples of the wall

Instead of cutting separate openings through a plate of wall, fill the whole window arch with glass and divide it by slender stone bars (mullions below, and circles, foils and sub-arches above) set out with compasses from the centres of the window’s own arches.

Date a Gothic window by its tracery: openings pierced through a plate (twelfth century); geometric bar tracery of circles and foils (thirteenth); flowing curvilinear tracery of ogees and daggers (fourteenth); and, in England, perpendicular panel tracery with mullions running straight up into the arch (late fourteenth to sixteenth).

Design a rose by dividing the circle into equal sectors; fill each with the same traceried unit (a light with a cusped pointed head, an oculus above) radiating from a central hub, so that the whole is generated from one sector by rotation.

n petals · each sector 360° ⁄ n

End the choir in a semicircle of piers; wrap an aisle (the ambulatory) around it, and open a ring of chapels off the ambulatory, each with its own altar and windows, so that the east end steps down in tiers of roofs and buttresses.

Within a generation the nave vaults of the French cathedrals rose from about 37 m to 48 m while their spans stayed near 15 m: the ratio of height to width climbed from about 2.3 to over 3, until at Beauvais part of the choir vault fell.

Chartres ≈ 37 m · Reims ≈ 38 m · Amiens ≈ 42 m · Beauvais ≈ 48 m

Do not lay the stylobate dead level: raise it gently toward the middle of each side, for a perfectly flat platform looks to the eye as if it sagged.

Parthenon: rise ≈ 11 cm over 69.5 m (flanks), ≈ 6.5 cm over 30.9 m (fronts), about 1 : 600

Make the corner columns thicker than the rest by a fiftieth of their diameter, because, seen against the open sky, they look thinner than they are.

D corner = D × (1 + 1⁄50); Parthenon: 1.95 m against 1.905 m

Set the outer columns so that they incline slightly inward, toward the cella wall, the middle columns of the fronts plumb on their own axes and the corner columns leaning both ways.

Parthenon: lean ≈ 6–7 cm over 10.43 m (≈ 1 : 150); the axes would meet some 2 km above the ground

Give the shaft a continuous convex curve from base to necking, departing from a straight taper by only a few hundredths of a diameter, so that the column seems to flex under its load rather than to pinch.

Parthenon: greatest departure ≈ 1.7 cm on a shaft 9.6 m high (≈ 1⁄110 D)

A Doric frieze cannot obey all three of its rules at a corner: triglyphs over the column axes, a triglyph turning the corner, and metopes as broad as they are high. One of them must give way.

displacement of the corner triglyph = e − t⁄2 (e = half-depth of the architrave, t = triglyph width)

To end the frieze with a triglyph at the corner and keep every metope square, move the corner column inward: make the last intercolumniation narrower than the others by the amount the corner triglyph is displaced.

contraction c = e − t⁄2; Parthenon: 4.29 m normal, 3.68 m at the corners (c ≈ 0.61 m)

Keep every triglyph centred over a column and every intercolumniation equal, and finish the frieze at the corner with a half-metope half a module wide.

corner semi-metope = ½ M (Vitruvius); equal intercolumniations throughout

Let one simple ratio govern the whole: at the Parthenon the lower diameter is to the axial spacing, the width of the stylobate to its length, and the height of the order to the width of the front, all very nearly as 4 is to 9.

D : axial spacing ≈ 1.905 : 4.296 ≈ 4 : 9; stylobate 30.88 : 69.50 ≈ 4 : 9

Plan a Greek theatre by inscribing three squares in the orchestra circle; the side nearest the scene fixes the front of the stage (proscaenium), and the scene is set back to a line tangent to the circle.

proscaenium at r cos 45° ≈ 0.71 r from the centre; logeion 10–12 ft high

Build walls of squared stone in regular courses with the vertical joints of each course falling over the middle of the stones below: either with all courses of one height (isodomum) or with courses of alternating heights (pseudisodomum).

In a thick wall, do not build two separate skins with loose filling between them: lay stones lengthwise into the thickness at intervals, and set single stones right through from face to face, so that the wall acts as one mass.

Never let a vertical joint stand over another in the course below: interchange the direction of the stones, lengthwise and across the wall, so that every joint is covered and every stone rests on two.

overlap ≥ ¼ of a unit (brick: a quarter-brick lap)

Divide the height from floor to ceiling into three and a half parts and give two and a half to the opening; make a Doric door five and a half twelfths as wide as it is high (an Ionic one two-fifths), and narrow it slightly at the top.

h = 2½⁄3½ H; Doric w = 5½⁄12 h, Ionic w = h ÷ 2½; top narrowed by ⅓, ¼ or ⅛ of the jamb as h ≤ 16, ≤ 25, ≤ 30 ft

Put the hall at the heart of the house and enter it at one end, by a passage running through the house from front door to back door, screened from the hall. Open the buttery, the pantry and, between them, the way to the kitchen off that passage, at the hall’s lower end; put the family’s parlour, and the great chamber over it, beyond the hall’s upper end, where the high table stands.

In a timber-framed house, let the floor joists of an upper storey run out beyond the wall below and carry the upper wall on their ends: the upper storey oversails the lower by a foot or two, gaining room upstairs and sheltering the wall beneath.

Plan the house in two departments, the family’s and the servants’, with their own rooms and passages (and, as the house grows larger, their own stairs), so that neither sees nor hears the other in the ordinary course of the day: the kitchen and its offices to one side or behind, reached by their own corridor, and nothing of the work of the house carried through the family’s rooms.

Give each room the aspect that suits the hours it is used: the drawing room to the south-east, the morning room to the east, the library a little east of south-east, the dining room to the north or north-east, cool and out of the evening sun, and the kitchen and offices to the north; the best aspect of all is south-east, the worst south-west.

Let the principal room of the front stand out in a bay window, canted or square, under its own gable: more light, a view along the street as well as across it, and a front that shows where the best room is.

Make the hall, not a passage, but the house’s largest and most used room: a great fire with a seat beside it, the other rooms opening off it, and the stair kept out of it or enclosed, so that it does not make the hall a thoroughfare.

Turn the house to the sun rather than to the street: the living rooms on the sunny side, lit from more than one direction, the kitchen and stores on the cool side, and the fewer and larger rooms the better.

Take the plan, the orders and the details from the builders’ pattern books, simplified for timber and for the carpenter: a symmetrical front, a central hall and stair with rooms either side, a doorway under a pediment or fanlight, and mouldings drawn to be made with the plane.

Give the country house in a hot summer climate a veranda, or piazza, some seven or eight feet deep: a roofed porch along the front on light posts, stopping short of the ends of the front, shading the windows and the walls and giving the household a room out of doors.

Do without the parlour kept for show, and reduce the entrance hall to a small vestibule or a nook of the living room: let the front door open, through it, into a living room built round a large fireplace, open widely into the dining room, with seats, bookcases and cupboards built into its walls.

Build a deep porch, eight or nine feet, usually recessed under the main roof, on sturdy posts or piers, big enough to sit and eat in: in summer it is the living room of the house.

Orient the prayer hall so that its principal wall, the qibla wall, stands square to the direction of Mecca, and mark the middle of that wall with a niche, the mihrab.

qibla wall ⟂ direction of the Kaaba; mihrab on the axis

Build the congregational mosque as an enclosed courtyard (sahn) with a covered prayer hall (haram) on its qibla side and shaded porticoes (riwaq) on the others.

Make the prayer hall of identical bays of columns and arcades, so that it can be enlarged by adding aisles or bays without changing its character.

hall = n aisles × m bays of one module

Where the available columns are too short for the height you want, stand a pier on each column and carry the roof on a second, upper tier of arches, bracing the piers with a lower tier of horseshoe arches.

height ≈ column + pier + upper arch; lower arch = brace

Carry the curve of the arch beyond the semicircle, so that the opening at the springing is narrower than its greatest width.

springing width = 2R cos β, where the arc continues β below the centre line

Alternate voussoirs of two colours (red brick and pale stone, or black and white stone) to make the structure of the arch legible as ornament.

Frame an important arch with a rectangular moulding or band rising from its springing, so that the curved opening is set within a square field.

Build the tower for the call to prayer in the form of the regional tradition: square and stepped in the Maghreb and Spain, cylindrical and tapering in Iran and Central Asia, needle-like and polygonal in the Ottoman lands.

To draw a star pattern, cover the surface with a tiling of polygons, draw two lines through the midpoint of every edge at the same angle to it, continue each line until it meets another, and erase the tiling.

pattern = tiling + contact angle θ at every edge midpoint

Keep the grid and vary the angle at which the lines cross the edges: a small angle gives broad, blunt stars; a large one gives slender, sharp stars.

Choose the symmetry of the principal star (six-, eight-, ten- or twelve-fold) and build the field from a repeat unit that tiles the plane; stars that cannot tile by themselves are completed by secondary shapes in the gaps.

Fill a half-dome, squinch or cornice with tiers of small concave cells, each tier corbelled forward from the one below and each cell set over the junction of two cells beneath.

height = tiers × module; each tier projects inward

Draw the muqarnas as its plan (the projection of every tier on the floor) and build it up from the plan tier by tier with a fixed height to each tier.

Place an iwan (a vaulted hall walled on three sides and open on the fourth) at the middle of each side of the courtyard, the one on the qibla side leading to a domed chamber before the mihrab.

Set the iwan arch in a tall rectangular screen that rises above the roofline, framed by bands of inscription and flanked by minarets.

Build two domes, one above the other: an inner shell proportioned to the room below and an outer shell raised on a tall drum and proportioned to the city outside.

outer crown ≫ inner crown; the gap may be a third of the total height

Cover the prayer hall with a single central dome on four piers and extend the unified space by semidomes on two or all four sides, with smaller half-domes (exedrae) in their corners.

To make the domed space as nearly one room as possible, set the dome on eight piers inscribed in a square, and fill the four corners with semidomes on the diagonals.

Let the exterior show the structure: pile semidomes, exedrae, corner domes and weight towers round the central dome so that the masses rise in steps to the crown.

Give an ordinary mosque one minaret with one balcony; reserve several minarets, and several balconies (şerefe) on each, for the foundations of the sultan and his family.

Balance the tomb so that the dome with its finial rises as high above the pishtaq as the pishtaq rises above the plinth, and let every element be answered by its mirror image on both axes.

pishtaq ≈ dome + finial ≈ 33–35 m; total ≈ 73 m (approximate)

Cover the tomb chamber with a low inner dome and raise over it, on a tall drum, a bulbous outer shell whose swelling profile is visible from every direction.

Mark the corners of roofs, minarets and gateways with chhatris: small open domed kiosks on columns with a projecting eave.

Lay out the garden as a walled square divided into four by raised walks with water channels on the two axes, and divide each quarter again.

1 → 4 → 16 beds; water on the axes

Stand the minarets free at the corners of the plinth, so that they frame the tomb in every view without crowding it.

Fix the bay (the distance between column centres) and derive every other member from it by fixed ratios: the column from the bay, the brackets and beams from the column, the rafters and their spacing from the brackets, the eaves from the rafters.

Set out the posts of a house on a square grid of one ken (six shaku, about 1.82 m) and size rooms, doors, screens and mats to fit the grid.

1 ken = 6 shaku ≈ 1.82 m · 1 tsubo = 1 ken² ≈ 3.3 m² = 2 mats

In the Kyoto system (kyō-ma) fix the mat (6.3 × 3.15 shaku, about 1.91 × 0.955 m) and set the posts outside it; in the Edo system (edo-ma) fix the post grid at 6 shaku and cut the mats to fit between the posts, making them smaller.

kyō-ma ≈ 191 × 95.5 cm · chūkyō-ma ≈ 182 × 91 cm · edo-ma ≈ 176 × 88 cm

Lay the mats of a room so that their joints form T-junctions: no point where the corners of four mats meet, and no continuous seam running straight across the room.

Place the tokonoma, a shallow raised alcove for a hanging scroll and flowers, in the wall opposite or beside the entrance, with staggered shelves beside it; the place of honour is the seat in front of it.

Build the visible eaves as low and wide as the design requires, and above the visible ceiling raise a second, steeper roof on cantilever beams (hanegi) to shed the rain.

Build the sanctuary of unpainted cypress on posts set directly in the ground, with a raised floor and veranda, a straight thatched gable roof entered on the long side, crossed finials (chigi) and ridge billets (katsuogi), and a free-standing post at each gable carrying the ridge.

Every twenty years build an exact replica of each sanctuary on the alternate site beside it, transfer the deity at night, and dismantle the old buildings.

cycle = 20 years · two sites used alternately

Raise the god’s dwelling high on thick pillars planted deep in the bedrock, with crossed finials reaching up to heaven.

Make each storey of a pagoda narrower than the one below and derive all from the bay of the first storey; in the oldest Japanese pagodas the top storey is about half as wide as the first.

top storey ≈ ½ first storey (Hōryū-ji)

Run a single great pillar up the centre of the pagoda to carry the spire, but do not let the storeys rest on it: each storey stands on its own frame, stacked around the pillar.

Build the shinmei torii with a straight lintel, upright posts and a tie beam that does not project; build the myōjin torii with an upcurved double lintel, posts leaning slightly inward, and a tie beam that passes through them.

A new material calls for new forms: iron should be used in slender members that show its strength in tension and bending, not cast in imitation of stone columns and vaults.

In a framed building the columns and beams carry every load; the outer wall can be reduced to a light cladding hung on the frame, floor by floor, and need grow no thicker as the building rises.

load-bearing wall: thickness grows with height · frame: cladding thickness constant

Treat a tall office building as base, shaft and top: an open, accessible ground storey for shops and entrances; an indefinite stack of identical office floors expressed as one; and a distinct attic that closes the composition.

base (1–2 storeys) + shaft (n identical storeys) + attic

The outward form of a building should arise from, and express, what the building does and how it is made, as the form of every living thing expresses its life.

Show each material as what it is (brick as brick, timber as timber) and let the plan, rather than a symmetrical front, decide where the rooms, windows and chimneys go.

Instead of applying ornament to a structure, shape the structure itself, especially exposed iron, into lines that are both constructive and decorative.

Omit applied ornament from buildings and everyday objects: it wastes work, money and material, dates quickly, and adds nothing that fine material and exact workmanship cannot give.

Build the house as a standard skeleton of flat concrete slabs, a few columns set back from the slab edges, and a stair; everything else (walls, windows, partitions) is infill that can be placed freely.

3 slabs + 6 columns + 1 stair

With a concrete frame, a house should stand on pilotis, have a free plan and a free façade, be lit by horizontal ribbon windows, and carry a garden on its flat roof.

pilotis · free plan · free façade · ribbon window · roof garden

Raise the main storey on slender columns so that the ground beneath flows under the building (for circulation, cars, gardens or public space) and the rooms are lifted into light, air and view.

Since columns carry the floors, partitions can be placed wherever the use requires, independent of the structure and different on every storey.

Set the columns back from the edge of the slab, so that the outer wall carries nothing and can be opened, closed or glazed at will.

cantilever of slab beyond column line ≈ ⅕–⅓ of the bay

Run the windows in a continuous horizontal band from one end of the façade to the other, lighting the room evenly across its whole depth.

Make the flat roof a garden and terrace, returning to the house the ground that the building has occupied.

Check the proportions of a façade with a few geometric lines (diagonals of similar rectangles, the golden section) so that its parts share one order and nothing is arbitrary.

Size everything from a single scale: a red series in which each measure is φ times the one before, anchored on a man 183 cm tall, and a blue series of its doubles, anchored on his raised hand at 226 cm.

red: … 27, 43, 70, 113, 183, 296 … (×1.618) · blue: … 53, 86, 140, 226, 366 … (= 2 × red)

House a whole community in one tall slab: stack duplex flats that run through the block with light on both sides, put shops and services on an interior street half-way up, and give the roof to everyone.

Leave concrete as it comes from the formwork, marked by the grain of the boards and the joints of the shuttering, and let the structure be seen.

Shape a building to its place so that it seems to grow out of the ground: take its lines from the landscape, its materials from the region, and its plan from the life it will hold.

On flat land, keep the house low and let its lines run parallel to the ground: low-pitched hipped roofs with deep overhangs, continuous sills and copings, bands of windows, and terraces extending into the garden.

roof pitch ≈ 15°–25° · eaves projecting 1.5–3 m

Plan the house around a great fireplace at its heart, and let the wings of the plan extend from it in a cross or a pinwheel.

Open the corners, extend the roofs and terraces beyond the walls with cantilevers, and let the space of the house flow outward rather than being enclosed in rooms like boxes.

Let each material reveal its own character (the grain of wood, the stratification of stone, the plasticity of concrete) and design each part in the way its material wants to be used.

Train artists as craftsmen in workshops, and unite architecture, sculpture, painting and design in the common work of building.

Treat architecture as volume enclosed by thin planes rather than as solid mass; order it by the regular rhythm of the structure rather than by axial symmetry; and avoid applied decoration.

Reduce a building to the fewest, most exactly made elements (a regular structure, a clear enclosure, an open and adaptable space) and let their proportion and detail carry the whole effect.

Hang the outer wall from the edges of the floor slabs as a continuous skin of glass and metal, independent of the frame behind it, and express its grid of mullions.

skin = vision glass + spandrel panel per floor, on a module of ≈ 1.5 m

At the street line a building may rise to a multiple of the street’s width; above that it must stay behind a plane drawn from the centre of the street, stepping back as it rises.

street wall H₀ = k × street width (k = 1 … 2½) · above it, 1 m back for every 2k m up

Over one quarter of the lot area, a tower may rise to any height, free of the setback planes.

tower footprint ≤ 25% of lot area · height unlimited

Judge every building against the first shelter: four trees for columns, branches laid across them for the entablature, and a pitched roof whose end makes the pediment. These three are the essential parts; all else is added.

Let columns stand free, round and upright, carrying a straight entablature directly; do not engage them in walls, replace them with pilasters, or set them on pedestals.

Avoid the ornaments that have no model in the hut: broken or curved pediments, pediments on the long side of a roof, orders piled storey upon storey on a church front, arches carried on columns, and twisted or cabled shafts.

Seek in every building only fitness (solidity, healthiness, convenience) and economy (symmetry, regularity, simplicity); a building that has these will please, and ornament added for its own sake is waste.

Compose any building on squared paper: first draw its principal axes, then a grid of equal squares; place the walls on the axes and the columns at their intersections; only then open the walls and combine the parts.

grid module = the interaxis of the columns; walls on grid lines; supports on grid points

Build every type (temple, museum, college, hospital, barracks) from the same small set of elements, combined about axes parallel, perpendicular or concentric to one another.

Begin a design by choosing its parti (the one governing idea of the plan) and commit to it in a first sketch; everything afterwards develops that choice.

Arrange the principal spaces as a sequence along the main axis (steps, portico, vestibule, great hall, and beyond) so that the visitor’s progress through the building is itself the composition.

Draw the solid masonry of a plan in black, filling every pier and thickened wall, so that the rooms read as shaped voids carved from a mass.

Let there be no feature about a building that is not necessary for convenience, construction or propriety; and let all ornament be the enrichment of the essential construction.

Give a pointed building a steep roof, the gable close to an equilateral triangle, which throws off rain and snow and completes the vertical character of the design.

pitch ≈ 55–60° (equilateral gable = 60°)

Commit no architectural deceit: do not suggest a structure other than the real one, do not paint or carve one material to look like another, and do not pass off cast or machine-made ornament as handwork.

Understand every building as a combination of four elements (the hearth, the roof on its supports, the enclosure and the mound), each descended from a craft: ceramics, carpentry, weaving and masonry.

Every building must satisfy three demands at once: it must stand up (firmitas), serve its purpose (utilitas), and give pleasure to the eye (venustas).

architecture = firmitas + utilitas + venustas

Architecture depends on order (ordinatio), arrangement (dispositio), eurythmy, symmetry, propriety (decor) and economy (distributio).

Lay out a city’s streets so that they do not line up with the directions of the principal winds, which otherwise rush down them cold, violent and unhealthy.

A column is narrower at the top than at the bottom, but the taller the column the less it should be diminished, because distance already makes its top look smaller.

upper Ø : lower Ø = 5⁄6 (≤15 ft) · 11⁄13 (15–20 ft) · 6⁄7 (20–30 ft) · 13⁄15 (30–40 ft) · 7⁄8 (40–50 ft)

Keep the lowest third of the shaft cylindrical; above it, reduce the radius along a curve found by dividing a quarter-circle and the upper two-thirds of the shaft into the same number of equal parts.

r(k) = R · cos(α·k), cos α = r_top ⁄ R, k = 0…1 over the upper ⅔

Columns may be spaced 1½ diameters apart (pycnostyle), 2 (systyle), 2¼ (eustyle), 3 (diastyle) or more (araeostyle). The eustyle, with a wider central bay of 3 diameters, is the best for use and beauty.

clear space = 1½ · 2 · 2¼ · 3 · >3 D

In a peripteral temple the flanks have twice as many intercolumniations as the front: six columns in front means eleven along each side, counting the corner columns twice.

side columns = 2n − 1 (Vitruvius) · 2n + 1 (Greek practice)

Divide the whole front of the cornice into nine parts; give one of them to the height of the tympanum, measured at its centre.

tympanum height = cornice width ⁄ 9 (≈ 12½°)

Divide the front of a Doric temple into modules; the column’s thickness is two modules, its height with the capital fourteen, the capital one, the architrave one, and the triglyphs one and a half high and one wide.

M = ½ D; column = 14 M; capital = 1 M; architrave = 1 M; triglyph = 1 × 1½ M; metope = 1½ × 1½ M

Set a triglyph over the axis of every column and over the middle of every intercolumniation (or two, in a wider bay); make the metopes between them as wide as they are high.

axis spacing = n × (triglyph + metope) = n × 2½ M

The Corinthian capital is as high as the column is thick; its abacus is one seventh of that height, and the rest is divided in three: the lower leaves, the upper leaves, and the stalks from which the volutes spring.

capital = 1 D; abacus = ⅐; leaves : leaves : volutes = 1 : 1 : 1; abacus diagonal = 2 D

Make the Tuscan column seven diameters high, diminish it by a quarter at the top, and give its base and capital each half a diameter.

column = 7 D; upper Ø = ¾ D; base = capital = ½ D

An order consists of a column (base, shaft and capital) carrying an entablature (architrave, frieze and cornice), optionally raised on a pedestal (plinth, die and cap). Each part is proportioned to the lower diameter of the column.

order = [pedestal] + column(base, shaft, capital) + entablature(architrave, frieze, cornice)

Divide the whole height into nineteen parts: four for the pedestal, twelve for the column, three for the entablature. Without a pedestal, divide it into fifteen, giving twelve to the column and three to the entablature.

pedestal = ⅓ column; entablature = ¼ column

Make the Tuscan column 14 modules high, the Doric 16, the Ionic 18, and the Corinthian and Composite 20, the module being half the lower diameter.

Tuscan 7 D · Doric 8 D · Ionic 9 D · Corinthian 10 D · Composite 10 D

Rank the orders in a sequence from the plainest and strongest to the richest and most slender (Tuscan, Doric, Ionic, Corinthian, Composite), each one diameter taller than the last.

6 · 7 · 8 · 9 · 10 D

Make the Tuscan column 7 diameters high, the Doric 8 with its base, the Ionic 9, the Corinthian 9½ and the Composite 10; give Tuscan and Doric an entablature of about a quarter of the column, the richer orders a fifth.

7 · 8 · 9 · 9½ · 10 D; entablature ≈ ¼ → ⅕

Grade the orders by slenderness (Tuscan 7½ diameters, Doric 8½, Ionic 8¾, Composite 9¾ and Corinthian 10), placing the Composite, the “Roman” order, below the Corinthian.

When orders are placed one above another, put the stronger below and the more delicate above (Doric under Ionic under Corinthian), and make each upper column a quarter shorter than the one beneath.

upper column = ¾ lower column; order sequence: Tuscan → Doric → Ionic → Corinthian → Composite

Set arches on piers and frame each bay with an engaged column or pilaster carrying a continuous entablature, so that the arch does the work and the order gives the measure.

arch height ≈ 2 × arch width (Vignola); impost at ⅔ of the column

Build every profile from a small set of elements (fillet, torus, astragal, scotia, ovolo, cavetto, cyma recta and cyma reversa), alternating curved with flat and light with shadow.

Use the orders according to their character: the Doric for strength and gravity, the Ionic for a middle dignity, the Corinthian for delicacy and splendour.

Beauty is the reasoned harmony of all the parts within a body, so that nothing may be added, taken away or altered but for the worse.

Proportion rooms and façades by the same ratios that make musical consonances: 1:2 (octave), 2:3 (fifth), 3:4 (fourth), and their compounds.

short plans 1:1, 2:3, 3:4 · medium 1:2, 4:9, 9:16 · long 1:3, 3:8, 1:4

In all architecture the principal ornament is the column: a part of the wall left standing and pierced through.

Make rooms round, square, or with a length equal to the diagonal of the square, to a square and a third, a square and a half, a square and two-thirds, or two squares.

circle · 1:1 · 1:√2 · 3:4 · 2:3 · 3:5 · 1:2

Make the height of a vaulted room the arithmetic, geometric or harmonic mean of its length and breadth.

arithmetic (a+b)/2 · geometric √(ab) · harmonic 2ab/(a+b)

Place rooms on the right of the entrance equal and corresponding to those on the left, so that the house is alike on both sides and the walls bear the roof equally.

Give the front of a country house a portico with a pediment, as the ancients did, for it shows the entrance, gives grandeur, and makes a place to display the owner’s arms.

Combine a central arched opening with two narrower flat-headed openings on either side, the arch springing from the entablature of the side openings.

side lights ≈ ½ central span; arch springs from the lintel of the sides

Distinguish positive beauty, which pleases everyone for evident reasons (fine materials, size, precise workmanship, symmetry), from arbitrary beauty, which pleases through custom, including the exact proportions of the orders.

Find every member of an order by dividing the given height into equal parts with the compasses, and those parts again, so that no arithmetic is needed.

height → 5 parts: 1 entablature + 4 column (without pedestal)

Read the age of a Doric order from its proportions: the oldest columns are little more than four diameters high, the finest Greek ones five and a half to six and a half, the Roman seven and a half to eight; as the column rose, the architrave grew lower.

archaic ≈ 4 D → classical 5½–6½ D → Roman 7½–8 D

Where the architrave runs on over the end of a wall, widen the wall’s end into an anta at least as broad as the architrave, so that the beam has a full bearing and the wall looks able to take it, even if the wall itself is thinner.

Do not enlarge or reduce an order by one set of ratios: design its members for the size at which it will be built, the material and the distance from which it is seen. In a large order make the parts relatively finer, in a small one relatively bolder.

Draw the Ionic volute seven eye-diameters high and six wide, with the eye four diameters below the top; lay three turns about centres set out in the eye so that each winding is half as wide as the one outside it and the spiral runs into the top of the eye.

height = 7 e, width = 6 e; eye centre 4 e below the top; radii from the eye centre (e = 16 parts): 60, 52, 44, 36, 30, 26, 22, 18, 15, 13, 11, 9 → 8

Space the eggs from centre to centre at the height of the moulding; make each egg as wide as the space between two eggs, and divide that space into three, giving the middle part to the dart and the outer parts to the shell that encloses each egg.

pitch = height; egg width = gap; gap = shell + dart + shell (⅓ each)

Compose the Roman or Composite capital from two rows of Corinthian leaves round the bell with a full Ionic capital (echinus and four diagonal volutes) above them.

When a window or door narrows towards the top and has no cornice, let the ends of the lintel run out beyond the jambs as ‘ears’ (crossettes), so that the head is firmly marked.

Compose from the plan: base it on the elemental figures (circle, square, rectangle and their derivatives), arrange its parts symmetrically about axes, and develop section and elevation from it together.

Do not leave a large circular building as a bare cylinder: steady it with features that break the curve and fix its axes: a portico, stairs, a projecting bay or a recess on the axis.

Make a simple square room as high as it is wide, so that it encloses a cube; for oblong rooms the double cube (twice as long as it is wide and high) is satisfactory in almost any circumstances.

For a hall of assembly use the “D” or tribune plan, a half-circle closed by a straight side: it gives ample room to the audience, a central place to the speaker and scope for display in the architecture.

Use the elliptical plan chiefly for interiors, where the whole curve is seen at once; on the outside of any but a very large building its changing curvature looks like a distorted circle, and the circle is better.

If a room is no longer than twice its breadth, make it as high as it is broad; make a coved room higher than a flat one, with a cove a quarter to a fifth of the whole height; take a vaulted room to the crown at half the sum of its length and breadth.

flat: H = B (L ≤ 2B); coved: H = B + (⅕ to ⅓)(L − B), cove = ¼–⅕ H; vaulted: H = (L + B) ⁄ 2

Keep a corridor’s height to no more than one and a half times its width, and divide its length into bays about as long as it is wide, marked by pilasters and by the divisions of the ceiling or vault.

H ≤ 1½ W; bay ≈ W

Place the staircase by three tests: its relation to the principal rooms it serves; keeping the main axial communications free of obstruction; and light, for itself and, as a well of light, for the corridors and vestibules around it.

Give a portico three jobs: to shelter those who arrive, to mark the main entrance, and to give the building dignity and character with its depth of light and shade; proportion it, as a rule, like a temple front.

Between two numbers a and b, find the arithmetic mean (a + b)/2, the geometric mean √(ab) and the harmonic mean 2ab/(a + b); use them to find a third dimension that belongs to the other two.

for 6 and 12: arithmetic 9 · geometric 8.49 · harmonic 8; always A ≥ G ≥ H, and G² = A × H

Let every part of a building be measured by the same unit and relate to the whole by simple whole-number ratios, so that one module generates all its dimensions.

e.g. 2 : 3, 3 : 4, 1 : 2: ratios of small whole numbers of modules

To make a rectangle of 1 : √2, draw a square and swing its diagonal down to extend one side; the result cannot be measured in whole units of the square, but it can be drawn exactly.

√2 ≈ 1.414; halving a 1 : √2 rectangle gives two more of the same shape

Swing the diagonal of a √2 rectangle to obtain a rectangle of 1 : √3, the proportion of an equilateral triangle’s height to half its base, used for setting out by triangles (ad triangulum).

√3 ≈ 1.732; height of an equilateral triangle of side 2 = √3

To divide a line in extreme and mean ratio, make the whole stand to the larger part as the larger part to the smaller; the ratio is φ = (1 + √5)/2 ≈ 1.618. Construct the golden rectangle by swinging the line from the midpoint of a square’s side to its far corner.

φ = (1 + √5)/2 ≈ 1.618; φ² = φ + 1; 1/φ = φ − 1 ≈ 0.618

Treat claims that a building embodies the golden section with suspicion unless the designer’s own writings or drawings say so and careful measurement agrees; a ratio near 1.6 is found somewhere in almost anything.

Take the height of a man (183 cm) and of his raised hand (226 cm) and divide each by the golden section, again and again, to obtain two interlocking series of dimensions for every part of a building.

red series … 27 · 43 · 70 · 113 · 183 cm; blue series … 53 · 86 · 140 · 226 cm (each ≈ previous × φ)

Let the rooms of a house share dimensions, so that the length of one is the breadth of the next and each stands to its neighbours in the ratios of the scale; the whole plan then forms one proportional system.

e.g. rooms of 12 × 12, 12 × 18 and 18 × 30 feet, sharing 12 and 18

Make each bay of an arcade as wide as its columns are high and as deep as it is wide, so that the loggia is a row of equal cubes of space.

bay width = column height = depth (Innocenti: ≈ 10 braccia ≈ 5.8 m)

Cover a square bay with a hemisphere whose diameter is the bay’s diagonal, cut off by the four vertical planes of its sides; the cut edges are then the four semicircular arches of the bay.

vault radius = ½ × diagonal = s / √2; the side arches are semicircles of span s

Draw the curve of a pointed dome with arcs whose radius is four-fifths of the span, their centres standing on the springing line one-fifth of the span in from the opposite side.

r = ⅘ S; centre at ⅕ S from the far springing; rise ≈ 0.74 S

Build a great dome as two shells (a thick inner one that carries, a thin outer one that protects and gives the silhouette) joined by ribs and locked at the top by a ring under the lantern.

Florence: inner shell ≈ 2 m at the base; outer ≈ 0.9 m; 8 angle ribs + 16 intermediate ribs

Lay the brickwork of a large dome in horizontal rings, each ring locked by bricks set on end in a herringbone pattern, so that every stage of the work stands by itself without a timber frame beneath it.

Set the columns of a round temple on a circle over a stepped base, carry the entablature round the circle, and cover the cella with a dome whose height is half the diameter of the whole.

dome height ≈ ½ diameter (Vitruvius IV.8, for the peripteral round temple)

Where the purpose allows, plan the church about a single centre (a circle, a square or a Greek cross) so that every part stands in the same relation to the middle, under a dome.

Where a church has a tall nave and low aisles, give the front a lower storey as wide as the whole church and an upper storey as wide as the nave, and join the two with great scrolls.

Lay out the ideal city as a regular star inscribed in a circle (two squares turned 45° to one another), with gates in the re-entrant angles and towers at the points.

Sforzinda: 8 points, 8 gates, 16 radial streets

Run the principal streets straight from the central square to the gates, so that every gate is visible from the centre and every quarter lies on a direct route to it; set secondary squares along the radials.

Give a fortified town a polygonal wall with an angled bastion at every corner, so that each face of every bastion and every stretch of curtain wall can be swept by fire from its neighbours.

Palmanova: a nine-sided trace with nine bastions and three gates

Where an old structure fixes bays of unequal width, keep the central arch of every Serliana the same and let the flat-headed side openings take up the differences.

Raise the principal rooms of a house on a basement storey that holds the kitchens, cellars and service rooms, and reach them by an external stair; put the lesser rooms above them.

Place the stair where it does not spoil the rooms and yet is easily found; light it well; and make the steps no higher than half a foot and the treads at least a foot deep.

riser ≤ ½ ft (≈ 17 cm); going ≥ 1 ft (≈ 35 cm) in Vicentine feet

Where space is short, wind the stair round a newel or an open well; measure the going on the walking line, about halfway along the tread, where it should be as generous as on a straight flight.

Make the height of the rotunda to the top of its dome equal to its diameter, so that the interior would exactly contain a sphere resting on the floor: the drum rises to the centre of the sphere and the dome is its upper half.

Pantheon: Ø = height = 43.3 m (150 Roman feet); drum = ½ Ø

Divide the inner surface of the dome into sunk panels arranged in rings and meridians; let them diminish as they rise with the converging meridians, and stop them short of the crown so that a smooth zone surrounds the eye.

Pantheon: 5 rings × 28 coffers, each stepped in several recessions

Build a great concrete dome thickest and heaviest at its haunches and thinnest and lightest toward the crown, grading the aggregate from dense stone at the base to light volcanic pumice at the top.

Pantheon: thickness ≈ 6 m at the springing to ≈ 1.5 m at the oculus; aggregate travertine and tuff → brick → pumice and scoria

Expect a hemispherical dome to be in compression along its meridians everywhere but in tension round its horizontal rings below about 52° from the crown; restrain that lower zone with weight, thickness, rings or chains, or it will crack along the meridians.

thin hemisphere under its own weight: hoop force changes sign at ≈ 51.8° from the crown

Make the sweating room of a bath circular, as high to the springing of its dome as it is wide, with an opening at the crown closed by a bronze disc that can be raised and lowered on chains to regulate the heat.

height to springing = width; oculus with a movable bronze shield

Face a concrete wall either with small irregular stones set at random (opus incertum), which is strong though not handsome, or with small square-faced stones set diagonally in a net (opus reticulatum), which is handsome but, its joints running every way, more likely to crack.

Build a Roman wall as two facings of brick or small stone with a core of concrete poured between them in layers; tie the facings into the core with bonding courses of large tiles laid through the whole thickness at intervals.

bonding courses of bipedales (2 ft tiles) every 1–1.5 m of height

Mix lime with the volcanic ash of the Bay of Naples, and the mortar will bind rubble into a solid mass that hardens even under the sea.

Lay the channel of an aqueduct with a steady fall of not less than a quarter of an inch in every hundred feet, cover it with a vault against the sun, and carry it through hills in tunnels and across valleys on arches.

minimum fall ≈ 1 : 4,800 (Morgan’s reading); Nîmes aqueduct: ≈ 12.6 m in ≈ 50 km ≈ 1 : 4,000

Cross a deep valley on superimposed tiers of arches, the piers of each tier standing directly over those below, the spans matched from tier to tier, and the smallest arches at the top where they carry only the channel.

Pont du Gard: tiers ≈ 22 m, 20 m and 7 m high; 6, 11 and 35 surviving arches; largest span ≈ 24.5 m

Draw the orchestra circle and inscribe in it four equilateral triangles at equal intervals; the side nearest the scene fixes the front of the scaena, a parallel through the centre fixes the front of the stage, and the angles on the auditorium side give the directions of the stairways.

12 vertices at 30°: 7 fix the stairs of the lower cavea, 5 fix the doors and passages of the scene; stage depth = r⁄2

Make the scaena twice as long as the orchestra is wide; raise its podium a twelfth of the orchestra diameter, its lower columns a quarter, their entablature a fifth of the columns; above, a parapet half the podium and columns a quarter shorter than those below.

L = 2D; podium D⁄12; columns D⁄4; entablature ⅕ col.; upper parapet ½ podium; upper columns ¾ lower

Roof the colonnade at the top of the seats at the same height as the top of the scene building, so that the voice rises evenly to the highest rows and is not checked on its way.

Give the atrium one of three shapes: width to length as 3 : 5, or 2 : 3, or a length equal to the diagonal of the square on the width; and make its height to the girders a quarter less than its width.

L = 5⁄3 W, 3⁄2 W, or √2 W; height to girders = ¾ W

Proportion the rooms round the atrium from it: the alae from its length (a third for atria of 30–40 feet, a quarter for 50–60), the tablinum from its width (two-thirds for a 20-foot atrium, half for 30–40 feet, two-fifths for 40–60), and the entrance passage from the tablinum.

alae = L⁄3 … L⁄5; tablinum = ⅔ W (20 ft), ½ W (30–40 ft), ⅖ W (40–60 ft); fauces = ⅔ or ½ tablinum

Leave an opening in the atrium roof not less than a quarter nor more than a third of the atrium’s width, with a length in proportion; beneath it set a basin to receive the rain.

compluvium width = ¼ to ⅓ W

Make the peristyle a third longer across than it is deep, its columns as high as the colonnades are wide, and the spaces between them not less than three nor more than four diameters.

width = 4⁄3 depth; column height = colonnade width; intercolumniation 3–4 D

Make a basilica between a third and a half as broad as it is long; make each aisle a third of the width of the central space, and the columns as high as the aisles are broad.

W = ⅓ to ½ L; aisle = ⅓ nave; column height = aisle width; upper columns ¼ smaller

Pierce a massive rectangular pier with one arch, or with a tall central arch between two lower ones; frame the openings with columns on pedestals carrying an entablature, and crown the whole with a high attic for the dedicatory inscription.

Arch of Titus: 13.5 × 15.4 m, one opening 5.4 × 8.3 m; Arch of Constantine: 25.9 × 21 m, openings 6.5 × 11.5 m and 3.4 × 7.4 m

Lay out the church as an Early Christian basilica (a tall nave lit from above, lower aisles on either side, a sanctuary ending in an apse to the east) and cross it with a transept, so that the plan becomes a Latin cross with a tower over the crossing.

Carry vaults on walls thick enough to absorb their thrust within their own mass; keep the windows small and set them low, below the springing of the vault, and strengthen the walls outside with shallow pilaster buttresses.

walls commonly 1½–2 m thick; buttresses shallow, window openings narrow and splayed

Let the square of the crossing set the measure of the whole church: the nave and transept arms are multiples of it, each nave bay is a square of the same size, and each aisle bay a square one quarter its area.

nave bay = crossing square; aisle bay = ¼ of it; nave length = n squares

Where one nave vault spans two bays of the arcade, give the principal piers, which receive the vault, a heavier compound section with shafts running up the wall, and make the intermediate supports lighter: columns or simpler piers.

A–B–A (pier, column, pier) · or A–B–B–A in Saxon churches

Run a shaft or pilaster from the floor, up the face of the pier and the wall above, to the roof or vault, so that the long nave wall is divided into a sequence of vertical bays, each containing one arch of the arcade and the windows above it.

Strengthen a barrel-vaulted nave with transverse arches at every bay, springing from shafts on the piers; they stiffen the vault, mark the bays and serve as ribs on which the vault can be built bay by bay.

Build galleries (tribunes) over the aisles and cover them with quarter-circle (quadrant) vaults that lean against the nave wall just below the springing of its barrel, so that the gallery vaults carry the barrel’s thrust down to the outer walls.

Multiply altars at the east end by opening apses off the transept arms and the choir, stepped back from the main apse in echelon, each with its own small semidome and roof.

Articulate plain rubble walls outside with shallow vertical strips (lesenes) linked at the top by a frieze of small round arches on corbels: the Lombard band.

Mark the west end with a massive, towered block containing an entrance hall below and a chapel or gallery above, rising higher than the nave: a church in front of the church.

Raise a thatched roof by a third and a tiled roof by a quarter: the rougher and more absorbent the covering, the steeper the roof must be.

thatch: rise = ⅓ · tile: rise = ¼

Pitch a roof no lower than its covering allows (roughly 45° for thatch, 35° for plain tiles, 20–25° for slate, 15–20° for interlocking tiles, a few degrees for sheet metal) and steeper where snow lies long or rain is heavy.

thatch ≥ 45° · plain tile ≥ 35° · slate ≥ 22½° · pantile ≥ 17½° · metal ≥ 5° · membrane ≈ 1–2°

Over the vaults of a northern church raise a timber roof of about 55–60°, its section close to an equilateral triangle, so that snow slides off and the slates or tiles overlap generously.

equilateral roof: pitch = 60°, rafter = span

Put a pediment only where the roof actually ends in a gable, on the short sides of a building, never along its length or stacked one above another, because the pediment represents the gable of the roof.

To roof a wide span, join each pair of rafters to a tie beam at their feet so that the three form a triangle that cannot spread; hang the tie beam from the apex by a king post, or by two queen posts on larger spans.

Instead of a triangle, stack beams one above another, each shorter than the one below and raised on short posts, and lay a purlin on each beam-end; the heights of the purlins then set the section of the roof, which may be curved at will.

Combine a hipped lower roof with a gabled upper roof, the triangular gable set back from the end of the building; count one main ridge, four gable ridges and four hip ridges: nine in all.

Break the slope of the roof on all four sides: a steep lower slope, nearly a wall, which holds a storey of rooms lit by dormers, and a shallow upper slope, often invisible from the street.

lower slope ≈ 60–75° · upper slope ≈ 15–30°

Make the roof flat and use it: a garden and terrace that gives back to the building the ground it covers, drained inward and kept watertight by a continuous membrane.

Classify a roof by how its planes meet: one slope (shed), two meeting at a ridge (gable), four meeting at hips (hip, pyramid), a hip below a gable (hip-and-gable), two pitches each side (gambrel, mansard), or a curved surface (barrel, cone, dome).

Set a vertical rod at the centre of the levelled site, draw a circle round it, mark where the tip of its shadow crosses the circle before and after noon, and join the two marks: that line runs east–west.

E–W = chord through the morning and afternoon shadow points; N–S ⟂ through its midpoint

The cord stretched along the diagonal of a rectangle produces as much area as the cords along its two sides together; use this to set out right angles and to make squares of any required area.

d² = a² + b²; e.g. cords of 3, 4, 5 or 5, 12, 13 units

Divide the square site into a grid of equal squares (padas), most often 8 × 8 = 64 or 9 × 9 = 81, and place every part of the building according to the grid.

64 padas (maṇḍūka) or 81 padas (paramaśāyika)

Imagine the vāstupuruṣa lying face down on the diagram with his head in the north-east and his feet in the south-west, held down by the gods who occupy the padas; do not place walls, pillars or doors on his vital points.

Read the diagram from the centre outward: Brahmā at the heart, then a ring of the gods, a ring of human concerns, and on the outside the lesser spirits; the holiest function goes at the centre and the most worldly at the edge.

81 padas = 9 (Brahmā) + 16 + 24 + 32

Place the garbhagriha (the small dark chamber that houses the image) over the Brahmasthana, with its walls on the ring of the gods and its door on the axis, usually to the east.

Build the stupa as a solid hemispherical mound (anda) over relics, on a circular terrace (medhi), and crown it with a railed enclosure (harmika) and a mast of parasols (chattra).

Sanchi: Ø ≈ 36.6 m, height ≈ 16.5 m (without the crown)

Provide a path for circumambulation (pradakshina) in the clockwise direction, keeping the sacred object on the right, and enclose it with a railing that separates the sacred ground.

Open the railing at the four cardinal points and mark each opening with a torana: two posts carrying three curved architraves with scrolled ends, crowned with Buddhist emblems.

Break each face of the square sanctum into projections (rathas), three, five, seven or more, that step forward towards the central one, and carry them up unbroken through the wall and the tower.

tri-ratha 3 · pancha-ratha 5 · sapta-ratha 7 projections per face

Raise over the sanctum a tower whose faces curve inward towards the top (rekha), divided into storeys (bhumis) that are marked on the corners by small ribbed discs (bhumi-amalakas).

Crown the shikhara with a great ribbed stone disc, the amalaka, on a short neck, and set over it the kalasha, a water-pot finial.

Lay the halls along the east–west axis in front of the sanctum (porch, hall, great hall, vestibule), each larger and more lofty than the one before, so that the roofs rise in a continuous range to the shikhara.

Multiply the shikhara: attach smaller spires (urushringas) against its faces, and smaller ones again against those, so that the tower becomes a cluster of peaks.

Build the shrine in six parts from the ground up: base (adhiṣṭhāna), wall or pillars (pāda), entablature (prastara), neck (grīva), crowning roof (śikhara) and finial (stūpi).

Raise the vimana in receding storeys (talas), and edge each with a parapet (hara) of miniature shrines: square domed kutas at the corners, oblong barrel-roofed shalas in the middle, and small panjaras between.

Thanjavur: 13 talas; vimana ≈ 66 m

Classify the temple by the shape of its crown: square is nāgara, octagonal drāviḍa, round (or apsidal) vesara.

In the later temple towns of the south, raise ever taller gateways (gopurams) in the successive enclosure walls, until the outer gates dwarf the shrine at the centre.

Build the state temple as a stepped mountain with five towers in a quincunx (one at the centre and four at the corners), representing the five peaks of Mount Meru, surrounded by walls and a moat for the mountains and ocean that ring the world.

Enclose the central mountain in rectangular galleries, one within the other, each on a higher terrace, with gopuras at the cardinal points and pavilions at the corners.

Angkor Wat: galleries ≈ 187 × 215 m, 100 × 115 m, 60 × 60 m

Angkor Wat reverses the usual eastward orientation: its entrance, causeway and principal approach face west.

Enclose the gathering place with a circular bank of chalk and dig the ditch on the inside of it, so that the earthwork marks off a sacred interior rather than defending it; leave causeways of undug ground for the entrances.

Build the circle of the living in timber, which decays, beside the settlement; build the circle of the ancestors in stone, which endures, apart from it; and join the two by avenues and the river.

Run a straight, banked or surfaced avenue from the entrance of each monument down to the river, on the monument’s solstice axis, so that the approach itself is aligned on the sun.

Set up forked stakes, weave twigs between them and daub the walls with mud; roof them with reeds and leaves, steep enough to shed the rain.

Build the town for a royal pyramid in one campaign, as a walled rectangle on a grid of straight streets; wall off the quarter of the labourers’ small houses from the quarter of the officials’ mansions, and give each rank a standard house.

Turn the principal rooms and the columned portico of a house to face north, towards the cool prevailing wind, and keep the walls to the south and west blank against the sun.

Find a granary’s capacity from its measures: multiply length, breadth and height in cubits (or, for a round bin, the square of eight-ninths of the diameter by the height), and add half as much again to turn cubic cubits into the grain measure, the khar.

1 cubic cubit = 1½ khar

Lay the city out square, walled and moated, with the river through its middle; run the streets straight, some along the river and others across it at right angles, each ending at a gate in the wall along the bank.

Build the temple tower in seven levels on a square base, the first two stages tall and the upper four low, with the god’s house on top, so that its total height equals the side of its base.

base 15 nindan (≈ 90 m) = height; stages 5½, 3, 1, 1, 1, 1 and temple 2½ nindan high

Raise and pave the processional road from the city gate to the god’s temple, wall it on both sides, and build the gate of glazed brick bearing the beasts of the gods, with cedar roofs and bronze-clad doors.

Build the house blank to the street, enter it through a bent passage, and arrange its rooms round a single open court, with the principal room broad and on the south side so that it faces north into shade.

Raise the royal seat on a new terrace cut from the mountain and built out over the plain, walled as a fortress, and make it secure, beautiful and fitting.

Dig the foundations down to bedrock and fill them with rubble; then build with the best material of every province, each worked by the people who know it best.

Make the hall square and fill it with columns on a square grid; put a portico two columns deep across the front, between solid corner towers; build the walls of mud brick and keep stone for columns, door and window frames.

Wall the royal garden, water it by channels, and plant its trees at equal distances in straight rows, so that every line and angle is regular.

Wall the whole hill; put the ruler’s megaron on its summit, reached by a ramp from the main gate; keep stores, shrines and the ancestral graves inside the walls, and bring water within reach by a hidden stair.

Dig the tholos tombs of later rulers into the hillsides outside the walls, along the roads that approach the city, each entered by a long open passage.

Set the main gate back in a narrow court between the wall and a projecting bastion placed on the right hand of those who approach, so that an enemy at the gate has his unshielded side to the defenders.

Build the house on a levelled terrace with a socle of rubble stone, walls of mud brick in a timber frame, and a flat roof of reeds and clay; store oil and wine in the basement rooms against the hill.

Divide the city into equal rectangular blocks by straight streets crossing at right angles, and assign whole blocks to the public buildings; but let walls follow the ground, not the grid.

Keep a free agora for public life, clear of merchandise and near the magistrates, the temples and the gymnasium; put buying and selling in a separate market place convenient for goods.

Make the Greek agora square, surround it with spacious colonnades, double if possible, with close-set columns and architraves of stone or marble, and walks above on an upper floor.

Put the court on the south side of the house and the principal rooms, fronted by a porch, on its north side facing south, so that the winter sun enters the porch and the summer sun passes over the roof; build the north side high and the south low.

Set the temples where they can be seen from the whole city and approached apart from its traffic: on a raised, commanding site above the agora.

Build the palaestra as a square or oblong court with colonnades all round, the side facing south doubled, and rooms for bathing, oiling and teaching opening off them; beside it lay out a running track with covered and open lanes.

Put the temples and their plazas at the centre of the town; round them the houses of the lords and priests, then those of the leading people, and the houses of the common people at the outskirts.

Build the house of wood and daub under a steep roof of thatch or palm, to throw off the rain; divide it along its length by a wall, with a whitewashed front room for receiving guests and a back room for sleeping.

Close the great plaza on the east and west with temple-pyramids facing each other across it, the ancestors’ acropolis on the north and the palaces on the south, so that the plaza lies on the east–west path of the sun.

Join the great groups of the city by raised, plastered causeways (sacbeob), running straight from plaza to plaza.

At the end of each katun of twenty years, build a plaza with two flat-topped pyramids facing east and west, each with four stairs, a walled court on the north for the king’s stela and altar, and a range of nine doorways on the south.

Build a household as three or four houses on low platforms round a small patio, with the family’s shrine, over the ancestors’ graves, on the east side.

Divide the city into two halves, Hanan (upper) and Hurin (lower), each with its own lineages, its own shrines and its own place in the rites.

Let the four royal roads to the four quarters of the empire begin in the great square of the capital, so that the city is the centre from which the whole realm is divided.

Settle the lords of every province in a ward of the capital lying on the side of the city toward their own province, so that the city holds the order of the whole empire.

Surround the great square with the houses of the kings, each opening onto it by a great hall large enough to hold a festival under cover when it rains.

Build every household, palace and temple as a kancha: a rectangular walled enclosure with a single doorway to the street, its single-room houses set against the walls and facing a central court; pair the kanchas back to back to make the blocks of the town.

Place the monastery where it is neither too far from the town nor too near: easy to come and go for all who wish to visit, not crowded by day, quiet at night, apart from the noise of people, fit for seclusion.

Give the monks dwellings of cells round a court with a verandah, a single gateway, and a wall round the whole; add halls for assembly and worship, wells and a bath-house, as they are needed.

Enshrine the relics of a Buddha, like those of a universal king, in a stupa raised where four great roads meet, so that all who pass may honour them.

Let monks and pilgrims raise small stupas for merit close round the great stupa, so that its terrace fills over the centuries with lesser monuments.

Choose the plan of a village from eight types named for their figures (daṇḍaka, sarvatobhadra, nandyāvarta, padmaka, svastika, prastara, kārmuka and caturmukha), according to its size, its site and who is to live in it.

Make the sarvatobhadra village square, walled, with a gate in the middle of each side; run the main streets from gate to gate to cross at the centre, and put the temple there, with streets round it and round the inside of the wall.

Set the chief temple at the centre facing east; put Śiva’s temple in the north-east, the quarter of Īśāna; house guardian gods at the gates; and keep the burning ground outside, toward the south, the quarter of Yama.

Settle the four varṇas in four quarters of the town: Brahmins to the north, Kṣatriyas to the east, Vaiśyas to the south and Śūdras to the west.

Give each house a breadth according to the rank of its owner, the largest for the king and for each varṇa less than the one above: thirty-two cubits for a Brahmin, then twenty-eight, twenty-four and twenty.

breadth = 32 − 4 k cubits (k = 0 … 3)

Build temples near water and gardens (by lakes, rivers and lotus tanks, in groves and pleasure gardens), for the gods delight to dwell in such places; where there is none, make a tank beside the temple.

Surround the sanctum with up to five walled enclosures (prākāra), each with gateways on the axes, and name and size the gateways by their enclosure, from the innermost dvāraśobhā to the outermost mahāgopura.

Lay out the streets of a temple town in concentric squares round the temple, parallel to its walls, with the main streets on the axes of its gateways, so that the processions of the god circle the whole city.

Make one ring of streets round the temple wide and straight enough for the temple car (ratha), on which the image is drawn round the town at the great festival.

Line the streets nearest the temple with continuous rows of narrow, deep houses for the priests and learned families, each with a raised verandah on the street and a court within.

Enclose the capital in a square wall with a moat outside it; open five gates, crowned with the four faces of the Buddha, and bring the roads over the moat on causeways lined with giants holding the serpent.

Set the state temple at the exact centre of the city, where the avenues from the gates cross, and the king’s palace to its north.

Roof the palace with lead and glazed tile; let officials tile only their family temple and their principal hall, sizing their houses by rank; and let common people roof with thatch.

Divide the ground inside the walls by embankments into a regular grid of blocks, each with its ponds and its raised mounds for houses.

Make the avenues within the city nine chariot-tracks wide, the ring road that runs inside the wall seven tracks, and the roads in the country outside the gates five.

avenue (jing tu) 9 gui · ring road (huan tu) 7 gui · country road (ye tu) 5 gui · 1 gui = 8 chi

Set the ridge of the palace gates at five zhi, the corner towers of the palace at seven, and the corner towers of the city wall at nine, a zhi being one zhang (ten feet) in height.

palace gate : palace corner : city corner = 5 : 7 : 9 zhang (≈ 11.5 : 16 : 21 m)

Within the palace build nine chambers for the nine consorts; outside it build nine chambers where the nine ministers attend court; and divide the realm into nine parts for the nine ministers to govern.

Divide the wards between the main streets with narrow lanes running east–west, and line both sides of each lane with courtyard houses whose main halls face south.

great street 24 bu · small street 12 bu · lanes (hutong) between · lanes c. 70–80 m apart

Divide the capital into blocks (machi) forty jō square; gather sixteen blocks into a ward (bō) and the wards into rows (jō); separate the blocks by lanes of four jō and the wards by avenues of eight to twelve, and make Suzaku-ōji, the central avenue, twenty-eight.

machi = 40 jō (c. 120 m) · kōji 4 jō · ōji 8–12 jō · Nijō 17 jō · Suzaku-ōji 28 jō (c. 84 m) · city 1,508 × 1,753 jō

Put the palace enclosure at the north end of the central avenue, facing south down its length to the city gate; govern the half of the city on the emperor’s left (east) and the half on his right (west) as two capitals.

Grant residential land in whole blocks and fractions of blocks according to rank: several blocks to the highest ministers, one or two to officials of high rank, and to ordinary households a half, a quarter or less, according to their size.

Minister of the Right 4 machi · high ranks 2 · middle ranks 1 · households 1, ½ or ¼ (691); in Heian-kyō, ordinary lots of 1/32 machi (c. 15 × 30 m)

Leave a court of white sand six or seven jō deep between the main hall and the pond, eight or nine at a palace, for the rites held there; bring the stream in from the east, past the front of the house, and let it leave toward the south-west.

south court = 6–7 jō (c. 18–21 m); at the imperial palace 8–9 jō

Seat the capital in a basin closed by mountains: a guardian mountain behind (north), a lower range on the left (east) as the Azure Dragon and on the right (west) as the White Tiger, a facing hill in front (south); let a stream cross the basin and a great river flow beyond.

Bugaksan (behind) · Naksan (left, Azure Dragon) · Inwangsan (right, White Tiger) · Namsan (in front) · Cheonggyecheon within, the Han River without

Run the city wall along the crests of the four inner mountains, so that the land itself is the rampart, and open the great gates where the ridges dip to let the roads in: east, west, south and north, with four lesser gates between.

c. 18 km of wall (1396) · 4 great gates + 4 lesser gates

Build long ranges of shops at public expense along both sides of the main east–west street, let them to licensed merchants trade by trade, and ring the bell at the crossing to open and close the city.

Limit houses by the number of bays (kan): a king’s son by his queen to sixty, other princes and princesses to fifty, officials of the second rank and above to forty, those of the third rank and below to thirty, and commoners to ten.

grand prince 60 kan · prince 50 · rank 2+ 40 · rank 3− 30 · commoner 10 · 1 kan = the square between four posts

Lay out the capital as a circle with four gates at equal intervals, each facing the region its road leads to, and put the ruler’s palace at the centre, so that no part of the city is further from him than any other.

4 gates on the diagonals (Khurasan NE, Basra SE, Kufa SW, Syria NW); palace at the centre

Defend the circle in depth: a moat, an outer wall, an open space between the walls, and a massive main wall with towers; bring each road in through a gate that turns, and set a domed chamber over the main gate from which the country can be watched.

Leave the centre of the city open: set the palace and the congregational mosque side by side in a great court, and allow round its edge only the guard, the police and the offices of state: no private houses and no markets.

palace 400 × 400 cubits; mosque 200 × 200 cubits; Green Dome 80 cubits high

Keep trade out of the ruler’s city: move the markets to a suburb outside the walls and give each trade its own street or row, with the noisome trades at the far end.

Where people disagree about the width of a public road, make it seven cubits, and let no one build out into it; lanes that serve only the houses on them may be narrower.

public road ≥ 7 cubits (≈ 3.5 m)

Build as you will on your own land, provided you do your neighbour no harm: open no window, door or roof terrace that looks into his house or court, and do not set your door opposite his across a lane.

Treat a dead-end lane as the joint property of the houses that open on it: no one may open a new door into it, build over it or narrow it without the consent of all of them, and they may close its mouth with a gate.

Let the market inspector (muhtasib) keep each trade in its own row, forbid shopkeepers and householders to build benches, awnings or walls out into the street, see that drains are covered and streets cleaned, and keep trades that make smoke, stench or noise away from where they do harm.

Found a city where it can be defended (on a steep height or inside the bend of a river) and where it has healthy air, fresh water close at hand, pasture, farmland and wood; a city that lacks these will not last.

Make a great square the new heart of the city and ring it with a uniform two-storey arcade of shops; on its four sides place the four powers: the royal congregational mosque, the palace gate, the court mosque and the gate of the royal bazaar.

Maidan ≈ 510 × 160 m

Set a mosque’s portal square to the street or square it opens from, but turn the building behind it to face Mecca, and let a vestibule or corridor make the turn between them.

Isfahan: qibla ≈ 225°, i.e. 45° to the Maidan

Build the bazaar as one continuous covered street, roofed by a dome over every bay, linking the city’s squares, and open caravanserais, colleges, baths and mosques off it along its length.

Run a broad avenue from the palace to the river, with a water channel, pools and fountains down its middle and rows of plane trees along its walks, and line it with walled gardens, each with a pavilion over its gate.

Found the mosque together with the institutions that serve it and its neighbourhood (colleges, a hospital, a public kitchen, a guest-house, a caravanserai, a bath, a school and the founder’s tomb), endowed as one charitable trust and grouped round the mosque’s outer court.

Set an imperial mosque on the crest of a hill so that its domes and minarets command the skyline, level the site on terraces and vaulted substructures, and step its dependencies down the slope.

Divide the city into neighbourhoods (mahalle), each gathered round its own small mosque, whose imam answers for its people; keep the workshops and markets in the market quarter, apart from the houses.

Build the house with a masonry ground storey and timber-framed upper storeys, and jetty the upper rooms out over the street in bays (çıkma) on brackets, squaring them over the irregular plot and turning their many windows to light, air and view.

Put the fortress-palace on the river bank, its private pavilions along the water and its public gates towards the town, and let the walled city spread round it on the landward side, its main streets starting from the fortress gates.

Run the principal street straight from the fortress gate to a mosque at its far end, lead a canal down its middle between trees, open an octagonal square with a pool halfway along, and line both sides with arcades of shops.

Raise the Friday mosque on the highest ground of the city and on a lofty plinth, approached by broad flights of steps on three sides, with its main gate facing the palace and its prayer hall facing Mecca.

Build the houses of the great round courts and gardens, with pools and fountains, rooms below ground for the heat of the day and terraces for the nights; the ordinary houses crowd behind the streets, many of them of mud and thatch.

Build the house of assembly long, with its head toward the east and rooms for the clergy on either side of the east end; seat the bishop in the middle of the head with his presbyters about him, and the people in order down its length.

Wall the whole precinct; open a great gateway toward the approach; before the church set a court open to the sky with porticoes on four sides and fountains in it, where those who enter may wash before they go in by the three doors.

Build the baptistery, and its font, with eight sides, because the eighth day, the day of the resurrection, is the day into which the baptised are born.

plan and font = octagon (8 = 7 + 1, the day after the Sabbath)

Bury no one inside the city; keep bodies, urns and sarcophagi outside the walls, even the relics of apostles and martyrs. So the great shrines of the martyrs rise in the cemeteries along the roads, not in the town.

Repair the public buildings of the city before beginning new ones; and pull down the booths, sheds and houses that private people build into the public porticoes or against the walls of public buildings.

Leave at least twelve feet between a new house and its neighbour; build no balcony lower than fifteen feet from the ground or nearer than ten feet to the house opposite; do not block a neighbour’s view of the sea within a hundred feet; and keep the shops between the columns of the great porticoes small and faced with marble.

gap ≥ 12 ft (≈ 3.7 m) · balcony ≥ 15 ft up, ≥ 10 ft apart · sea view protected within 100 ft

Assign each guild its quarter: the silversmiths to work in their shops on the Mese in full view; the perfumers to keep their stalls between the Milion and the Chalke gate, so that their fragrance rises to the holy image over the gate and fills the palace porches; the livestock trades to the outer fora.

Set ovens, baths, kilns, dyers, fullers, tanners and threshing floors at a fixed distance from their neighbours’ houses, and where the wind will not carry their smoke, smell or chaff into them.

Raise the principal church of the capital on its highest ground at the head of the city, so that it rises above the houses like a ship riding the waves and is seen from the sea on every side.

Before the palace gate make a square ringed by public buildings (the Great Church, the senate house, the palace vestibule), and set in it a column carrying the emperor, facing east toward the empire’s enemies.

Build the monastery, if possible, so that everything it needs (water, a mill, a garden and the various crafts) is inside the enclosure, so that the monks need not wander outside, which does their souls no good; and set a wise old monk at the gate, with a cell beside it.

Let all the monks sleep in one place if their number allows, each in his own bed, clothed and ready to rise, with a light burning until morning; place the dormitory where they can go straight to the church for the night office.

Receive every guest as Christ, the poor and pilgrims with the greatest care; give the abbot and the guests a kitchen of their own, so that guests arriving at unexpected hours do not disturb the brothers; lodge them in a guest house apart, in the care of a God-fearing brother.

Give the sick a room set apart and a brother to serve them, with baths when they need them; give the novices a place of their own where they study, eat and sleep under a senior.

Put the church in the centre with its altar to the east; the cloister against its south side, ringed by dormitory, refectory and cellar; the abbot, the school and the guests to the north; the novices, the sick and the gardens to the east; the kitchens, workshops, mills, barns and animals to the south and west; the hospice for the poor by the entrance.

Lay out a new town with the cord on open ground: straight streets crossing at right angles, running through from gate to gate, blocks of equal size between them, and a square in the middle; then, when the town can afford it, enclose the rectangle with a wall.

Give every settler a house plot of the same size in the town, at the same yearly ground rent, with a garden plot and a share of fields outside; so the blocks are divided into narrow lots of equal frontage, each house on the street with its yard behind.

Leave one block of the grid open as the market square; surround it with houses standing on arcades, so that the market may be held under cover on all four sides; put the market hall in the middle, with the standard measures.

Give the church a block of its own next to the market square, near a corner of it, so that the market keeps its open ground and the church its churchyard.

Do not build party walls: leave a narrow gap between neighbouring houses, to carry off rain and waste and to stop fire running along the row.

Set the kremlin on the high ground where two rivers meet, so that water guards two sides; wall it with towers at the corners and gates, dig a moat along the landward side, and put within it the cathedral, the prince’s palace and the metropolitan’s court.

Keep a broad strip before the kremlin wall free of houses and churches, so that fire in the wooden town cannot reach the fortress and defenders have an open field of fire; let the market be held on it.

Lay the posad out in streets radiating from the market square, and give every household a fenced yard with its house, barn, bath-house and garden, so that houses stand apart among gardens rather than in rows.

Build the house of whole logs laid horizontally and notched into one another at the corners, their ends left projecting; stop the joints with moss; roof it with boards or birch bark on a steep pitch; raise the living room on a lower storey where the ground is damp.

Build the principal cathedral after an honoured prototype: study the old cathedral that is to be the model and keep its form (five domes, walls divided into bays each crowned with a rounded gable, three apses to the east) while building it with the best new craft.

Lay the whole island out in a single rectangular grid regardless of the ground: avenues 100 ft wide running its length, cross streets 60 ft wide across it, and blocks 200 ft deep between them, the houses to be straight-sided and right-angled because such houses are cheapest to build and most convenient to live in.

avenue 100 ft (30 m) · street 60 ft (18 m) · block 200 ft (61 m) deep, 600–900 ft long

Divide the city into residence, business and unrestricted districts, and admit to each only the uses that belong there, so that lofts and factories cannot invade the shopping streets and houses.

Treat the largest mass the zoning law allows as a block to be sculpted: cut it into setbacks that give usable, well-lit floors, and gather the remaining height into a tower that ends in a crown.

Concentrate the business centre in a few very tall towers set far apart on a regular grid, so that they cover only a small fraction of the ground and the rest can be park: density and open space at once.

twenty-four cruciform towers of about sixty storeys; a small fraction of the centre built on

Replace the street lined with buildings by roads for traffic alone, sorted by speed and on separate levels, crossing at a central interchange; give the ground between them to pedestrians and parkland.

House the population in continuous slabs raised on pilotis and folded back and forth round large open courts, so that every flat looks over the park and the ground runs unbroken beneath the buildings.

Analyse the city into four functions (dwelling, work, recreation and circulation) and plan a separate, appropriate place for each, joined by a network of roads classified by speed.

Where density is needed, build tall blocks widely spaced, so that every dwelling has sun, air and a view, and the ground freed between them becomes green space.

Run housing slabs in parallel rows, north to south so that the flats have morning and evening sun, and space them by a fixed angle of light: then the taller the rows, the more dwellings the same ground holds, or the more ground is left free for the same number.

gap = H ÷ tan α (α the chosen angle of light, e.g. 30°: gap ≈ 1.7 H)

Place industry and other large work places in their own zone, served by road and rail, near the housing but separated from it by a belt of green.

Give every family at least an acre of its own ground (enough for a house, a garden and an orchard) and let the house open to that ground rather than to a street.

1 acre ≈ 4,050 m² ≈ 64 × 64 m

Dissolve the centralised city: with the car, the telephone and the radio every service can be spread along the highways, the market beside the road, the factory near the homes, the few tall buildings standing alone in parkland.

Build the family house on one level on a heated concrete slab, on a planning grid, round a brick core of hearth and kitchen; wall it in wood towards the street and in glass towards the garden; roof it flat with deep eaves, and shelter the car in a carport instead of a garage.

Give high housing a deck every few floors, wide enough to be a street, on which front doors open and neighbours meet, and join the decks from block to block into a continuous network across the city.

Run the decks the whole length of a continuous, snaking slab and use the fall of the site so that every deck arrives at ground level somewhere: the roofline stays level while the number of storeys rises as the ground drops.

Park Hill: decks 10 ft (3 m) wide every third floor; 4 to 13 storeys under one level roof

In a dense centre, give the ground to vehicles and service, and build a continuous pedestrian level above it (a podium and highwalks joining the buildings) so that people never meet the traffic.

Do not design the buildings; write a code. Give each lot a building type, and for each type fix on one sheet where the building stands, how high it rises, where its porch and outbuilding go, whether it may have a tower and where cars are parked; then let any architect design within it.

Bring every house forward to the street with a front porch, and mark the lot line with a white picket fence, each house with a different pattern of fence; build in wood with pitched metal roofs, and allow a small tower where the code permits.

Make neighbourhoods compact, pedestrian-friendly and mixed in use, with a clear centre and edge, so that daily needs lie within a short walk: about a quarter of a mile, five minutes, from centre to edge.

centre to edge ≈ ¼ mile ≈ 400 m ≈ 5 minutes on foot

Give civic buildings and public places the important sites (the heads of streets, the edges of squares, the ends of vistas) so that they reinforce the identity of the community.

Plan the permanent things (the structure, the service routes, the roads) as a lasting framework, and treat the enclosures plugged into it as a kit of standard parts, each renewed at its own rate.

Do not compose the park as one picture: lay down independent systems (points, lines and surfaces), each complete in itself, one over another, and let them collide where they meet.

Set a red folie at every intersection of a 120-metre grid across the whole site; make each a 10.8-metre cube on a 3.6-metre grid, then subtract parts and add stairs, ramps, drums and canopies so that no two are alike.

grid 120 m · cube 10.8 m = 3 × 3.6 m

Cross the park with two straight covered galleries at right angles, joining its gates and landmarks, and wind a promenade through it like a film strip, linking a sequence of themed gardens.

Leave large open surfaces (lawns and gravel) for games, crowds, markets and whatever the city brings, defined only by their shape.

Take the lines of the roads that arrive at the site from its surroundings and bundle them into a grid that bends and swells to meet them, instead of imposing a rigid orthogonal grid.

Make the building types a continuum governed by parameters: let the perimeter block morph by gradual steps into a block with a tower, and into towers alone, so that neighbouring blocks differ by degree, never by collage.

Govern the height and density of every block by a continuous field that rises towards the main axis and the centres, so that the skyline is a smooth landscape rather than a set of isolated peaks.

Lay out the rows east–west so that the long, glazed face of every house turns to the south (in the northern hemisphere), taking the low winter sun and shading easily from the high summer sun.

Set the gap between rows so that the shadow of each row at noon on the chosen winter day falls short of the glazed face of the row behind it.

gap = H ÷ tan α, with α = 90° − latitude − 23.4° at the winter solstice (at 48° N, α ≈ 18.6°: gap ≈ 3 H)

Make each neighbourhood a local transport area: cars reach it by a ring or edge road and park at its boundary, while inside the quickest way from place to place is on foot or by bicycle.

Put an open public green within about three minutes’ walk of every house, large enough to be used and not just looked at.

≤ 750 ft (≈ 230 m) from every house

Face the homes south behind glazed sunspaces; put workspaces on the north side, in the shadow of the row in front, where even light suits them; cover the roofs with sedum and gardens; and ventilate through wind-driven cowls with heat recovery.

At the centre of the city lay out the main piazza as a rectangle twice as long as it is wide, with the cathedral at one end and the prince’s palace at the other, and put a merchants’ square and a market square beside it.

Sforzinda: main piazza 300 × 150 braccia (about 175 × 90 m)

Partway along the sixteen streets that run from the centre to the gates and towers, open a secondary square in each, and give half of them parish churches and the other half markets, so that every quarter has its church and its market close at hand.

Make a public square twice as long as it is wide, surround it with porticoes, and keep the buildings round it in proportion to the open space: about a third of its breadth high, and never less than a sixth.

length = 2 × breadth; height of the buildings = ⅓ to ⅙ of the breadth

Set the principal temple where it can be seen, standing free in its own open space, and raise it on a podium reached by steps, so that it stands above the ordinary business of the town.

In a great and powerful city make the main streets broad and straight, lined with porticoes and with houses of the same height and design, so that the street itself becomes an ornament; in a small town let the streets wind instead.

Give a new city a regular polygonal outline with a bastion at every angle, put the principal piazza at the centre with the chief church and the seat of government on it, and distribute lesser squares for the markets through the quarters.

Make the central square a regular hexagon from which six streets leave, three to the gates and three to bastions, so that the garrison assembled in the middle can reach any gate or any point of the walls directly.

Build the gates of a fortress in rustication (rough, heavy blocks bound into a Tuscan or Doric order) so that the gate shows the strength of the walls it pierces.

Line the principal street with the houses of great persons (palaces with regular fronts, cornices and pediments) receding to a monument, as in the stage set for tragedy; leave the humbler houses, shops and taverns of the comic scene to the side streets.

Build the country house by a navigable river if you can, for cheap transport of the harvest and for beauty, or else by a main road; on raised ground open to the air, away from standing water and deep valleys, and in the middle of its own land.

Put the owner’s house at the centre and join the farm buildings to it with covered porticoes (barchesse), so that one can go everywhere under cover; give granaries, cellars, stables and threshing floor their proper places, in proportion to the estate.

Make the streets of a city straight and wide enough for their traffic; pave the middle for carts and animals and raise the sides a little for people on foot, and where the climate is hot or wet, line them with porticoes so that citizens go about their business under cover.

Make the roads outside a city straight, broad and raised above the fields, and plant trees along both sides, to shade the traveller and delight the eye.

Leave piazzas in several parts of the city for markets and assembly, the principal one in the middle; ring it with porticoes, and set round it the palace of the prince or the signoria, the treasury, the prisons and the basilica where justice is done, with the principal church close by.

breadth : length = 2 : 3 (the Latin forum)

Join the principal churches of the city by straight streets cut across hills and empty land, so that a pilgrim can walk from one to the next directly, and mark the end of every street with an obelisk visible from far away.

Where the main road enters the city, open a piazza inside the gate and send three straight streets from it into the town, fanning from a common point marked by an obelisk, with a pair of matching churches in the angles between them.

Before the mother church lay out an oval piazza enclosed by free-standing colonnades four columns deep, so that the church seems to embrace the faithful; set an obelisk at the centre and a fountain on each side along the cross axis, and join the oval to the church front by straight corridors.

Bring water back to the high, empty parts of the city by aqueduct; let it arrive at a monumental fountain and feed public fountains at the crossings of the new streets, so that people will come to live there.

Put the prince’s palace at the centre of a circle of thirty-two straight rays; give nine of them, between the wings of the palace, to the town, and send the other twenty-three through the forest as rides, so that every street and every ride ends on the palace tower.

32 rays at 11¼°: 9 streets + 23 forest rides

Give settlers free plots and privileges, but oblige them to build to the prince’s model designs, with the same height, the same eaves line and the same mansard roof, so that the streets read as continuous fronts leading to the palace.

Lay the parterres next to the house, low and flat so that nothing blocks the view from its windows; set bosquets and groves beyond them, and lead long allées out toward the horizon, so that the garden always seems larger than it is.

Plan a town as a park is cut through a forest: with many straight streets, squares from which several streets branch, and broad entrances, so that there is regularity in the details and variety, even a certain disorder, in the whole.

Give a pleasure house a single principal storey of rooms in enfilade round a central salon that projects into the garden, open on to the parterres through tall windows, and set it where its axis continues into the garden’s allées.

Lay out a town as a gardener lays out a forest park: wide, straight streets as its avenues, squares where they meet, well-marked entrances, and within that order enough variety of buildings that the whole never becomes monotonous.

Do not leave the fronts of houses to the whim of each builder: let the authorities fix their line, height and design so that a street or square is composed as a whole, and vary the designs from one street to the next so that uniformity does not become tedium.

On a ridge, run the principal street along the crest and make it the widest; set a square at each end of it; lay two more streets along the flanks, and narrow lanes between them for coach-houses, shops and lesser houses.

Build the outer streets on one side only, so that every house looks across open gardens to the landscape beyond, and keep the far side of the street free of building.

End each long street on a monument: put a church on the axis of the main street at the far side of each square, so that the view along it finishes on a portico and a dome or steeple.

Let a town’s churches and public buildings, not its engines of utility, shape its skyline: a spire to each parish, halls for its guilds, a cross in its market place.

House the poor as a Christian community: small dwellings, each with its own door and hearth, round a garden court, with a chapel over the gate, not in a walled workhouse built for surveillance and punishment.

Orientate a church with its altar to the east; give it a chancel clearly distinct from the nave (lower, narrower and at least a third of its length); a porch, preferably on the south; a steep roof; a tower at the west or beside the nave; open benches instead of pews; and details of the Middle Pointed style.

Arrange a building for its uses first and let its outline follow: gables, bays, chimneys and towers go where the rooms behind need them. Never contrive irregularity for effect, and never force a symmetrical front on an unsymmetrical plan.

Cut new straight streets through the old city to join its railway stations to one another and to its centre, and end each one on a monument (a station, a church, an opera house) so that it has an object as well as a use.

Tie the height of every house front to the width of its street: on streets 20 m wide or more the cornice may stand at 20 m, on narrower streets lower; above the cornice the roof must stay inside a line rising at 45°.

street ≥ 20 m → cornice ≤ 20 m; street < 20 m → cornice ≤ c. 17.5 m (older rule); roof inside 45°

Build every house on a new street to the same alignment, in dressed stone, with its floors, balconies and cornice at the levels of its neighbours, and keep the fronts clean, so that the street reads as a single wall.

Where many avenues meet, make a round place with a monument at its centre, bring the avenues to it at equal angles, and wall the place with uniform buildings between them.

Line the great streets with rows of trees and wide pavements, and give every quarter a planted square within a few minutes’ walk.

Build a town of limited size (about 32,000 people on 1,000 acres) in the middle of an estate of 6,000 acres owned by the community, and keep the surrounding 5,000 acres as a permanent agricultural belt; when the town is full, found another beyond it.

town 1,000 acres + belt 5,000 acres; 30,000 in the town + 2,000 on the land

Plan the town as rings round a central garden and cut it into six wards by six boulevards; put the great public buildings round the garden, a park beyond them, houses in the middle rings, and factories and warehouses on the outer ring beside a circular railway.

Enclose the central park with a wide glass arcade, the “Crystal Palace”, to hold the shops and serve as a winter garden, so that everyone can shop and walk under cover within a few hundred yards of home.

Run a ring avenue about 420 feet wide through the middle of the residential rings as a continuous park, and set in it the schools, with their playgrounds, and sites for churches of every denomination.

Build no more than about twelve houses to the acre, in groups rather than endless rows, each with a front garden and a long back garden; the saving on roads pays for the extra land.

≈ 12 houses per net acre (≈ 30 per hectare), against 40 or more in by-law streets

Take the forms of a new architecture from modern life, modern materials and modern construction, not from past styles: what is not practical cannot be beautiful, and a building should show plainly what it is for.

Begin every design from its construction; the architect’s task is to turn the construction into art-form, never to hide it behind forms borrowed from another material.

Accept that the modern city is made of rental apartment blocks of equal height and similar floors; line its streets with them in long uniform rows, and give it variety and dignity through monuments, squares and parks rather than through fancy dress on each house.

Regulate the river, run the city railway beside it, vault the river over and build a great boulevard on the vault, so that engineering and city beauty are planned together.

Cap the height of new buildings (at 130 feet from 1893) so that the business district does not become a canyon of towers; the cap was raised to 260 feet in 1902 and lowered to 200 in 1911.

1893: 130 ft (≈ 40 m) · 1902: 260 ft · 1911: 200 ft

Treat the city’s railways, elevated lines and streetcars as a single system serving the business centre, and plan their stations and routes together, relieving the elevated loop with subways and through routes.

Keep the whole shore of the lake as public park, with lagoons, beaches and harbours, and set the city’s museums and libraries along it as its cultural forecourt.

Widen one street into the city’s main axis, running from a monument on the lakefront to a domed civic centre on high ground inland, and let the rest of the plan hang on it.

Size the forum to the population, neither cramped nor deserted, and make its breadth two-thirds of its length, so that the oblong suits the shows held in it.

breadth : length = 2 : 3

Put the temples of the city’s protecting gods (Jupiter, Juno and Minerva) on the highest ground; Mercury in the forum; Apollo and Bacchus by the theatre; Hercules by the circus; Mars and Venus outside the walls.

Project the towers beyond the line of the wall, no further apart than a bowshot, so that attackers are exposed on their flanks; bring the roads to the gates obliquely, so that an enemy’s unshielded right side faces the wall.

Build the theatre on a healthy site and do not let its curved auditorium face south, where the sun would heat the enclosed air and harm the audience sitting through the plays.

Site the baths where it is warmest, turned away from the north and north-east, and light the hot and tepid rooms from the south-west, since people bathe from midday to evening.

A barrel (tunnel) vault is an arch extruded along its axis. It thrusts outward all along both sides, so it must rest on continuous thick walls, which can be pierced only sparingly, and only below its springing.

thrust per metre of wall ≈ constant along the whole length

Intersect two barrel vaults of equal height at right angles: the result is a groin vault, whose weight and thrust are gathered along the diagonal groins to the four corners, leaving the sides free for openings.

surface = max(barrel₁, barrel₂) · groins on the diagonals

Where the groin vault is the union of two barrels, the cloister (domical) vault is their intersection: four curved cells rise from all four walls and meet in valleys along the diagonals. It needs walls on every side.

surface = min(barrel₁, barrel₂) · valleys on the diagonals

Two semicircular barrels of different spans rise to different heights. To groin-vault an oblong bay the narrower barrel must be stilted or flattened, and the groins then wander out of plane: the difficulty that pushed builders towards ribs and pointed arches.

Build the arches of the vault first (transverse, wall and diagonal ribs) on light centring, then fill the triangular cells between them with thin webs of stone laid course by course from rib to rib.

Over an oblong bay, make the diagonal ribs semicircular and strike the transverse and wall arches as pointed arches of the heights needed to bring every crown nearly level with the central boss.

diagonal: r = ½ diagonal · transverse & wall: pointed, rise ≈ half-diagonal

Cover two bays of the nave with a single square vault divided into six cells by its diagonals and an extra transverse rib through the centre. The main piers carry three ribs each, the intermediate piers only one, so the supports may alternate strong and weak.

Raise the crown of each bay well above the tops of its boundary arches, so that the vault becomes a shallow dome on ribs whose thrust falls more steeply onto the walls.

crown ≈ half-diagonal above the springing; boundary arches ≈ half-span

Add to the diagonal ribs further ribs (tiercerons) that rise from the same springers to points on the ridge rib, so that each springer sends up a sheaf of ribs and the vault reads as a palm.

Add short ribs (liernes) that spring from neither the springers nor the central boss but run from rib to rib, linking them into star and net patterns across the vault.

Make each springer the foot of an inverted half-cone (a conoid) whose ribs all have the same curve and are cut across by horizontal rings; where the conoids stop, fill the centre with flat panels.