Rules · Cities · Walls · Geometry
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
Gunpowder artillery made high medieval walls obsolete in a few decades after 1500. Italian engineers answered with low, thick earth-backed ramparts and with the angle bastion, a pointed platform projecting from the wall whose flanks allowed defenders to fire along the faces of the next bastion; there was no “dead ground” where an attacker could shelter. The trace italienne spread across Europe, and when the Venetian Republic founded Palmanova as a frontier fortress in 1593 it gave the new town the complete geometry: a regular nine-pointed star, radial streets and a central piazza, uniting the humanist ideal city with the engineer’s fortress. A second and third ring of outworks were added in the seventeenth century and under Napoleon.
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
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.
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
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)
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.
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
Cite this page Chris Paton, “The bastioned trace”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/bastioned-trace. Text, drawings and models licensed under CC BY-NC 4.0.