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
Hankin’s observation explains why Islamic star patterns look inevitable: every line is generated by the grid underneath. An octagon produces an eight-pointed star, a square a small four-pointed cross, and where the lines from neighbouring polygons cross the shared edge they continue straight on, so the whole surface is woven from a few long straight bands.
The model below does exactly this, including the pairing of lines where they first meet. Show the construction grid to see where each star comes from; it is the kind of grid that the draughtsmen of the Topkapı Scroll left faintly visible behind their drawings.
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.
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
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.
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
Cite this page Chris Paton, “Patterns from polygons in contact”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/girih-polygons-in-contact. Text, drawings and models licensed under CC BY-NC 4.0.