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.
Rules · Ornament · Geometry · Modules
Symmetry and the repeat
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.
Squares, triangles and hexagons tile the plane alone; octagons, decagons and dodecagons do not, and leave gaps: squares between octagons, bow-tie shapes between decagons, triangles between dodecagons. The pattern-maker treats those gaps as part of the design: they receive smaller stars and rosettes, and the eye reads a hierarchy of primary and secondary figures. Ten-fold patterns, which cannot belong to any simple square or hexagonal grid, were a favourite display of skill in Seljuk and Timurid Iran.
The draughtsman worked with the smallest unit from which the whole field could be produced by reflection or translation, which is how the Topkapı Scroll presents its designs. A mason or tile-cutter could then transfer the unit to a wall of any size.
Related rules
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
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.
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
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.
Cite this page Chris Paton, “Symmetry and the repeat”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/girih-repeat. Text, drawings and models licensed under CC BY-NC 4.0.