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.
The contact angle sets the character
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.
For each tiling there are a few classic angles at which the lines meet neatly and the pattern closes into familiar figures: 67.5° for the octagon-and-square grid, 54° and 72° for the ten-fold grids, 60° for the hexagonal ones. Between them lie many less familiar but equally valid patterns. Pattern-makers chose among them for the effect they wanted: open, calm networks for large surfaces, dense sharp stars for the panels of a minbar or a door.
Related rules
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
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.
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
Cite this page Chris Paton, “The contact angle sets the character”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/girih-contact-angle. Text, drawings and models licensed under CC BY-NC 4.0.