Rules · Domes · Geometry · Structure
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
The squinch is older and simpler than the pendentive: it turns the corner in steps rather than in one continuous surface. It was the Persian solution, used in Sasanian palaces, and it spread into Armenian and Georgian churches, into middle Byzantine buildings such as the katholikon of Hosios Loukas and Nea Moni on Chios, and into Islamic architecture, where squinches were subdivided into the honeycomb vaults called muqarnas.
The octagon may carry the dome directly, or a second tier of smaller squinches may turn it into a sixteen-sided polygon, nearer still to a circle. Where the pendentive hides the transition, the squinch displays it.
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)
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.
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)
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)
Cite this page Chris Paton, “The squinch: from square to octagon”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/squinch-octagon. Text, drawings and models licensed under CC BY-NC 4.0.