Geometric Squinch to Octagonal Dome Transition Projection: Tusi Geometry, 2D Tusi Plans & 3D Stereotomy

In the history of structural architecture, one of the most celebrated geometric problems is the zone of transition. When an architect seeks to crown a building with a majestic hemispherical dome, a profound physical conflict arises: the room below is naturally square or rectangular, defined by four flat vertical walls meeting at ninety-degree corners, while the dome above is fundamentally circular. How does one transition from a square to a circle without creating awkward, unstable masonry overhangs?

Roman architects solved this with massive, unadorned concrete walls; Byzantine architects developed smooth, spherical triangular pendentives. In medieval Persia, however, architects chose a vastly more sophisticated and sculptural path: the squinch (gushvareh). By throwing diagonal structural arches across the four upper corners of the square chamber, the room was transformed into an eight-sided octagon. By multiplying these squinches at subsequent tier levels, Persian mathematicians transformed the octagon into a sixteen-sided polygon, then thirty-two sides, and finally into a seamless circular drum that carries the dome without structural strain.

The genius of Persian muqarnas lies in its translation of complex, multi-layered three-dimensional stereotomy into elegant, comprehensible two-dimensional plan drawings. For centuries, architectural historians wondered how medieval Islamic craftsmen could construct vaults of such staggering mathematical complexity without modern computer-aided design tools.

The revelation came with the discovery of historic architectural working scrolls, most notably the Topkapi Scroll (preserved in Istanbul, dating to the late Timurid/early Safavid era) and the nineteenth-century Mirza Akbar drawings (in the Victoria and Albert Museum). These scrolls reveal that master architects (ostads) designed muqarnas vaults entirely as two-dimensional plan projections. Drawn with black ink and red register lines, the plans resemble intricate geometric mandalas. Each drawing is composed of a standardized vocabulary of repeating polygonal modules: rhombuses with 45-degree angles, kites (deltoids), right-angled triangles, and bipyramids. Because each 2D polygon corresponded to a specific three-dimensional niche unit of standardized height and projection, the flat drawing was a complete, unambiguous construction blueprint for the master mason.

To translate the 2D plan into three-dimensional reality, the architect relies on three primary stereotomic components. The fundamental unit is the taas (niche cup). Corresponding to a rhombus or kite on the plan, the taas is a curved, quarter-spherical hollow that acts as the decorative canopy of each individual cell. Flanking the taas are the baleh (wings or cheeks)—triangular, flat-beveled surfaces that connect the curved niche to adjacent units, ensuring that no raw structural voids remain between cells.

At the corners and transition points sit the kousheh (squinchettes). These are convex, triangular pendentive facets that bridge horizontal and diagonal steps. Crucially, the entire system is governed by a uniform vertical rise increment. Every tier in a given muqarnas vault—regardless of how far it corbels forward into space—rises by the exact same vertical distance (typically 150 to 250 millimetres). This mathematical discipline ensures that the tops of all units in a tier form a perfectly level horizontal ring upon which the next tier can be assembled without height discrepancies.

The construction of muqarnas was not merely an intuitive folk craft; it was deeply rooted in the advanced mathematics of the Islamic Golden Age. The legendary Persian astronomer and mathematician Ghiyath al-Din Jamshid al-Kashi (1380–1429), working at the royal observatory of Ulugh Beg in Samarkand, dedicated an entire book of his seminal mathematical work Miftah al-Hisab (The Key to Arithmetic) to the stereotomy and mensuration of muqarnas.

Al-Kashi provided the first rigorous mathematical definitions of muqarnas elements, classifying them into standardized geometric families and providing exact trigonometric formulas to calculate their surface areas, volumes, and projection angles. Al-Kashi’s formulas enabled master builders to calculate precisely how much calcined gypsum plaster would be needed for a monumental dome, how many thousands of mosaic tiles would be required for the revetment, and the exact physical load that the cantilevered tiers would impose on the building’s foundations, uniting theoretical mathematics with architectural master-craftsmanship.