Welcome to the Overwhelming Beauty of a Muqarnas

Welcome to Henk Hietbrink's webpage on muqarnas. Muqarnas are a masterpiece of Islamic geometrical ornamentation. I call it works of art because it is so much more than geometry, although some mathematics can help to describe their structure. For a true understanding, I recommend that we listen to the architects and engineerings who take care of their heritage. For more information, please contact h.hietbrinkplanet.nl.

Isfahan Sheikh Lotfollah Mosque
© Henk Hietbrink

This webpage:

Works of Art
Muqarnas are Works of Art
Aim of my Research
My research aims to develop algorithms that facilitate the transformation of 2D tessellated floor plans into 3D-printed muqarnas scale models by assigning standardized, pre-designed units.

From 2D to 3D: Bridging the Gap

Read your Literature
Over the past fifty years, leading architects and scholars in Türkiye and Iran have published significant scholarly works on muqarnas.

Excel
Excel is my development environment to generate and analyze muqarnas floorplans.

GeoGebra

GeoGebra is a browserbased app in the cloud. GeoGebra's ability to define and modify macros is a powerful feature. It allows users to record macros and refine them later. Together with Chris Cambré, I developed a tool that uses buttons to activate macros, each of which draws one of twenty specific muqarnas units.

Rhino Grasshopper

Rhino (or Rhino3D) is a commercial 3D computer graphics and computer-aided design (CAD) application. My Grasshopper muqarnas model has four objects, a full unit, an intermediate unit, a flat wall, and a püskül (stalactite).

Related webpages on this website:

Aim
My research aims to develop algorithms that facilitate the transformation of 2D tessellated floor plans into 3D-printed muqarnas scale models by assigning standardized, pre-designed units.
Tiers
A definition: The muqarnas is a vaulting system based on the replication of units arranged in tiers, each of which supports another one corbeled on top of it.
Pre-designed Units
The replication of units gives rise to the idea of standardised, modular, pre-designed units. Could muqarnas units be assembled like Lego bricks?

Debate

Additional Units and Exceptions
We identified many more unit types than the twelve combinations of upper and base proposed by Harmsen.
al-Kashi and his followers
For centuries, mathematicians have been intrigued by muqarnas. Their story traces back to the 15th-century Iranian mathematician and astronomer al-Kashi, who computed the area of curved surfaces.

Ödekan Tuncer Uluengin
Three Turkish writers on muqarnas should be mentioned: Prof. Dr. Ayla Ödekan, Prof. Dr. Orhan Cezmi Tuncer and Mehmet Fatin Uluengin.
Sakkal
Mamoun Sakkal worked on Armenian muqarnas. I explored his drawings
Conclusions
Literature
Over the past fifty years, leading architects and scholars in Türkiye and Iran have published significant scholarly works on muqarnas.

Photos
A few photos and a long list of photo repositories are available
My Instagram
Here is a list of my Instagram stories. This page is searchable and offers direct links to each Instagram story.
NWD 2025 Workshop
How much math is hidden in a muqarnas? What about symmetry and similarity?
Bridges 2025 Contribution
The figures in my contribution deserve a better stage. High resolution images are available on this webpage.
Bridges 2025 Activity
At the Family Day, participants can build a muqarnas with my Lego-like building units.

Muqarnas as Works of Art

Have you ever stood before a mosque, gazed upward, and felt captivated by the beauty above you? You may have been admiring a muqarnas, a breathtaking work of art designed to leave you speechless. Muqarnas, found in portal hoods, beneath minaret balconies, or in the corners of prayer halls, are masterpieces of architectural ornamentation. With their variety of materials, designs, and interplay of light and shadow, these decorations have been enchanting visitors for nearly a millennium.
Photos and links to photo albums are on their own web page. My stories on Instagram include many photos.

Definition

The muqarnas is a vaulting system based on the replication of units arranged in tiers, each of which supports another one corbeled on top of it. To truly grasp the essence of muqarnas, one must first understand its fundamental elements: units and tiers. Depending on the material and method of construction, muqarnas can be conceived either as a standing ornament-such as a layered stacking of cut stone rising from the base, or as a hanging ornament, such as plasterwork suspended from above by ropes attached to a frame hidden behind the muqarnas. However it is conceived, analysis of the pattern usually begins with the central star at the top.

Mathematicians

For centuries, mathematicians have been intrigued by muqarnas. Their story traces back to the 15th-century Iranian mathematician and astronomer al-Kashi, who documented the craft of masons and their techniques for constructing muqarnas. Mathematicians dream of algorithms capable of seamlessly elevating (converting) a 2D tessellation plans into 3D muqarnas forms. It turns out that this kind of reverse engineering is as promising as it is disappointing.

Aim

My research aims to develop algorithms that facilitate the transformation of 2D tessellated floor plans into 3D-printed muqarnas scale models by assigning standardized, pre-designed units. I refer to this as a reverse engineering process, with the goal of integrating it into rule-based software. My tools are Rhino Grasshopper, GeoGebra, and Excel Visual Basic.

Pre-designed Units

The replication of units gives rise to the idea of standardised, modular, pre-designed units. Could muqarnas units be assembled like Lego bricks? Among others, Harb, Harmsen, Sakkal, and I had ideas.

"Pre-designed" and "pre-fabricated" are similar terms, but they're not exactly the same. The difference matters. In the figure above, the orange jug (top part) and the orange biped (bottom part) go together. Almost every jug has a biped underneath, with just a few exceptions. So, they form one pre-designed unit. But when it comes to production, only the top part (the jug) is made using a mold. The bottom part isn't. The same idea applies to the blue squares and red triangles. There's a mold for the top part (the pre-fabricated piece), but no mold for the bottom part.

Dold-Samplonius and Harmsen do "pre-fabricated", Sakkal does "pre-designed", Ödekan is more into "pre-designed" because of her geometric modules. Interesting to note that the Turkish architects Uluengin, Tuncer, and Senalp pay attention to both approaches.

Permutations

I wrote down all permutations in the octagonal grid and counted 28 configurations. Each unit is divided into two mirror-symmetrical triangles. Angles are multiples of 22,5° and 8 × 22.5° = 180°. Other configurations were rejected because they would be too large. The next question is whether these exceptional and additional units exist in real-world examples, and the answer is yes. Most of them can be found in Seljuk, Ottoman, and Armenian styles.

From 2D Floorplan to 3D printed Muqarnas: Bridging the Gap

The projection of a 3D muqarnas onto a plane forms a type of tessellation. However, a projection inherently contains less information than the original 3D structure, making it unclear whether a 2D tessellation can fully determine its 3D counterpart. The central challenge lies in how to "elevate" a tessellation: transforming a 2D floorplan into a 3D form by assigning pre-defined units.

Excel, GeoGebra, Rhino Grasshopper, and STLviewer

Excel is my development environment to generate and analyze muqarnas floorplans.
GeoGebra is a browserbased app in the cloud. GeoGebra's ability to define and modify macros is a powerful feature. It allows users to record macros and refine them later. Together with Chris Cambré, I developed a tool that uses buttons to activate macros, each of which draws one of twenty specific muqarnas units.
Rhino Grasshopper (or Rhino3D) is a commercial 3D computer graphics and computer-aided design (CAD) application. My Grasshopper muqarnas model has four objects, a full unit, an intermediate unit, a flat wall, and a püskül (stalactite).
Rhino and GeoGebra can output files in many formats. STL is commonly used for interfacing with 3D printers. ViewSTL offers an online webbased tool that shows the contents of an STL file. This tool supports rotation, panning and zooming.

Irregularities and Exceptions

Through my travels in Türkiye and Iran, I have observed hundreds of muqarnas. At first glance, the tiers appear symmetrical and uniform, as our definition suggests. Yet, appearances can deceive. Subtle irregularities are common: varying tier heights, uneven unit lengths, broken symmetries, or manipulated shapes. Seljuk and Ottoman muqarnas feature pentagonal and hexagonal shapes (püskül) within an octagonal grid. Mathematically, it's impossible to fit regular pentagons or hexagons into a strict octagonal grid. Yet, muqarnas defy these constraints, blending art and geometry. Fieldwork shows that the repertoire of units extends far beyond the twelve listed in Table 1. The coding has evolved historically and is therefore incomprehensibly illogical. Craftsmen, architects, and engineers designed muqarnas with artistic freedom, prioritizing the illusion of rules over strict adherence to them. Irregularity, rather than exception, is the norm in these intricate works of art.

Literature in Turkish

Western literature has provided little insight into understanding Turkish and Iranian muqarnas designs. Over the past fifty years, leading architects and scholars in Türkiye and Iran have published significant scholarly works on muqarnas. Three Turkish professors stand out in this field: Tuncer, Ödekan, and Uluengin. Their books deserve a wider audience and are worthy of translation. I paid attention to them in my Instagram stories. Three webpages are devoted to each of them.

Grid

Understanding the tessellation grid is crucial. In Seljuk, Ottoman, and Armenian muqarnas, the dominant grids are octagonal, decagonal, and hexagonal. In an octagonal grid, four lines of symmetry divide the muqarnas into four mirror-symmetrical parts. The fundamental angle in this grid is 45°. The floor plan consists of a tessellation of squares and rhombuses, with angles of 45° and 135°. If you cut a rhombus in half, you create two isosceles triangles, each with a 45° apex angle and a 67.5° base angle. It is often more practical to think of all angles as multiples of 22.5°, that is half of the apex angle. Some muqarnas use a decagonal grid, where five lines of symmetry are visible from the apex. The base angle is 36°, and all angles are multiples of 18°. A hexagonal grid, on the other hand, has angles of 30°, with all angles being multiples of 15°.

While these underlying grids can often be identified in the tessellation, they tend to appear only locally rather than governing the entire structure. A central grid for the entire tessellation is rare. Ödekan uses her geometric modules to link the local grids. Further research is needed to explore how the arrangement of these circles enhances our understanding of the interconnections between local grids.

Different Views to Pre-designed Units

A major aspect of muqarnas analysis is understanding how to identify tiers and units. One approach is to divide the muqarnas into horizontal tiers and then identify the elementary units within each tier. This method allows us to describe the muqarnas layer by layer. However, it has its drawbacks: objects that logically belong together may be split between two units, with one part on an upper tier and the other on a lower one. Harmsen follows this approach in her dissertation, where she distinguishes between full units (edge to point) and intermediate units (point to edge). Sakkal, using a more systematic approach, lists all possible permutations where angles are multiples of 22,5°. He combines full and intermediate units into a single object. This makes sense because each intermediate unit is always paired with a full unit above it. Sakkal mapped out all possible configurations in the octagonal grid and identified over twenty pre-designed units. Kazempour names the objects within the tiers: taseh (full), shaparak (intermediate), pabarik (intermediate), toranj (intermediate), and shamseh (radial star). Typically, the top of a muqarnas features a large radial star, almost horizontal, forming a wide opening. Smaller radial stars appear in the lower tiers along lines of symmetry.

Ödekan, Tuncer, and Senalp further emphasize the verticality of muqarnas. Dincer summarizes that in Anatolian Seljuk muqarnas, the units are generally classified into two main categories: yaprak (full) and kazayagi (bipeds), with a third category for püskül, small hanging ornaments in the center of stars.

In Ottoman muqarnas, Senalp identifies two additional elements: fitil (a slimmer version of the kazayagi) and badem (an alternative to the yaprak).

Do We Need Pre-designed Units?

So far, I have focused on pre-designed units with fixed angles. However, as noted earlier, Turkish elements like the yaprak and badem are not bound by strict angular constraints; their angles can approximate 30°, 36°, or 45°. This flexibility invites us to move beyond the concept of rigid pre-fabricated units and explore alternative approaches. Modern computer graphics offer promising possibilities. Historical techniques, such as Al-Kashi's use of squares and circles, can now be extended with advanced geometries like rectangles, ellipses, and smooth curves such as b-splines, made possible by modern drawing software. Future research will aim to highlight the advantages of parametric modeling over pre-designed units, paving the way for greater flexibility and creativity in muqarnas design.

Conclusion

The concept of pre-designed units within a system of horizontal tiers is undeniably appealing. However, as structures grow larger, the process of assigning pre-designed units to the tessellation becomes increasingly cumbersome, leading to a proliferation of unit types and no solution how to incorporate heptagonal stars in an octoganal grid. Exploring traditional concepts such as yaprak, badem, and taseh proved far more effective than rigidly categorizing units as full or intermediate. This shift in perspective paved the way for parametric design, which, in turn, simplified software development. Unlike fixed-angle units, parametric units allow for greater flexibility, making it possible to incorporate unconventional shapes, such as pentagonal and nonagonal püskül, as well as stalactites with polygonal bases.

Recommendations

For a deeper understanding of muqarnas, I encourage readers to visit Türkiye and Iran, and to engage with the foundational works of Ödekan, Tuncer, and Uluengin. AI can help in reading Turkish. I also recommend completing the full cycle of research, from interpreting a tessellation to 3D printing a miniature muqarnas. A physical model conveys understanding in a way that a digital rendering simply cannot.

Literature