Muqarnas Visualization

Project of the Visualization and Numerical Geometry Group



Historical Remarks

Click to see the fullsized image Muqarnas is the Arabic word for stalactite vault, an architectural ornament developed around the middle of the tenth century in north eastern Iran and almost simultaneously, but apparently independently, in central North Africa.
Two points about these new forms are of importance. One is that, from the late eleventh century on, all Muslim lands adopted and developed the muqarnas, which became almost as common a feature of an elevation as the Corinthian capital was in Antiquity. The second and far more important point is that, from the moment of its first appearance, the muqarnas acquired four characteristic attributes, whose evolution and characteristics form its history: it was three-dimensional and therefore provided volume wherever it was used, the nature and depth of the volume being left to the discretion of the maker; it could be used both as an architectonic form, because of its relationship to vaults, and as an applied ornament, because its depth could be controlled; it had no intrinsic limits, since not one of its elements is a finite unit of composition and there is no logical or mathematical limitation to the scale of any one composition; and it was a three-dimensional unit which could be resolved into a two-dimensional outline.
The fifteenth century Timurid mathematician Ghiyath al-Din Mas'ud al-Kashi (~1380 - 1429) defines the muqarnas in his practical way as: "The muqarnas is a ceiling like a staircase with facets and a flat roof. Every facet intersects the adjacent one at either a right angle, or half a right angle, or their sum, or another combination of these two. The two facets can be thought of as standing on a plane parallel to the horizon. Above them is built either a flat surface, not parallel to the horizon, or two surfaces, either flat or curved, that constitute their roof. Both facets together with their roof are called one cell. Adjacent cells, which have their bases on one and the same surface parallel to the horizon, are called one tier." [1]. In addition there are intermediate elements which connect the roofs of adjacent cells. (For a more detailed explanation, see the example in the following chapter.)
Al-Kashi distinguishes four types of muqarnas: The Simple Muqarnas and the Clay-plastered Muqarnas, both with plane facets and roofs, as well as the Curved Muqarnas, or Arch, and the Shirazi, in which the roofs of the cells and the intermediate elements are curved. The plane projection of a simple element (either cell or intermediate element) is a basic geometrical form, namely a square, half-square (cut along the diagonal), rhombus, half-rhombus (isosceles triangle with the shorter diagonal of the rhombus as base), almond (deltoid), jug (quarter octagon), and large biped (complement to a jug), and small biped (complement to an almond). Rectangles also occur.
The elements are constructed according to the same unit of measure, so they fit together in a wide variety of combinations. Al-Kashi uses in his computation the module of the muqarnas, defined as the base of the largest facet (the side of the square) as a basis for all proportions.
The muqarnas is used in large domes, in smaller cupola, in niches, on arches, and as an almost flat decorative frieze. In each instance the module as well as the depth of the composition is different and adapts to the size of the area involved or to the required purpose. The muqarnas is at the same time a linear system and an organization of masses.
There is a relatively unbroken tradition of architectural practice in the Islamic culture. From the Ilkhanid period until today, 700 years later, the elements mentioned above did not change, but at the same time more elaborate muqarnas were constructed in which we find elements like five-, six-, or seven-pointed stars.

Here is a link to the website with a great survey on plans of Muqarnas, ordered by there geographic and historic relations:

Elements of Bastam

In this  detailed picture of a  muqarnas at Bastam (an Ilkhanid shrine situated midway between Tehran and Mashad) you see a few cells and the correlating plan. These cells consist of two facets, or vertical sides, with their curved roof. In general a roof can be a flat surface, not parallel to the horizon, or two joint surfaces, either flat or curved. The roofs of two adjacent cells can be connected by intermediate elements consisting of one surface, or two joint surfaces. A row of cells, with their bases on the same surface parallel to the horizon, is called a tier.
On the lower tier we see from right to left (in the plan: beginning at the upper right corner of the non-shaded area): an intermediate element based on a small biped, a cell based on a rhombus, then two broken intermediate elements, a cell based on a quarter octagon, an intermediate element based on a small biped, a cell based on a quarter octagon, again an intermediate element based on a small biped and a cell, with only the right facet visible, based on a quarter octagon. On the tier above: a cell based on an almond, a cell based on a quarter octagon, three cells based on an almond with a fourth one being barely visible.

Basic Geometrical Forms

The plane projections of the two different types of elements, cell and intermediate element, are simple geometrical forms. There are six such forms that are used most often: square, rhombus, almond (deltoid), jug (quarter octagon), large biped (complement to a jug) and small biped (complement to an almond).
Without context there exists no one-to-one relationship between cells, intermediate elements and their plane projections (see the rhombus example).
In the following we present the basic geometrical forms according to Al-Kashi [2, 3] and possible realizations as cells or intermediate elements. The side measuring the length of the module always lies at the bottom of each picture.

  1. Square

    Square as a cell.

    Plane Projection
    [Animation] [VRML Model]

  2. Rhombus

    Rhombus both as a cell and as an intermediate element.

    Plane Projection
    cell: [Animation] [VRML Model] intermediate element: [Animation] [VRML Model]

  3. Almond (deltoid)

    Almond as a cell.

    Plane Projection
    [Animation] [VRML Model]

  4. Jug

    Jug as a cell.

    Plane Projection
    [Animation] [VRML Model]

  5. Large Biped

    Large biped as an intermediate element.

    Plane Projection
    [No Animation available] [No VRML Model available]

  6. Small Biped (complement to a jug)

    Small Biped as an intermediate.

    Plane Projection
    [Animation] [VRML Model]

South Octagon Vault of the Takht-i Suleyman

The Takht-i Suleyman (Throne of Salomon) is situated in a wide valley at an altitude of approx. 2000 m, ca. 30 km North of Takab, N.W. of Tehran. The Sassanian sanctuary here flourished in the 5th and 6th century and was one of the three main fire sanctuaries. After the Islamic conquest it remained important as a Zoroastrian temple until the 9th century. In the 13th century the Ilkhanid ruler Abaqa (1265-1281) constructed a summer palace on the southern part of the ruins, partly based on the same layout and using former construction material. As the Mongol court was used to living in tents, the construction of the palace was not made for eternity and soon fell to ruins. In the ruins of the western part of the palace a plate has been found, which was recognized as a construction plan for a muqarnas vault. In analogy to this plan, Ulrich Harb proposed a possible plan to reconstruct the much simpler south octagon vault [6]. This plan is the base of our construction.

[104 kB] look into the vault from beneath [108 kB] detail [84 kB]

This animation [180kB] blends between Harb's plan and our construction.



Magic of Muqarnas

Stalactite Vaults in Islamic Architecture

Yvonne Dold-Samplonius

Silvia Harmsen

Susanne Krömker

Michael J. Winckler

Duration: 18 min
May 2005

We have completed the DVD/VHS with an accompanying booklet in English. Audio traces are in Arabic, English, German, Persian, and Turkish.

Sorry, but we do not ship the video any more.
Since January 2015, the English version is available for free via the following link

Muqarnas - stalactite vaults - form an essential part of Islamic architecture. The first written reference on the geometrical concepts dates back to a treatise by the famous Islamic mathematician and astronomer al-Kashi (died 1429). The relation between layout and spatial arrangement is the topic of this video.

First, historical sites of muqarnas around the Mediterranian Sea and in Middle Asia are shown. Then follows an introduction to the basic concepts of muqarnas construction with the help of computer graphics. The various elements are visualized and the connection between erected vaults and their 2D projections is explained.

The focus point of reconstructing muqarnas lies in Iran. Turning to a still existing niche of the Friday Mosque at Natanz, the film continues with a reconstruction of a vault corresponding to the oldest known muqarnas design, found at Takht-i Suleyman. Finally, animations of illuminated muqarnas at Bastam reveal the fine art of both architecture and computer graphics.


Mathematical Concepts and Computer Graphics for the Reconstruction of Stalactite Vaults - Muqarnas - in Islamic Architecture

Grant 03WNX2HD

There are a lot of ground plans of existing Muqarnas. Some of these 3D-vaults are still in good shape, others broke down and have to be reconstructed from their plans. However, in many cases even such plans do not exist any more.

First we want to convert existing muqarnas plans into the computer in such a way that we can analyze their properties: What kind of elements occur, which elements can be connected and how, what are the possible heights of the succeeding tiers, what about regional differences, cultural differences, differences in time?
We aim at a computer program that is able to answer these questions on muqarnas plans automatically. The obvious material to start with are the Illkhanid muqarnas plans in Ulrich Harb's book. These can be compared with existing architecture and thus show limitations in computer possibilities.

In the second stage we want to apply these methods to plans which are known to have not been realized, the Topkapi Scroll being our first choice; and we want to apply these methods in reconstructing muqarnas vaults to ruins like Varamin, Iran. It is also planned to produce several 15 min. videos on muqarnas to be used for teaching.

Yvonne Dold-Samplonius
Silvia Harmsen
Susanne Krömker
Michael Winckler

International Cooperation Partners

Professor Dr. Gülrü Necipoglu
Sackler Museum
Aga Khan Chair for the History of Architecture, Harvard
485, Broadway
Cambridge, MA, USA

Professor Dr. Mohammad Al-Assad
Director of the
Center for the Study of the Built Environment
PO-Box 830751
Amman 11183, Jordan

Dr. Jan P. Hogendijk
Mathematical Institute
University of Utrecht
PO-Box 80 010
3508 TA Utrecht, NL

We are very sorry to hear about the unexpected death of our cooperation partner Professor Alpay Özdural as a result of a heart attack on 22 February 2003. His invitation to Heidelberg was planned for May 2004.

     Professor Dr. Alpay Özdural
     Eastern Mediterranean University, Gazy Magusa
     Faculty of Architecture
     PO-Box 95
     Mersin 10, North Cyprus, Turkey


[1] Al-Kashi, Key of Arihmetic, 1429.

[2] Yvonne Dold-Samplonius, Practical Arabic Mathematics: Measuring the Muqarnas by al-Kashi, Centaurus 35 (1992), pp. 193-242.

[3] Yvonne Dold-Samplonius, How al-Kashi Measures the Muqarnas: A Second Look, Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich. Ed. M. Folkerts, (Wolfenbütteler Mittelalter-Studien Vol. 10), Wiesbaden 1996, pp. 56-90.

[4] Josie Wernecke, The Inventor Mentor, Addison-Wesley, 1994.

[5] Gülru Necipoglu, The Topkapi Scroll - Geometry and Ornament in Islamic Architecture, Santa Monica, CA: The Getty Center for the History of Arts and the Humanities, 1995.

[6] Ulrich Harb, Ilkhanidische Stalaktitengewölbe, Dietrich Reimer Verlag Berlin, 1978.

Technical Hints


We used JPEG images, animated GIFs and VRML models for visualization on these pages. You can browse through the page without an appropriate plugin installed, but you won't see any 3D VRML models and you will miss this nice feature. Which plugin you have to download, depends on your platform and browser. For Irix, MacOS and Windows 3.1/95/NT the Cosmoplayer from Cosmosoftware is available.
For Linux and other Unix Systems you need the Xswallow Plugin and an X-based VRML viewer.

Contact the Authors

For more information about

3D-modelling of muqarnas vaults and their elements
Silvia Harmsen Tobias Illenseer, Christian Reichert, Kurt Sätzler

Visualization of architecture
Susanne Krömker

Michael Winckler

Islamic mathematics and architecture
Yvonne Dold-Samplonius