Procedural modeling of Muqarnas

Introduction & Motivation

Example of Muqarnas structure

Muqarnas is a type of 3D ornamentation used in Islamic Architecture. This form of ornamentation was first developed during the Abbasid Empire. It is a complex stalactite vaulted structure composed of different units arranged one on top of another and spanning several tiers. The primary purpose of Muqarnas was to create a transition between a circular dome and the square structure beneath it.

This project uses procedural modeling to recreate several types of 3D Muqarnas models based on fixed sets of rules.

The logic followed to recreate 3D muqarnas models is mainly based on the book Miftah al-Hisab, Volume II: Geometry written in 1427 by Jamshid Al Kashi, a 15th century Persian mathematician and astronomer. In this book, Al Kashi tries to simplify complex geometric calculations in a way that lets artisans who do not have a thorough mathematical understanding to efficiently and accurately perform calculations necessary for taking measurements of the structures they are constructing.

The third section of his book On the surface area of the muqarnas contains three key points interesting for our project:

1. Muqarnas Definition

2. Muqarnas elements

3. Muqarnas Types

Muqarnas Definition

In his book, Al Kashi gave the following definition of a Muqarnas:

“A muqarnas is a stair-like ceiling that has facets and a surface. Each facet intersects with its adjacent either on a right angle or half a right angle or the sum of one and a half right angles, or others. The two facets can be thought of as perpendicular to a plane parallel to the horizon. Built over these two facets is a plane not parallel to the horizon, or two planes, or two curved surfaces, which are the ceiling of the facets. The two facets along with their ceiling are called a cell. Adjacent cells with bases on the same plane parallel to the horizon are called a tier. The length of the base of the largest facet is called the module of the muqarnas.”

Facet and roof of a Muqarnas cell

Intermediate element connecting two adjacent cells

Tiers of a Muqarnas

Facet: vertical side of a unit

Roof: plane not parallel to the horizon, or two planes, or two curved surfaces on top of the facets

Cell: two facets along with their roofs

Intermediate element: surface, or two joint surfaces, connecting the roofs of two adjacent cells

Tier: Adjacent cells with bases on the same plane parallel to the horizon

Module: length of the base of the largest facet

Muqarnas elements

Al Kashi describes five types of elements. The two main ones are the square and the rhombus, whose sides are equal to the module defined. Other elements mentioned are the Almond, Biped and Barley-Kernel.

Muqarnas Types

1. Simple Muqarnas: the ceilings have plane surfaces only

2. Clay-plaster muqarnas: simple muqarnas but the tiers do not have the same height

3. Curved muqarnas: surfaces of the ceilings are curved

4. Shirazi-style muqarnas: like curved muqarnas but with a larger variety of elements

Elements defined by Al Kashi

Curved Muqarnas construction

Project Plan

The project is separated in three main goals: identify the historical document to use, develop the procedural modeling methods and perform the modeling. The following table provides the project plan:

Date	Task	Completion
By Week 3	Choose three interesting project ideas. Prepare presentation slides for each project idea.	✓
By Week 4	Initial tutorial: Introduction to Rhino and Grasshopper.	✓
By Week 5	Rhino and Grasshopper applied exercises. Think of a definition of the context, content and outcome of the project.	✓
By Week 6	Grasshopper implementation of Islamic 2D star pattern Researching Abu al Wafaa al Buzjani's manuscript "On those geometric constructions that are necessary for a craftsman"	✓
By Week 7	Find the steps traditionally followed by craftsmen to construct muqarnas.	✓
By Week 8	Understand the construction methods described by Al Kashi's manuscript "Methods of the masons'. Identify different components used in Al Kashi's book to construct Muqarnas. Learn how to calculate the curvature coefficient for curved Muqarnas. Understand how to calculate muqarnas surfaces using Al Kashi's method. Implement the basic shapes in 2D.	✓
By Week 9	Work on the grasshopper 3D model. Parametrize the 2D basic shapes with a 3 point model, which will be useful to create the 2D plan. Model the 3D cells of each shape for simple Muqarnas.	✓
By Week 10	Midterm presentation: Prepare slides for the midterm project presentation. Layout steps to create 2D muqarnas plans in a procedural way.	✓
By Week 11	Complete the parametric 2D plan on Grasshopper. Identify the position of basic shapes on the plan. Identify the outline of each tier of the Muqarnas.	✓
By Week 12	Start applying the 3D shapes on the 2D plan of the model. Calculate the surface of the Muqarnas following Al Kashi's method. Compare the results to the surface calculated on Grasshopper.	✓
By Week 13	Final changes and fine tuning of the model. Implement curved muqarnas (only changing the arc is necessary). Complete the final report and wiki.	✓
By Week 14	Final presentation.	✓

Future work

In the future, it would be interesting to have a fully automatic way to generate Muqarnas, where the user would be able to choose the type, a base shape, and the number of tiers as input and would get all the different muqarnas combinations possible as output.

The work could also be extended to cover Clay-plastered and Shirazi style Muqarnas.

Another interesting possibility would be to calculate the surface area of the Muqarnas based on the method that Al Kashi formulates in his book and compare it to the actual surface calculated on Grasshopper.

References

Literatures

• Al-Kashı-'s Miftah al-Hisab original manuscript: Qatar National Library
• Nuh Aydin, Lakhdar Hammoudi, Ghada Bakbouk, Al-Kashi's Miftah al-Hisab, Volume II: Geometry: Translation and Commentary 1st ed. 2020 Edition, Birkhauser
• Dold‐Samplonius, Yvonne. (2007). Practical Arabic Mathematics: Measuring the Muqarnas by al‐K¯ash¯i. Centaurus. 35. 193 - 242. 10.1111/j.1600-0498.1992.tb00699.x.
• Hamekasi, N. & Samavati, Faramarz & Nasri, Ahmad. (2011). Interactive Modeling of Muqarnas. Proceedings - CAe 2011: International Symposium on Computational Aesthetics in Graphics, Visualization, and Imaging. 129-136. 10.1145/2030441.2030469.

Tutorials

• Rhino and Grasshopper Tutorials
• Rhino and Grasshopper Tutorials/ A Practical Introduction to Muqarnas
• Making parametric Muqarnas using Grasshopper