Procedural modeling of Muqarnas: Difference between revisions
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=== Muqarnas elements === | === 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. | 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. | ||
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'''4. Shirazi-style muqarnas:''' like curved muqarnas but with a larger variety of elements | '''4. Shirazi-style muqarnas:''' like curved muqarnas but with a larger variety of elements | ||
{| class="wikitable" style="margin: 1em auto 1em auto;" | |||
| [[File:elements.png|thumb| Elements defined by Al Kashi]] | |||
| [[File:curvature.png|thumb| Curved Muqarnas construction]] | |||
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==Methodology== | ==Methodology== | ||
Revision as of 15:04, 21 December 2022
Introduction & Motivation
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: 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
Methodology
Create the shapes in 2D
Transform the shapes in 3D
For a simple Muqarnas
For a curved Muqarnas
Create the 2D plan and the 3D volume
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 |
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✓ |
| By Week 4 |
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✓ |
| By Week 5 |
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✓ |
| By Week 6 |
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✓ |
| By Week 7 |
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✓ |
| By Week 8 |
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✓ |
| By Week 9 |
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✓ |
| By Week 10 |
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✓ |
| By Week 11 |
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✓ |
| By Week 12 |
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✓ |
| By Week 13 |
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✓ |
| By Week 14 |
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✓ |
Results
3D shapes
3D shapes with curve
Simple Muqarnas
Curved Muqarnas
Limitations
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.
Github Repository
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.