Design of Persian Karbandi: The Problem of Dividing the Base from a Mathematical Viewpoint

Karbandi is the structure of a kind of roofing in Persian architecture. One of the main issues related to the design of karbandi is that, due to its geometrical structure, it is not possible to design any desired karbandi on a given base. Therefore, it is necessary for the designer to be able to discern the proper karbandi for a given base. The most critical stage in designing a karbandi is when the designer should recognize the number of sides of a proper karbandi for the given base. Therefore, the question that this paper is trying to answer is that from a mathematical point of view how an architect can find out the proper segmentation of the base to fit the geometrical structure of a karbandi. In this paper first a literature review of traditional methods for designing karbandis is conducted in which the methods of three master architects are examined. They include master Pirnia, Sha’rbaf, and Lorzadeh. The problem with these traditional methods is that no clear explanation is given for the way the designer can reach a proper division of the base. Only Pirnia speaks of a simple empirical formula that was used by traditional architects. This formula has its specific limitations too. It is so arranged that in order to produce a meaningful answer, the numbers pertaining to the length and width of the base rectangle have to be integer numbers. However, it is highly probable that a designer wants to design a karbandi on a rectangular base with non-integer dimensions. Besides its limited range of function, it is not mathematically clear how much accurate its answers are. Pirnia admits that this formula was not aimed to be quite accurate and most of the times the designers had to modify the original dimensions of the base in order to find an acceptable one. The less accurate the segmentation of the base rectangle, the more deviated will be the shamseh piece of the karbandi from a true circle towards an ellipse. Therefore, in the absence of a precise mathematical solution for this problem, it was probable that most of the times the designers had to content themselves with proximate solutions out of trial and error process or had to alter the original dimensions of the base in order to be able to build up a karbandi on it. In this paper this problem is formulated as a mathematical question and is solved. Then a program in Maple language is written to do the iterative calculations and print the relevant answers. Designers can easily run the program to find possible karbandis for any desired base. When the program is executed in Maple application, it prompts the user to determine whether a one-footed karbandi is required or a two-footed one. Then the length and width of the base rectangle and finally, the acceptable tolerance are asked. The acceptable tolerance is defined based on the ratio of the length to width of the rectangle. After entering these variants, the calculations are done and the answers which are within the domain of acceptable tolerance are printed orderly from the most exact answers to the least. In other words, the first answer belongs to the rectangle which is most similar to the original base. The output of the program includes the method for the segmentation of the base, the radius of the cutting circle, the proportion of the sides of the suggested rectangle, and the relevant tolerance for every printed answer. The advantages of this program include the high level of accuracy and the possibility to check an infinite number of options in a very short time (in the current version of the program the variants are set in order that the program checks 1000 options to produce its outputs but this number can be adjusted by the user). After introducing the mathematical solution to the problem and writing the relevant program to do the calculations, the paper continues to include one example of its application to design one-footed and two-footed karbandis on a given base to show its capability and convenience. After that, the program is used to test the credibility of the empirical formula mentioned by master Pirnia. Nine different bases that are mentioned by Pirnia to show the application of the formula are used to compare the results. Surprisingly, it is observed that the Pirnia's simple empirical formula in five cases out of nine cases produces the most exact answers produced by the program. Regarding cases in which the answer by the formula is not the most exact answer according to the program, it can be said that this difference reflects practical, structural, or aesthetical concerns of traditional architects. Therefore, the test shows that the formula used by Persian architects, though limited in its scope, was really a working formula that could be used as a useful basic guidance for designers of karbandis. This finding might be related to the historical fact that there were expert mathematicians who were interested in solving practical problems of professions like architecture and who devised simple mathematical solutions to be used by ordinary practitioners.