طراحی کاربندی: مسالۀ تقسیم‌بندی زمینه از منظر ریاضی

در طراحی کاربندی بر روی یک زمینۀ مشخص، تعیین تعداد اضلاع کاربندی مرحله ای حیاتی محسوب می شود. با توجه به خصوصیات هندسی کاربندی ها، هر کاربندی برای هر زمینه ای قابل استفاده نمی باشد. در پژوهش حاضر، روش طراحی کاربندی در آثار سه تن از اساتید و معلمان معماری سنتی ایران شامل استاد پیرنیا، استاد شعرباف و استاد لرزاده مرور و بررسی می شود. در هیچ یک از این منابع برای پاسخگویی به مسالۀ مورد اشاره، یک روش علمی دقیق مطرح نشده است و به نظر می رسد یافتن پاسخ مناسب برای این مساله بیشتر به آزمون و خطا موکول شده که به یافتن جواب های تقریبی می انجامیده است. تنها استاد پیرنیا از یک فرمول تجربی ساده برای تعیین تعداد اضلاع کاربندی های یک پا سخن گفته که این فرمول نیز محدودیت های خاص خود را داشته است. هر چه جواب‌های به دست آمده برای نحوۀ تقسیم بندی مستطیل زمینه با رواداری بیشتری حاصل آمده باشد؛ شمسه کاربندی دارای اعوجاج بیشتری خواهد بود و به جای دایره به سمت بیضی تمایل پیدا می نماید. پژوهش حاضر این مرحلۀ حساس از طراحی کاربندی را به صورت یک مسالۀ ریاضی صورت بندی می نماید و با نگارش برنامه ای به زبان میپل، ابزار علمی دقیقی برای حل آن در اختیار طراحان قرار می دهد. برای اجرای برنامه، تعیین یک‌پا یا دوپا بودن کاربندی مورد نظر، اندازۀ طول و عرض مستطیل زمینه و نهایتاً مقدار رواداری قابل قبول الزامی است. پس از ورود این اطلاعات، محاسبات انجام می شوند و پاسخ هایی که در محدودۀ رواداری قابل قبول قرار دارند؛ به ترتیب از دقیق ترین پاسخ ها تا کم دقت ترین آنها در خروجی برنامه درج می شوند. دقت بسیار بالا و امکان بررسی تعداد نامحدود گزینه ها در زمان بسیار کوتاه (در نسخۀ فعلی هزار گزینه که البته به سادگی نیز قابل تغییر است) از مزایای برنامه پیشنهادی می باشد. پس از معرفی روش ریاضی و نگارش برنامه مورد نظر، نمونه ای از کاربرد این برنامه جهت طراحی کاربندی یک پا و دوپا بر روی یک زمینه واحد به نمایش گذاشته می شود. در ادامه، فرمول تجربی استاد پیرنیا توسط برنامه مورد نظر به محک آزمایش گذاشته می شود. نتیجۀ آزمون نشان می دهد که این فرمول سنتی در حیطۀ کاربرد محدود و متعارف خود، می توانسته پاسخ های نسبتاً دقیق و قابل قبولی را تولید نماید. در نهایت، بحثی دربارۀ نتایج و کاربردهای پژوهش ارائه می شود.

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 rsquo;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 noninteger 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 onefooted karbandi is required or a twofooted 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 onefooted and twofooted 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.