Assessing the Influence of Geometric Principles on the Architectural Form of Traditional Bathhouses in Kurdistan Province: A Case Study of the Iranian Golden Rectangle and Arch Covering Systems

Abstract
Among the teachings of Pirnia concerning traditional Iranian architecture, there are clear references to the “Iranian golden rectangle” and “arched covering,” and their use by traditional architects. Many elders have stated that Iranian architects have employed these proportions to create architectural spaces in buildings. This research aims to analyze the effect of geometry on the massing and architectural form of the bathhouses of Kurdistan Province. The present research questions are as the following: 1. What has been the influence of geometry on the use of the “Iranian golden rectangle” in the bathhouses of Kurdistan Province? 2. What types of arched coverings were used in the various spaces of the studied bathhouses? Information was collected by means of library research, field survey, and drawing. Through the selection of 14 bathhouses in Kurdistan Province and the analysis of their architectural plans and main components, namely the bineh (dressing room) and hot house (caldarium), the hypothesis that this hidden geometric framework was applied in the plans of most bathhouses is confirmed. The designers of these bathhouses have sought to bring the proportions of the bathhouse spaces closer to the golden rectangle. Based on the findings of this research, it was revealed that geometry is an influencing factor on the architecture of bathhouses in Kurdistan Province. The plan of some of the studied bathhouses follows the Iranian golden rectangle or its derivative proportions. According to the coefficient of variation, the conclusion of the authors is that looking at the actual conditions and the plan pattern of the bathhouses in Kurdistan Province, the small amount of error in the number of the bathhouses is due to the separation and division of the land, and the designers therefore tried to bring the ratio of the important spaces of the bathhouse closer to the relative value of the “Iranian Golden Rectangle”. The covering of the main spaces, including the bineh and the hot house, often employed tavizeh vaults, while the infill between the tavizehs was of the colombo and karbandi type (rasmi-bandi and yazdi-bandi). Irrespective of any climatic conditions, the covering of the hot house is less spacious than the covering of the bineh, due to the preservation of heat within the hot house.
Keywords: Iranian Architecture, Iranian Golden Rectangle, Arch Covering, Bathroom, Kurdistan Province.
 
Introduction
In the study of Iranian architecture, scholarly discourse occasionally references the “Iranian golden rectangle,” a subject of considerable intrinsic interest. The late Pirnia, a leading authority on traditional Iranian architecture, explicitly addressed this concept on numerous occasions. In his well-known teachings on traditional building design, he remarked: “They made use of the Iranian golden ratio when designing architectural spaces. This golden ratio was derived from a rectangle inside a hexagon”. Correspondingly, the basins within the bathhouse were executed in Iranian hexagonal, dodecagonal, and rectangular shapes that exhibit golden proportions, matching the aesthetic canons of the golden rectangle. In many Iranian buildings, the plan and vertical section are organized within a modular framework of equilateral squares and triangles; their intersections define key fixed points—such as width, length, height, openings, and the position of inscriptions, among other features. Accordingly, the size of each part was proportionally related to every other part. Thus, a building was not a random collection of disconnected elements, but a harmonious ensemble of parts bound by proportional relationships that imparted movement to space and a sense of ordered repose to the eye. Iranian architects favored the hexagon because it is a form that cannot be drawn incorrectly, as its construction is based on the equilateral triangle. Assembling six such triangles yield a hexagonal shape; the honeycomb, a natural exemplar, is structured in an analogous manner, highlighting geometry’s integral role. Given the centrality of public bathhouses in Islamic urban life, the investigation of roofing systems in the bathhouses of Kurdistan Province—a significant architectural type of the Islamic period—is essential for understanding the characteristics of public buildings of that era in this region. The present research aims to examine the Iranian golden rectangle and the influence of geometry on the formation of the arch coverings in these bathhouses. In sum, the article commences with a theoretical framework divided into four sections; it subsequently describes the research method and process, and the approach to data selection is examined. The findings are then presented and discussed in the form of tables and graphs, and the paper ends with a concluding summary.
 
Iranian Golden Rectangle
In studies of Iranian architecture, the Iranian golden rectangle has been cited many times, and workers such as Pirnia have clearly and explicitly noted this concept. The Iranian golden rectangle is derived by inscribing a rectangle within a hexagon. In his description of the Iranian golden rectangle, Pirnia has mentioned the use of a rectangle enclosed within a regular hexagon yielding the proportions of 1.73, and the rectangle half of it with the proportions of 1.15, in the design of buildings; they also examined rectangles with the proportions 1.15, 1.73, and 2.31. Therefore, in this article, the ratio of these proportions in the historic bathhouses of Kurdistan Province has been investigated for the bineh space and for the hot house space. In the examined spaces, the amount of deviation is less than one tenth (0.1) in order to comply with the golden proportions in the bineh and the hot house. In the studied samples, the observance of the golden proportion was not applied in the bathhouses of Haj Saleh, Shihrah, Moshir, and Khalifa Fattah in any of the spaces (bineh and hot house), while in the rest of the bathhouses this proportion was used. It occurs in bineh, hot house, or both spaces. The conclusion of the authors is that, according to the actual conditions and the plan pattern of the bathhouses in Kurdistan Province, the small amount of error in some of the bathhouses is due to the separation and division of the land; despite this, the designers tried to bring the ratio of the important spaces, namely the bineh and hot house, of the bathhouse closer to the relative value of the “Iranian Golden Rectangle”.
 
Discussion
Important factors influencing the selection of the covering type include the size of the intended space, its spatial proportions, and the shape of the plan. In general, in the bathhouses of Kurdistan, the covering of the main spaces, the bineh and the hot house, is based on load-bearing ribs (tavizeh), while the infill between the ribs is typically of the colombo and karbandi type. The tavizeh ribs have functioned to bear and transfer the load from the roof to the columns and joists. These elements together provide a suitable foundation for the dome covering. In the studied bathhouses of Kurdistan Province, most of the arches employed are of the four-centered and pointed types; only in the Shisheh and Dokhazineh bathhouses are the arches of the simple curved type. The bineh covering in the bathhouses of Khan, Qaslan, Abdul Khaliq, Vakil, Dokhzineh, Shisheh, and Haj Saleh is higher than that of the hot house. To better transfer the load from the main opening to the adjacent openings, arches and vaults were used to cover the bathroom spaces.
 
Conclusion
From an examination of the diagrams obtained, the authors’ conclusion is that the designers of these bathhouses were concerned with creating spaces of precise dimensions grounded in geometry when determining the ratio of length to width of the bineh and the hot house. In the abundance chart related to the bathhouses, the highest abundance can be observed in the ratio they termed the “Iranian golden ratio” (such as √3=1.732). Based on the first assumption, with reference to Table 2 and the diagrams of the investigated spaces, the amount of difference is less than one tenth (0.1) in order to comply with the golden ratio, and the amount of difference greater than one tenth (0.1) is deemed non-compliance with the golden ratio of bineh and hot house. Based on the second hypothesis and with reference to Table 3, the predominant arch covering type employed in the bathhouses of Kurdistan Province, in the main and large spaces, namely the bineh and the hot house, was the colombo roof type, and it was surrounded by an arch and a fountain.