هندسه نظری بسط و گسترش کاربندی (نمونه مورد مطالعه: کاربندی 16)

The expansion of Karbandi is one of the methods of creating various covers in traditional architecture, and the expanded Karbandi has an important part in historical architecture of Iran. However, this type of Karbandi has not been properly considered in the existing typologies, and few, if any, targeted research has already been performed on how Karbandi is developed and expanded, and all that is provided is inadequate attempts within other works. To provide a better understanding of the traditional structures, this research has investigated the theoretical geometry of Karbandi development and expansion, the development of the base to achieve more diversity and the requirements of Karbandi implementation on new bases. The research has adopted a descriptive-analytic approach and logical reasoning to examine geometric propositions. Information collection was done through library research and computer modeling, and Sixteen-sided Karbandi was selected as the study sample to verify the research results. The results showed that the Expanded Karbandi is a compound type of Karbandi produced by the intersection of two or more identical or non-identical mother bases to provide various forms according to the two components of base and connection distance. The Expansion Karbandi base can be shaped by three methods: peripheral geometry, non-elimination geometry and elimination geometry. Therefore, this type of Karbandi can be implemented on all the bases of regular polygons, irregular polygons with axial symmetry, and irregular polygons without axial symmetry in its specific conditions. In peripheral geometries, the connection distance conditions are similar to those of Karbandi on a mother base; however, in elimination and non-elimination geometries, the number of dividing points between two adjacent bases with the greatest distance from each other determines the minimum connection distance in them. On the bases with non-elimination geometry due to the increase in the number of columns and hence the decrease in the distance between adjacent columns, the connection distance limit is reduced compared to simple Karbandi, and therefore this type of Karbandi can be implemented in more situations.
Keywords: Karbandi, Expanded Karbandi, Sixteen-sided Karbandi, traditional structures

Introduction
Due to its chord geometry, Karbandi has more variety and flexibility in covering different areas than other traditional coverings such as domes and arches. This feature has attracted the attention of many researchers, and much research has thus been done on Karbandi’s geometry and typology. Nevertheless, there has been no targeted research on how Karbandi is developed and expanded, and all that has been observed with this theme is few citations within studies. With the aim of providing a deeper understanding of traditional architectural structures, the current research has investigated the theoretical geometry of expansion and deployment of Karbandi (apart from execution issues) and focused on the development of bases to achieve more diversity. In this regard, first, Karbandi geometry and its formation were investigated using library resources and documents, and then, based on the definitions of expanded Karbandi in the literature, Karbandi expansion and development method was discussed. To verify the results, different expansion states for Sixteen-sided Karbandi were investigated. By logical reasoning and collecting information through library research and computer modeling, the research aims to answer the following questions:
• What is the position of expanded Karbandi in typology and how is it formed geometrically?
• How can we develop the base shape and create more diverse bases by expanding Karbandi?
• On the Expansion Karbandi bases, what are the conditions for implementing Karbandi’s geometry?
Conclusion
A literature review showed that Karbandi structures are variously named and that inadequate attention has been paid to Karbandi typology. This study provides a typology of Karbandis based on their formation geometry, discussing the theoretical geometry of their expansion. The results showed that Expansion Karbandi includes numerous Karbandis on regular, symmetric irregular, and asymmetric irregular with specific geometry bases and that its frequency of forms on a particular base can be much more than what is found in traditional architecture. Many Karbandis in traditional architecture are under this category. Todays, by the increased use of software in designing problems and the diversity of materials in covering applications, it is possible to execute various forms derived from Karbandi geometry. Therefore, recognizing the geometry of expansion and development of Karbandi can prepare the ground for the creation of new forms in the discussion of updating Karbandi.
Two main problems in the discussion of expanding Karbandi are the way the geometry of the base is formed and the requirements of the connection distance on the new base. The final shape of the base is caused by the intersection of the bases, their being identical or non-identical, and the drawing methods used (peripheral, non-eliminative and eliminative geometries), and thus various shapes of bases are attainable. Besides, a base with non-eliminating geometry is more abundant in traditional architecture. All kinds of irregular and asymmetric shapes can be formed through elimination geometry, which is less regarded and less implemented in practice in traditional architecture. The connection distances in an Eexpansion Karbandi are determined by the base geometry. On the bases with non-eliminative geometry, due to the significant expansion of the coverage level of Karbandi and the increase in the number of bases and hence in the distance between the neighboring bases, the connection distance limit is decreased, compared to simple Karbandi, and therefore the Karbandi can be implemented in more cases. In peripheral geometry, the connection distance limit is similar to the mother base.