The purpose of this paper is to develop the Islamic geometric patterns from planar coordinates to three or higher dimensions through their repeat units. We use historical plane methods, polygons in contact (PIC) and point-joined, in our deductive approaches. The mentioned approach makes use of a novel method of tessellation that generates 3D Islamic patterns called “interior polyhedral stellations”. The outputs showed that both the PIC and point-joined methods have strengths and weaknesses. Point-joined stellations are more efficient for regular repeat units and PIC is suitable for complex designs. These two methods can produce a large range of patterns and can be employed simultaneously. This study effectively answers the question regarding the gap between planar design from Muslim achievements and contemporary demands in modern art and architecture. We also propose techniques for constructing aperiodic three-dimension Islamic geometric patterns tessellation and two-point family.

中文翻译：

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更高维度的伊斯兰几何图案

本文的目的是通过其重复单元将伊斯兰几何图案从平面坐标发展到三个或更高维度。我们在演绎方法中使用历史平面方法、接触多边形 (PIC) 和点连接。所提到的方法利用了一种新颖的镶嵌方法，该方法可以生成称为“内部多面体星座”的 3D 伊斯兰图案。结果表明 PIC 和点连接方法都有优点和缺点。点连接星号对于规则重复单元更有效，PIC 适用于复杂设计。这两种方法可以产生大范围的图案并且可以同时使用。这项研究有效地回答了穆斯林成就的平面设计与现代艺术和建筑的当代需求之间的差距的问题。我们还提出了构建非周期性三维伊斯兰几何图案镶嵌和两点族的技术。