On this page are enclosed my recent papers on geometric patterns and geometry of patterns. Some of them were published in my favorite journal – The Electronic Journal of Mathematics and Technology (various years and volumes). You can find it at https://php.radford.edu/~ejmt/ . Some other were presented as lectures during the ATCM conferences (see https://atcm.mathandtech.org). Each of these papers was written in English and each of them was peer reviewed and SCOPUS indexed.
Papers in the series “Understanding Geometric Pattern and its Geometry”
- PART 1: Understanding Geometric Pattern and its Geometry – In this paper, we explore selected mathematical concepts of art with geometric patterns as its major component. Such art includes carving patterns in stone; various types of geometric mosaic; or assembled wooden structures, known as kundekari art, into a sophisticated decoration of doors, window shutters, or furniture. We discuss here examples of real patterns taken from medieval and modern architecture. We show their constructions and geometry hidden behind these patterns.
- PART 2: Decagonal Diversity – In the first part of papers in this series, we discussed selected geometric concepts related to the class of patterns that were referred to as gereh. The creation of these patterns followed the rules of gereh (axioms). Constructions of such patterns comprised three steps: construction of the contour for the pattern template, construction of tessellation inside the contour, and designing a pattern for each tessellation tile. In this part, we will discuss some concepts of decagonal gerehs, i.e., gerehs with tessellation tiles derived in some way from the geometry of a regular decagon. We will discuss different types of decagonal patterns, as well as tessellations used to create decagonal designs.
- PART 3: Using Technology to Imitate Medieval Craftsmen Designing Techniques – The medieval artists produced incredibly complex geometric art using very basic tools – a ruler, simple compasses, and several templates drawn on a parchment or paper. This was all that they had and all that they needed. They did not have computers, AutoCAD, or printers. With all the modern tools, we still have problems reconstructing the old geometric art correctly, and our easy-to-use tools do not help much. In this paper, we will explore one of the possible ways of creating geometric patterns using simple geometry software and following the old XV century methods.
- PART 4: Geometry from the Mughal’s land – In this paper, we discuss selected geometry concepts used in geometric art from the Mughal Empire. We show here how selected geometric designs from Mughal architecture were constructed and what geometry was employed there. All examples are presented as step-by-step constructions with Geometer’s Sketchpad (GSP), a school geometry software. One can use GeoGebra or Cinderella. Occasionally we will mention the creation of custom tools in geometry software. However, the main emphasis will be placed on geometry and geometric pattern design.
- PART 5: Patterns on tessellations with regular tiles – We discuss selected aspects of geometric patterns created on tessellations with regular and convex polygons only. We demonstrate how one can use such tessellations for designing a variety of geometric patterns. Various approaches are discussed. The whole discussion is based on historical geometric patterns from Muslim communities known as gereh designs.
- PART 6: Using Geometer’s Sketchpad for designing sizeable geometric projects – This document aims to explain a few aspects of dealing with geometric objects in Geometer’s Sketchpad that are not entirely related to geometry. We discuss how one can reduce the amount of data needed to create bulky geometric constructions without losing the quality and accuracy of the final design. We use a real complex geometric pattern to demonstrate our conclusions.
- PART 7: What can go wrong? – In this part, we will discuss why some patterns are considered incorrect and what are the fundamental features of a geometric pattern that make it acceptable?
- PART 8: Designing patterns with alternative tessellations in decagonal geometry – In this paper, we discuss an alternative approach to the construction of decagonal geometric patterns. We show how one can construct geometric structures for a pattern using alternative tessellations. For the sake of simplicity, we deal here with decagonal geometry and one type of pattern only. However, most of our discussion can be continued with some changes for geometric patterns in other geometries (e.g., octagonal or dodecagonal).
- PART 9: On walking pentagons and Isfahani inflation – We introduce Isfahani inflation, a simple inflation technique for decagonal tessellations. We discuss the creation process of geometric patterns using Isfahani inflation. Using examples from Bukhara and Iran, we show how one can reconstruct them with this inflation and produce numerous variants. Some extensions of Isfahani inflation will also be mentioned.