# Creating Self Similar Tiling Patterns and Fractals using the Geometric Factors of a Regular Polygon

Stanley Spencer
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture (2014)
Pages 279–284

## Abstract

A regular polygon with n sides can always be decomposed into isosceles triangles chosen from a set of k non-similar isosceles triangles where k is the integer part of (n-1)/2. It appears that the set displays a property I have, in previous papers, called preciousness. This implies that each triangle in the set can be decomposed into assemblies of uniformly scaled triangles chosen from the set. This process can be repeated and forms the basis of recursive tilings and fractals depending upon the details of the process. The value of n has an effect on the symmetry of the design. The relationship between two different regular polygons where the number of sides of one is a factor of the other is also explored. Isaac Newton once famously said that he could see further because he was able to stand on the shoulders of earlier giants. As a theme for decorating the designs I have created pictures of his giants. I have also included pictures of Newton and Einstein as being giants in their own right.