Geometry and Design of Equiangular Spirals

TL;DR

This paper explores the geometry and design principles of equiangular spirals, focusing on their self-similarity and potential for digital-inspired design applications.

Contribution

It analyzes the geometric conditions for fitting similar quadrangles and triangles into equiangular spirals, proposing new design possibilities based on these principles.

Findings

Identified conditions for similar polygons within equiangular spirals

Demonstrated potential for digital-inspired spiral designs

Explored geometric properties enabling continuous shape growth

Abstract

In an equiangular spiral, "the whorls continually increase in breadth and do so in a steady and unchanging ratio... It follows that the sectors cut off by successive radii, at equal vectorial angles, are similar to one another in every respect and that the figure may be conceived as growing continuously without ever changing its shape the while" as stated by Sir D'Arcy W. Thompson. The mathematical modeling of them is a very attractive topic of study and research and more specifically, the geometrical conditions under which any quadrangle or triangle can be fitted into similar copies of itself and form an equiangular spiral. This formation gives the impression of a digital form of spiral, where every digit is a triangle or quadrangle following similarity laws, which can allow a multiplicity of design capabilities. The study of these capabilities is presented in the present article and is related with the geometry and the design characteristics of equiangular spirals.