Geometry and Design of Equiangular Spirals

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.