One of the key mathematical ideas discussed in On Growth and Form is the logarithmic spiral (also known as the equiangular spiral). In this type of spiral, the distance between the whorls continually increases. This is due to twisting at a constant rate but growing at a constant acceleration. This happens in nature, for instance, in the Nautilus shell. In such cases the new, growing tissue is added to the end of an old, already hardened, one. The new tissue does not change the shape of the shell, the shape of it remains continually similar during all of its growth. This allows the shell to grow asymmetrically (it grows at one end only), while retaining its unchanging form.
The logarithmic spiral differs from the more commonly seen Archimedes spiral where there is a constant distance between the whorls. The difference can be seen in the diagrams below.
The logarithmic spiral was first discovered by the French scientist René Descartes in 1638. On Growth and Form concentrates on the natural occurrences of this spiral. It is seen in the form of animal horns, teeth and tusks, as well as in the shells of molluscs and spiders’ webs.
In his second edition chapter on the Equiangular Spiral (which Geoffrey Thomas Bennett suggested be renamed as such), D'Arcy discusses gnomons and their relationship to the Spiral. A gnomon is any figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original. A numerical example is the square numbers having the successive odd numbers for their gnomons. Adding 3 and 1 gives the square of 2, adding 5 and 4 gives the square of 3, etc.
D'Arcy describes that in any triangle, one part is always a gnomon to the other. As an example, in an isosceles triangle ABC with one angle 36 degrees and the other two 72 degrees, we can bisect one of the base angles. This leaves us with a subdivided large isosceles with two isosceles triangles as subdivisions, one of which is similar to the whole figure and the other being its gnomon.
A "simple but novel construction" which D'Arcy lifts from his correspondence with Bennett, and includes on page 762 of On Growth and Form, is as follows: Taking the above figure and adding to it in succession a series of gnomons, i.e making it into larger and larger triangles all similar to the first, D'Arcy notes that the apices (plural of apex) of all of these triangles have their locus on an equiangular spiral.
If in this triangle we take corresponding median lines of the successive triangles, by joining C to the midpoint, M, of AB and D to the midpoint, N, of BC then the pole of the spiral is the point of intersection of CM and DN.
Introduction

Overview & D'Arcy's Life

On Growth and Form

Heilmann & Shufeldt

Maths of Transformations

Correspondence

D'Arcy and Mathematics

Coordinate Transformations

Logarithmic Spirals

Forms of Cells

Forms and Mechanical efficiency

Shrinkage

Wartime and D'Arcy

The Leg as a Pendulum

Recreational Maths

Fibonacci Sequence

CellAggregates

Claxton Fidler

Eric Harold Neville

John Marshall

Alfred North Whitehead

Charles Robert Darling

Peter Guthrie Tait

William Peddie

Geoffrey Thomas Bennett

Dorothy Wrinch



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