Correspondence about the Logarithmic Spirals


About

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.

Spiral of Archimedes
Spiral of Archimedes
Equiangular Spiral
Equiangular Spiral

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.

isosceles
Isosceles triangle with gnomonic division

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.

isoscelesspiral
Construction with spiral
spiralconstruct
A drawing of Bennett's from ms26140

which includes M and N

Correspondents

William Peddie

Geoffrey Thomas Bennett

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

   

Cell-Aggregates

   

All Correspondence Links

Claxton Fidler

   

Eric Harold Neville

   

John Marshall

   

Alfred North Whitehead

   

Charles Robert Darling

   

Peter Guthrie Tait

   

William Peddie

   

Geoffrey Thomas Bennett

   

Dorothy Wrinch

   


Main Index Biographies Index


The support of The Strathmartine Trust towards this website is gratefully acknowledged    

Alice Gowenlock & Indre Tuminauskaite © June 2018
Copyright information
Cammy Sriram and Edd Smith © June 2019
Creative Commons LicenceExcept where otherwise indicated, the text in this work by Cammy Sriram and Edd Smith is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
School of Mathematics and Statistics
University of St Andrews, Scotland

The URL of this page is:
http://www.mcs.st-and.ac.uk/~dat/