By Dr Andrew French
Mathematicon 2014 is a book of mathematical imagery based upon a series of art exhibitions in Winchester, UK.
Mathematicon is a series of nineteen images grouped into four themes: Spherium, Harmonograph, Circularis, Julia. Each theme corresponds to a bespoke software tool designed to enable an intuitive exploration of the curves and surfaces associated with a particular set of mathematical relationships.
The concept of this exhibition is to illustrate the possibility of a harmonious synthesis between human and machine, whereby the idea generation capability of the mind is hybridized with the extraordinary power of precision calculation repetition offered by modern computers. The nature of the interface between these two very different worlds is critical for humans to be empowered, rather than constrained, by technology.
Spherium (exhibits 01 to 12) is a three dimensional digital sculpturing tool which incorporates lighting, transparency and texture mapping effects. A surface is created from a mathematical relationship between spherical polar coordinates, that is range, azimuth and elevation. Other variants include Ammonites, where an ellipse with periodic disturbances is extruded around a spiral.
A Harmonograph (exhibit 17) is a Victorian curiosity consisting of two or three coupled pendulums connected to a sheet of paper and a pen. Particularly interesting traces can be obtained by setting the paper in a clockwise circular motion and the pen in an anticlockwise sense. In the digital Harmonograph, the four control parameters (amplitude ratio, phase, rotation frequency and damping) are set dynamically by a set of sliders reminiscent of the graphic equalizers attached to the exterior of pre-iPod stereos. Unlike the mechanical version, the digital Harmonograph generates the pen trace almost immediately. This enables the dependence upon control parameters to be investigated dynamically.
Circularis (exhibit 16) is a computer program for curve stitching, that is the drawing of straight lines between equally spaced points along a set of curves. In exhibit 16, the first epicycloids are drawn (Cardoid, Nephroid, Epicycloid of Cremona).
Julia (exhibits 13,14,15,19) is a mechanism for the generation of surfaces formed from repeated transformations of the Argand diagram, that is a two dimensional space metricated by real and imaginary number lines. These infinitely detailed fractal forms, seemingly biological in nature, are determined from very simple equations. The classic Mandelbrot, plus many variations, are a somewhat psychedelic conclusion to Mathematicon.
Dr Andrew French teaches Mathematics and Physics at Winchester College. He was formerly a 400m runner and a radar engineer.