Extreme Maths: Infinity - The Art of the Infinite

Is this ideal divine or diabolic? The star-shaped pentagram, sign of the Black Arts and trap for Mephistopheles, is made up of segments in just this golden ratio. And is it angels of light or of darkness who stand behind the mathematician in his daring leap into the infinite? One answer would be to look at Escher's self-portrait, surrounded by his mannerist art with its staircases that both ascend and descend, inside-out reflections, space tiled with birds seen one way, fish the other – all planned with the most cunning mathematics.

with thanks to Professor Christopher Leaver Sunflowers

But Escher no more stands for all of modern art than Parmigianino stands for all of classical art. Art and mathematics, are both dependent on balance, and balance in mathematics is stored in what most people find frightening: equations. Yet what is there to be frightened about? These expressions, pivoting on the slender fulcrum of an equals sign, are just different ways of seeing the same thing; equations are the Cubism of mathematics: here is this profile, here is this face seen front on – and they are the same!

Take for example the five Platonic Solids: those polyhedra from pyramid through cube to icosahedron, whose nested totality Kepler saw as emblematic of the universe. You find them everywhere in nature and art: the building-blocks of space. How different they are from one another - yet a most remarkable equation relates them all. Just count the number of corners (vertices, to give them their technical name) on any one of them - call the result V. Now add up the number of edges and call it E, and the number of faces, F. What do you find?

For the pyramid, V = 4, E = 6, F = 4.

And for the cube? V = 8, E = 12, F = 6.

Not much in common here. Find V, E, and F for the octahedron, dodecahedron, and icosahedron as well - a bunch of numbers. Yet in every case, V - E + F = 2!

The Five Regular Polyhedra or Platonic Solids 1 - Tetrahedron/Pyramid 2 - Cube 3 - Octahedron 4 - Dodecahedron 5 - Icosahedron

How wonderful! Mathematics ferrets out what is universally true in the infinite scatter of seemingly haphazard phenomena, and expresses it in eternal equations. This art of the infinite lies behind all of our arts (think of music, whose harmonies are the audible expression of ratios), but it then goes beyond them.