The true mysteries of mathematics lie at the
limits of our thinking - infinity. Reach beyond what you think is
possible and you start to explore the wonders of maths at the extremes.
by Robert and Ellen Kaplan
When you look down
the tracks you won't see a train, since these days the trains never
seem to be running - but you will see infinity. Follow that point
where the tracks meet - it must be no more than a mile away; but
when you get there, where has it gone? Vanished again.
The Renaissance painters
made sense of pictorial space by organising it around this vanishing
point. That wonderful Florentine, Leon Battista Alberti, described
how he translated three dimensions into two: "I make a veil,
loosely woven of fine thread, dyed whatever colour you please, divided
up by thicker threads into as many parallel squares as you like,
and stretched on a frame. I set this up between the eye and the
object to be represented, so that the visual pyramid passes through
the loose weave of the veil."
gave rise in time to two vanishing points – but can you picture
a space with an endless number of points at infinity – one
for each point on the horizon? It takes outrageous imagination and
an Art of Extremes to do this - and that is exactly what mathematics
is. Just as heard melodies are sweet but those unheard are sweeter
still, so what the eye can be trained to see is marvellous - but
the mind's eye sees wonders incomparably greater.
A painting, a piece
of sculpture, a work of architecture are organised above all by
gracefully balanced masses - and again the infinite in mathematics
lies in the hidden, central room of this art of proportion. Which
rectangle has the most pleasing ratio of length to width, for example?
Experiment for yourself by making a rigid frame with a sliding fourth
side, and test where your friends think best to halt it. Was the
ratio of width to length close to 2 x 3, 3 x 5, 5 x 8 - the standard
sizes of index cards and photos? Or to some other pair of adjacent
numbers in the sequence:
1, 1, 2, 3, 5, 8,
13, 21, 34, 55, ...?
Leonardo of Pisa, nicknamed
Fibonacci, was an Italian mathematician at the start of the thirteenth
century, who showed how to form this elegant sequence of numbers,
which is linked to our understanding of what makes pleasing proportions.
Have you discovered how to make new terms of his sequence? The 'secret'
is to add the previous two. But why should that sequence have anything
to do with pleasing proportions?
The great Piero della
Francesca wrote a book, "On the Divine Proportion", and
in his paintings framed their parts and the whole in rectangles
with these ratios. Leonardo da Vinci saw that tree branches, as
they spiral up the trunk, are spaced in these proportions too. Virtually
all artists work on these principles, whether they realize it or
not. Pine cones and nautilus shells, antlers, the crossing rows
on a sunflower's head - again and again these Fibonacci ratios appear
in nature. But it is only in mathematics, the art of the infinite,
that these ratios are carried beyond the visible to their invisible
extreme: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8 and so on approach a certain
number, called The Golden Mean (roughly 1.618...): which describes
the ideal proportion that the finite ratios in art and nature only
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.
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!
with thanks to Professor Christopher Leaver
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!
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.
Regular Polyhedra or Platonic Solids
1 - Tetrahedron/Pyramid
2 - Cube
3 - Octahedron
4 - Dodecahedron
5 - Icosahedron
"I sent my soul
through the invisible," wrote the eleventh century Persian
mathematician and poet Omar Khayyam. Let's take only one excursion
to illustrate this. That cube we just spoke of had 8 vertices, 12
edges, and 6 faces. What can we say about four-dimensional cubes?
What can we say at all about four dimensions? Astonishingly enough,
we can say a lot. We can't see a four-dimensional cube, but we can
think it, and it has16 vertices, 32 edges, and 24 faces! Shall we
go on? A seven-dimensional cube has 672 faces. A ten-dimensional
cube has 5,120 edges. We could go on...
We could go on,
because the true home of mathematics is at the limit of all our
thinking. Calculus, that wonderful invention of Newton (and independently,
of the German diplomat, philosopher, and mathematician, Gottfried
Leibniz), comes to grips with change as Seurat and the Impressionists
did, by reducing reality to flickering points, and then rebuilding
the world more profoundly.
Artist’s live flamboyant lives,
mathematicians tend to be thought of as reclusive and dry. How could
they possibly be of the same breed? Mathematicians come in all the
usual flavours: some are playful, some arrogant, some devious, and
others direct. John Maynard Keynes pictured Newton as the last of
the Magi, trying to wrest its secrets from the universe alone in
his tower at night. Da Vinci - half artist, half mathematician -
comes down to us as ambiguous as his Mona Lisa's smile.
Like the structures
they probe for, what is essential to mathematicians often lies concealed
- for not only does nature hide itself, as Heraclitus said long
ago, but it is art to hide art. And it is the infinite that lies
hidden in the least grain of sand.
into the maelstrom...
Text Copyright 2003 - Robert
and Ellen Kaplan
and other mysteries are unravelled in the Kaplan’s'
new book, "The Art of the Infinite", published by
Allen Lane/Penguin on August 28th, 2003.
The authors plunge into
these mysteries of maths with their students in The Maths
Circle (begun nine years ago in America and soon to start
up in the UK), where conversation takes the place of lectures
and pleasure replaces fear.)
Available to buy from Amazon.co.uk