Can you lend shape to your thoughts?

Marcus Du Sautoy

How good are you at visualising shapes? Here's a little exercise. Imagine picking up a cube with your thumb placed at one corner of the cube and your finger at the furthest corner on the other side of the cube.

You should be able to spin the cube between your finger and thumb.

Now cut the cube exactly down the middle between your finger and thumb. You have two pieces each with a new face. What is the shape of that new face?

I'll give you a little time to play around with the cube in your mind's eye before I tell you the answer. Ever since the very first dice carved in ancient times, people have found many ways to explore the shape of the cube. Its perfect symmetrical form makes it ideal for playing a game of chance. Renaissance artists discovered the power of perspective to capture the three-dimensional cube on the two-dimensional canvas. For a hundred years, OXO have used the shape to bring us flavour to our cooking. The cube became the shape of the manufacturing age thanks to its ability to pack space so efficiently. The architect Frank Lloyd Wright attributes his architectural style to the colourful cube-shaped building blocks that his mother gave him to play with as a child.

Some have found more esoteric ways to play with the cube. In music, composers enjoy playing around with variations on a theme. Shifting the theme up a few notes, playing the theme backwards or even turning the music upside down. The Greek composer Iannis Xenakis realised that these are all very simple symmetries. So why not exploit some more complicated symmetries of an object like the cube? In his piece, Nomos Alpha, the variations of the main theme correspond to the 24 different rotations of the cube.

Xenakis might have tried to make us hear what a cube sounds like but a few people can actually report what cubes taste like. Synesthesia is a neurological condition where cross-wiring in the brain causes senses to be mixed up. For example, a sound can trigger a very particular smell. In one of the rarest forms of synesthesia, a patient's visual and taste worlds get mixed up. Oxo might fill their cubes with the flavour of beef or chicken but what is the taste of an abstract unadulterated cube? A friend of mine who suffers from this cross-wiring described to me in graphic detail how imagining the shape of the cube would always result in the strong taste of mint in his mouth. Of course our language indicates that we are all synesthetic to a certain extent: that a lemon tastes sharp or a wine is smooth is an indication that we are all happy to link the visual and taste worlds.

Mathematicians discovered that seeing, hearing or tasting cubes was not the best way to negotiate the world of shapes. As Descartes warned: "sense perceptions are sense deceptions". Instead, the language of mathematics helps us to explore nature's geometry. Using coordinates, Descartes found a way of translating shapes into numbers. Geometry became a language with a grammar of rules to help you manipulate the shapes you might have difficulty visualising. Translating shapes into numbers opened up a whole new world of cubes in higher dimensions. A square is a shape with corners at coordinates (0,0), (0,1), (1,0) and (1,1). A cube is the three-dimensional version of the square with corners got by putting 0s and 1s in the three coordinate positions. A four-dimensional cube, or hypercube, is the same again but using four coordinate positions. Of course we will never be able to see a hypercube but there are ways to get a glimpse of this mystical four-dimensional shape. At La Defense in Paris there is a huge arch commissioned by Francois Mitterrand. The arch is the shadow of a four-dimensional cube realised in three dimensions. Just as Renaissance artists drew a square inside a square to bring to life a cube on the canvas, architect Johann Otto von Spreckelsen has constructed a cube within a cube to give the architect's perspective on the four-dimensional world.

But if you really want to get your head round the hypercube then you need to talk to a mathematician. It is the language of mathematics which provides the glasses to see in four dimensions. Translating geometry into numbers is the only way to see that a hypercube has 16 corners, 32 edges, 24 square faces and is built from 8 cubes. But I can't tell you what it tastes like.

For those still battling with cutting up a common or garden 3D cube, the shape you get on the face of the cut cube is a hexagon. The cut goes through all the faces (and none of the corners). There are six faces which each contribute a side to the new shape hence the cut produces a hexagonal face. If you got that one, try working out the shape you get when you cut up a hypercube. Who said maths was dull?

Professor Marcus du Sautoy will present 5 Shapes on BBC Radio 4 starting on Tuesday at 9.30

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Marcus Du Sautoy

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