# A 4 dimensional cube in Paris

## Introduction

## Description

To celebrate the 200th anniversary of the French Revolution, the president of France, François Mitterrand, commissioned the Danish architect Johann Otto von Spreckelsen to build something special in La Défense, the financial district of Paris. The building would line up with several other significant Paris buildings – the Louvre, the Arc de Triomphe and the Egyptian needle – in what has become know as the Mitterrand perspective.

The architect certainly didn’t disappoint. He built a huge arch, called La Grande Arche, which is so large that the towers of Notre Dame could pass through the middle, and weighs a staggering 300,000 tonnes. Unfortunately von Spreckelson died two years before the arch was completed. It has become an iconic building in Paris, but perhaps less well known to the Parisians who see it every day is that what von Spreckelsen actually built is a four-dimensional cube in the heart of their capital.

Well, it isn’t quite a four-dimensional cube, because we live in a three-dimensional universe. But just as the Renaissance artists were faced with the challenge of painting three-dimensional shapes on a flat two-dimensional canvas, so the architect at La Défense has captured a shadow of the four-dimensional cube in our three-dimensional universe. To create the illusion of seeing a three-dimensional cube while looking at a two-dimensional canvas, an artist might draw a square inside a larger square and then join the corners of the squares to complete the picture of the cube. Of course it’s not really a cube, but it presents the viewer with enough information: we can see all the edges, and visualize a cube. Von Spreckelsen used the same idea to build a projection of a four-dimensional cube in three-dimensional Paris, consisting of a small cube sitting inside a larger cube with edges joining the vertices of the smaller and larger cubes.

Question: A Square has 4 corners. A 3D cube has 8 corners. How many corners do you think a 4D cube has?

Answer: 16. You can see them all in the shadow of the 4D cube in Paris.

Question: How many edges does a 4D cube have?

From each corner there are 4 edges coming out, one in each dimension. So that gives us 16x4 edges – or does it? No, because we’ve counted each edge twice: once as an edge emerging from the vertex at one of its ends, and again as an edge emerging from the point at its other end. So the total number of edges in the four-dimensional cube is 16x4/2=32. Can you count all these edges on the shadow in Paris?

Cubes don't stop in 4 dimensions. You can move into five, six or even more dimensions and build hypercubes in all these worlds. For example, a hypercube in N dimensions will have 2^{N} vertices. From each of these vertices there will be N edges emerging, each of which I am counting twice, so the N-dimensional cube has Nx2^{N}^{-1} edges.

You can find out more about the mathematics of hyperspace in my book The Number Mysteries.

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