Every day thousands of travellers take the Eurostar to a strange and foreign land. No, not Paris; the Fourth Dimension. Although many visitors to Paris don't realise it, at the heart of the city is a portal to hyperspace. As you emerge from the Paris subway into the financial district at La Défense you are greeted by a huge four-dimensional cube.

Well, it's not quite a four-dimensional cube. The architect of the Grande Arche, Johann Otto von Spreckelsen, was restricted to building in the

three-dimensional world in which we live. What he has done instead is to create a shadow of a 4-D cube. The Danish architect used a trick developed by Renaissance artists to depict a three-dimensional cube on a 2-D canvas. Artists would draw a square inside a larger square and, by joining up the corners, they would create the illusion of seeing three dimensions. Von Spreckelsen built a smaller cube inside a larger cube and joined up the corners of the two cubes to create this hypercube with its 16 corners.

He wasn't the first artist to experiment with the mysteries of the hypercube. Salvador Dali used a different method to unravel the fourth dimension in our 3-D world. If you want to build a 3-D cube, you can start in two dimensions by cutting out a net consisting of six squares joined in a cross shape and fold up the shape to make a cube.

In Dali's painting, Crucifixion (Corpus Hypercubus), he depicts Christ being crucified on what a 4-D cube would look like if it was unfolded into our 3-D world. It consists of eight cubes, four stacked in a tower and four arranged around the faces of the third cube in the stack. It looks like two intersecting crosses. For Dali the sense of a dimension beyond our physical world resonated with the spiritual world that he was trying to capture.

But if you want to see in four dimensions, it isn't the world of the architect or artist that will give you that vision but the world of mathematics. Descartes' invention of Cartesian geometry creates a dictionary that turns geometry into numbers. And it's using these numbers that ultimately allows the mathematician a passport to hyperspace.

We are all familiar with how this dictionary works. Every position on the surface of the planet can be turned into a set of two numbers. One number indicates the location's longitude, the other its latitude. So, for example, the co-ordinates (48.892778 N, 2.235833 E) give the precise location of the Arche at La Défense. I can use this dictionary of numbers to describe a 2-D square by numbers. It is a shape whose corners are located at the positions: (0,0), (1,0), (0,1) and (1,1). If I want to describe a 3-D cube then I need to add an extra co-ordinate to keep track of the height off the ground. So the eight corners of the 3-D cube are located at (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1) and (1,1,1).

So what's a 4-D cube? Since we have only three physical dimensions, the geometric side of Descartes' dictionary runs out. I can't physically construct a 4-D cube. But the numbers on the other side of the dictionary don't run out. To move into the fourth dimension I simply add another co-ordinate to keep track of how far I'm moving in this new direction. So the 16 corners of a 4-D cube? They start at (0,0,0,0), which is connected to (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1) and then we move all the way through to the extremal point at (1,1,1,1).

Using the pair of 4-D spectacles provided by Descartes' language of co-ordinates, mathematicians can explore the geometry of these shapes in hyperspace. But if you want a glimpse of this world, a Eurostar ticket will start you on your journey.

But a little warning. Whenever I've visited the Arche there seems to be a howling wind that sucks through the centre of the arch, even on a calm day. By building a hypercube at the heart of Paris it feels sometimes as though the architect has opened a gate into the fourth dimension. You begin to wonder whether it really is the suburbs of Paris that you see through the centre of the arch. So approach the hypercube with caution. You never know where you might end up when you pass through the Arche.

Marcus du Sautoy takes Alan Davies into the fourth dimension in Horizon, BBC Two, Tuesday, March 31, 9pm