Cartesian Coordinates

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We live in a three-dimensional space.

But what would a four-dimensional space look like?

It's difficult to visualise, but the simplest description of the fourth dimension is that it is all space reachable by travelling at right angles to three-dimensional space.

So, imagining the fourth dimension isn't easy.

Describing Dimensions

But just as modern technology lets two-dimensional pictures take on a three-dimensional look, computer simulations can help mathematicians begin to visualise a four-dimensional space.

The number of dimensions in a space tells us how many coordinates are needed to specify a point within that space.

In one dimension, only one coordinate is needed.

So a line is one-dimensional because just one coordinate, x, specifies a point on it.

In two dimensions the two coordinates x and y show the horizontal and vertical position of a point.

With three dimensions, the three coordinates, x, y and z, give the horizontal, depth and vertical position of a point.

Mathematicians therefore use four coordinates w, x, y and z to describe a four-dimensional space.

The one-dimensional line had two obvious points at either end.

The two-dimensional square had four, and the three-dimensional cube had eight.

The sequence suggests that a four-dimensional object would have sixteen points where its lines meet called vertices.

The result is a hypercube, called a tesseract, with sixteen vertices, twenty-four faces, and thirty-two edges.

Tesseract

Every one of its edges is straight, and all angles between them are right angles, which is hard to depict with two or three-dimensional imagery.

But the fourth dimension's abstract nature hasn't stopped architects from trying to work with it.

La Grande Arche de la Défense
Paris, France

In Paris, La Grande Arche was built with a resemblance to a tesseract.

Its peculiar structure gives us a glimpse into a four-dimensional world.