4-dimensional Hypercubes

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The picture above is a drawing of a 4-dimensional hypercube. In general, we can define an n-dimensional hypercube for n a non-negative integer. The first 4 of these are

0: A point
1: A line segment
2: A square
3: A cube

We construct higher dimensional hypercubes from lower dimensional ones. We start with a 0-dimensional hypercube: a point.

Start with a point A and draw a line to a different point B. Let’s say the distance between A and B is d. This is a 1-dimensional hypercube: a line.

Now draw a line of length d, orthogonal to AB, from B to a point C, and a line of length d, orthogonal to AB, from A to a point D. This is a 2-dimensional hypercube: a square ABCD.

Now draw lines of length d respectively from each corner of the square ABCD, orthogonal to both AB and BC. Call the terminal points E,F,G,H. This is a 3-dimensional hypercube: a cube ABCDEFGH.

And so on for higher dimensions.

Drawing these on a piece of paper, a 2-dimensional plane, becomes more difficult as n increases. For example, a picture of a 7-dimensional hypercube looks like

How do we draw these pictures? In the 4-dimensional case, we can draw a hypercube from these steps. To get a better idea of the hypercube, we can watch the hypercube rotate

or rotate it ourselves.

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