## 4D models

What does a 4-dimensional shape look like? How can you draw/make one in 3 dimensions? To understand what they look like, it's easiest to consider what happens when you go from a point to a line to a square to a cube and... beyond. Suppose that when you move a point, it traces a line. As Paul Klee said: " A line is a dot that went for a walk!"If you extend a point in the x-direction (giving it 1 dimension), you create a line. You can think of this as having 2 ends (vertices) and 1 edge. If you slide this line in the y-direction (2 dimensions) you create a unit square, with 4 vertices and 4 edges (and 1 face). |

Now if you slide this square in the z-direction you create a unit cube, with 8 vertices, 12 edges (2x4 for the two squares + 4 for the vertex traces) and 6 faces (2 for the two squares and 4 for the edge traces).

Here it is easy to see that if you slide this cube in the 4th dimension, you create a unit 'hypercube' with:

You can keep going! Can you work out how many of everything a 5D cube has? Here is a picture of one I made below (distorted to be demonstrable in 3 dimensions). The yellow lines represent the vertex traces into the 5th dimension.

Here it is easy to see that if you slide this cube in the 4th dimension, you create a unit 'hypercube' with:

- 16 vertices
- 32 edges (2x12 for the two cubes + 8 for the vertex traces)
- 24 faces (2x6 for the two cubes + 12 for the edge traces)
- 8 cubes (2 for the two cubes and 6 for the face traces) - called 'cells'.

You can keep going! Can you work out how many of everything a 5D cube has? Here is a picture of one I made below (distorted to be demonstrable in 3 dimensions). The yellow lines represent the vertex traces into the 5th dimension.

## working out V, E, F, C, etc...

You can work out the number of vertices etc for each model in a different way. For example, consider the cube. It has 2^3 = 8 vertices. Each vertex has 3 edges but each edge is shared by 2 vertices, so the number of edges = 8 x 3 / 2 = 12. Each vertex is the meeting of 3 faces with each face shared by 4 vertices, so the number of faces = 8 x 3 / 4 = 6.

The hypercube has 2^4 = 16 vertices. Each vertex has 4 edges with each edge shared by 2 vertices, so the number of edges = 16 x 4 / 2 = 32 edges. Each vertex is the meeting of 6 faces shared by 4 vertices, so the number of faces = 16 x 6 / 4 = 24 faces. 4 cube cells meet at each vertex, with each cube cell shared by 8 vertices, giving 16 x 4 / 8 = 8 cube cells.

If you do a similar calculation for the 5D cube, you will notice that there is a pattern to the number of each type of cell shared at each vertex - the binomial expansion...

## shadow representations

You can represent higher dimensional shapes in a different way by considering what they look like from a particular view.

To see how, again consider a line (1 dimension). If you look at it along the x-axis (ie along its length) it will look like a point. It looks like it has only one vertex from this direction, but in fact it has 2 (one is hidden).

Consider the square. If you look at it along the y-direction (ie side on), it looks like a line - 1 edge with 2 vertices - but of course you know it has 4 vertices (double those you can see) and 4 edges (double those you can see + one for each vertex trace).

Now if we look at the cube face-on, we see a square. But you know there is another face directly behind it and 4 other faces from the edge traces, giving 6 faces.

So.. consider the hypercube viewed 'cube-on'... You will just see a cube! But you know there is one 'behind it' and 6 others from the face traces, giving 8 cube 'cells'. You could use this to analyse the number of vertices and edges too.

So the 5D cube will look like a hypercube, but with one hidden and 8 more hypercubes from the cube traces giving 10 hypercube cells... I think you get the idea!!

## Other 4-D shapes

There are 6 4-dimensional polytopes (the general name for shapes) of which the hypercube is one. One of them is based on the dodecahedron called the 120-cell, so-called because it comprises of 120 dodecahedron cells, with 4 cells meeting at each vertex in the same way as the hypercube. As there are 20 vertices for each cell, there are 120 x 20 / 4 = 600 vertices!

You can demonstrate it in 3 dimensions using the shadow idea above. Here is one made by my year 10 students!