Or put more formally, this is an elementary construction of the convex regular polytopes of every dimension. This is the presentation I gave at MathFest 2010. For the last part you’ll need some red/blue glasses. These are normally used to create 3d effects, but here we’ll be seeing in 4 dimensions.
I plan on making a video out of this that you may enjoy more.
A Platonic Solid is a 3 dimensional object that has the following property: select a face, then an edge on that face, then a vertex on that edge. This selection is called a ‘flag’. All the flags look the same and can be sent to each other by symmetries. This definition is really easily generalizable to work in any dimension: for example, if we’re in 4 dimensions, first you select a 3d cell, then a 2d face, then a 1d edge, then a 0d vertex.
The way to construct these objects is to make them in one dimension lower and try to fit them around a vertex (in the case of 3d, an edge if you’re making 4d things, a face if you’re making 5d things, etc) and ‘fold them up’ into the next dimension. To see if you can fit them, look at their interior angle (that’s the angle listed next to each of the polytopes). Any multiple of them from 3 upwards (you need at least 3 so that when you fold them up they don’t just fold flat) that has total angle less than 360 degrees will work. For example, with the triangle, which has interior angle 60 degrees, you can put 3, 4, or 5 of them around a vertex, making the tetrahedron, octahedron, and icosahedron, respectively.
The last section, with the photos, deals with the Universal Regular Polytopes, the ones that exist in every dimension. The Simplex, on the left, is n+1 (n is the dimension you’re in) points that are equidistant from each other. The Cube, in the center, is the Minkowski Sum of the standard basis vectors-that is, you move one of the vectors along another and look at the shape you trace out. The Cross Polytope, on the right, is the convex hull of the positive and negative standard basis vectors. For the full effect on the 4d versions you should wear red/blue glasses. For the 4d simplex you should see a tetrahedron with another point floating in front of it: all of those points are the same distance from each other. For the cube, the ‘big’ cube is not actually bigger than the ‘small’ cube, it’s just closer. For the cross polytope, the blue lines are in a single 3d hyperplane, while the orange lines traverse the 4th dimension: one point is closer to you, one is further away.
Thomas Eliot 