Now let’s start thinking about 4d Platonic solids. We’ve seen the 4-cube… what else is there? Well, in 3d we can take a cube and build an octahedron as shown here. The same trick works in any dimension. In n dimensions, we get something called the n-dimensional cross-polytope, which has one corner at the center of each (n-1)-dimensional ‘face’ of the n-cube.
Puzzle 1. What’s a 2d cross-polytope?
It’s worth noting the relationship between cubes and cross-polytopes is symmetrical. In other words, we can also build an n-cube by putting one corner at the center of each face of the n-dimensional cross-polytope! For example:
But now let’s think about the 4-dimensional case. Since the 4-cube has 8 faces (each being a cube), the 4d cross-polytope must have 8 corners. And since the 4-cube has 16 corners, the 4d cross-polytope must have 16 faces. This is why it’s also called the 16-cell.
It also has other names. Amusingly, the Simple English Wikipedia says:
In four dimensional geometry, a 16-cell is a regular convex polychoron, or polytope existing in four dimensions. It is also known as the hexadecachoron. It is one of the six regular convex polychora first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. Conway calls it an orthoplex for ‘orthant complex’, as well as the entire class of cross-polytopes.
Simple English, eh? 
That would really demoralize me if I were a non-native speaker.
But let’s sidestep the fancy words and think about what the 4d cross-polytope looks like. To draw a cross-polytope in n dimensions, we can draw the n coordinate axes and draw a dot one inch from the origin along each axis in each direction. Then connect each dot to every other one except the opposite one on the same axis. Then erase the coordinate axes.
In 3 dimensions you get this:
It may not look like much, but it’s a perspective picture of the vertices and edges of an octahedron, or 3d cross-polytope.
Puzzle 2. How many line segments going between red dots are in this picture? These are the edges of the 3d cross-polytope.
Puzzle 3. How many triangles with red corners can you see in this picture? These are the triangular faces of the 3d cross-polytope.
Now let’s do the same sort of thing in 4 dimensions! For this we can start with 4 axes in the plane, each at a 45° angle from the next. We can then draw a dot one inch from the origin along each axis in each direction… and connect each dot to each other except the opposite one on the same axis. We get this:
If we then erase the axes, we get this:
This a perspective picture of a 4d cross-polytope!
Puzzle 4. How many line segments going between red dots are in this picture? These are the edges of the 4d cross-polytope.
Puzzle 5. How many triangles with red corners can you see in this picture? These are the triangular 2-dimensional faces of the 4d cross-polytope.
Let’s say that 4d polytope has:
• 0-dimensional vertices,
• 1-dimensional edges,
• 2-dimensional faces, and
• 3-dimensional facets.
In general the facets of an n-dimensional thing are its (n-1)-dimensional parts, while the parts of every dimension below n are often called faces. But in 4d we have enough words to be completely unambiguous, so let’s use the words as above. And in 3d, let’s use face in its traditional sense, to mean a 2d face.
So, as long as I talk only about 3d and 4d geometry, you can be sure that when I say face I mean a 2-dimensional face. When I say facet, I’ll mean a 3-dimensional face.
Puzzle 6. What shape are the facets of the 4d cross-polytope?
On top you see the 4-cube. At right, the 4d cross-polytope. Both are projected down to the plane in the same way.
So, the 4d cross-polytope has
vertices: one centered at each 3d cubical face of the 4-cube. To see how this works, mentally move the cross-polytope up and put it on top of the 4-cube.
On the other hand, the 4d cross-polytope has
faces: one for each corner of the 4-cube.
And this is a general pattern. The n-dimensional cross-polytope has one vertex in the middle of each face of the n-cube, and vice versa. For this reason we say they are Poincaré dual to each other, or simply dual. The n-cube has
vertices and
faces, but for the n-dimensional cross-polytope it’s the other way around.
The picture of the octahedron in cube and cube in octahedron are from Frederick J. Goodman, who has written a book about this stuff called Algebra: Abstract and Concrete.
The other images are on Wikimedia Commons, and all have been released into the public domain except this one:
which was made by Markus Krötzsch.
For more on cross-polytopes, see this:
• Cross-polytope, Wikipedia.
]]>