Introduction to polytopes

For more advanced details, see Polytope.

We can try to define a polytope by saying "its elements must be polytopes of lower dimension." This means that 2-dimensional polygons, as well as 3-dimensional polyhedra, are polytopes. This in turn tells us that line segments (formally referred to as "dyads") and points must be polytopes too.

(The "elements" you'll need to worry about are the vertex (which is a point), the edge (which is a dyad), and the face (which is a polygon).)

The lowest-dimensional polytope, the one at the base of the definition of all others, is the –1-dimensional nullitope. Visualizing it is not important, but it serves as a placeholder in advanced operations.

For why we need the nullitope, see Introduction to abstract polytopes.

The Platonic solids

Interest in polytopes often begins with the Platonic solids, shapes commonly found in dice. They are special because for each of them, each element is the same as each other element of that type. This property is known as regularity.

For instance, each of the square faces of the cube connects to four other identical faces and four other identical vertices, each edge lies between two identical faces and two identical vertices, and each vertex connects to three identical faces and three identical edges.

On a side note, there are other possible regular polyhedra, but they are not convex - that is, they are spiky or they have holes, and wrapping a hypothetical rubber sheet around them would give you a different shape than them. (This is not true of the Platonic solids.) They may be made of "star" polygons instead of convex polygons, but each one still has only one type of vertex, one type of edge, and one type of face

These are the small stellated dodecahedron, great stellated dodecahedron, great icosahedron, and great dodecahedron.

Even more regular polyhedra are possible, but they all have infinite amounts of faces. The mucube shown here is a relatively simple example: each vertex has six square faces, each edge is between two identical vertices, and each face is surrounded by a saddle-like arrangement of other faces. Even if it goes on forever, it still qualifies as regular.

Other infinite regular polyhedra are possible, that further stretch the bounds of what is possible by having infinite polygons as faces. All the polygons in the polyhedron are the same, and all the vertices in the polygon are the same, so why can't they be regular too? This video is a good place to learn about those.

The Archimedean solids

The Platonic solids can be modified in many ways. When these modifications produce a convex polyhedron that has identical vertices (like the restriction of regularity, but only applying to the vertices - it is formally called vertex-transitivity) and a single edge length, the result is called an Archimedean solid. The modifications (often called operations) that produce Archimedeans typically involve cutting off vertices or edges, or pulling faces away from one another.

Some Archimedean solids

• 

  Cutting off the vertices of a cube gives you a truncated cube.

• 

  Cutting off the vertices of a cube all the way to the middle of the edge gives you a cuboctahedron. You can also get it by doing the same thing to the octahedron - it's like the cuboctahedron is halfway in between them.

• 

  Truncated dodecahedron

• 

  Pulling the faces of a dodecahedron apart and then inserting new faces in between them gives you a rhombicosidodecahedron (or "small rhombicosidodecahedron"). The same thing can be done to an icosahedron to get it. One could also cut off the edges of either of those to get it.

The Archimedean solids are a small subset of the uniform polyhedra, and there are 13 of them in total.

Some things that are NOT Archimedean solids

• 

  Even though prisms (and antiprisms) meet all the requirements for being Archimedeans, they aren't counted as such.

• 

  The chamfered cube looks a lot like the Archimedean truncated octahedron, but upon closer inspection it has two different kinds of vertex, disqualifying it. Its hexagonal faces are also not regular.

• 

  Even though the ditruncated cube just truncates the truncated cube again, the result is clearly not Archimedean.

The Johnson solids

Pieces of Platonic and Archimedean solids can be cut-and-pasted to form Johnson solids, although some are "elementary" and are not related to any other shape. There are 92 of these in total.

Some Johnson solids

• 

  The square pyramid is half an octahedron.

• 

  The triangular cupola is half a cuboctahedron.

• 

  The pentagonal gyrobicupola is two cupolas from the rhombicosidodecahedron attached together. (Note that there are two possible orientations - triangle-to-square, or triangle-to-triangle which would be a pentagonal orthobicupola.)

• 

  The augmented dodecahedron is a dodecahedron with a pentagonal pyramid attached to it.

• 

  The gyrate rhombicosidodecahedron has one of its cupolas rotated. (Look at the top - some triangle faces are connected to pentagons instead of the usual squares.)

• 

  The disphenocingulum has no strong relations with any other polytope, although the belt of triangles around the middle looks a bit like a hexagonal antiprism.

Higher dimensions

For more information, see Introduction to higher dimensions.

What if we were to make a 4-dimensional polytope out of 3-dimensional elements, just like the 3-dimensional polytopes (polyhedra) are made of 2-dimensional elements?

The 3-dimensional element is called a "cell," and in polychora, two cells join together at a shared face just like two faces join at an edge in a polyhedron.

In this view of the tesseract (which is regular, just like the Platonic solids), we see eight cubes (one in the middle, six around the sides, and one encompassing the whole thing.) Three cubic cells meet at each edge, and four cells meet at each vertex.

All of the previously discussed elaborations and modifications are possible in 4 dimensions and above, as well as some that weren't discussed, and ones that aren't even possible in dimensions as low as 3. This is the scope of the study of polytopes.