After one learns that in three space only five regular polyhedra exist, one might feel that there is not much else in the subject of polytopes to think about. Well it is both interesting and frustrating that this is not the case.
One of the amazing aspects of moving into higher dimensions is that you can i) ask more questions about your objects and ii) do more things to your objects. For example, as cubes move into successive dimensions they are accompanied by a new "perspective" or angle at which to examine them. This "new perspective" always occurs at the vertex of the cube and it presents phenomenon not encountered before (granting that the other perspectives will still be "new", but their phenomenon will have a greater correlation with a phenomenon already seen in the dimension below). Take a look at slicing cubes starting from a square (from edge to vertex), then cube (from face to edge to vertex), then 4 cube (from solid to "face" to edge to vertex) (pp. 43-4 9). The "new" phenomenon each time occurs w/ slices starting at the vertices.
To begin another discussion, a collection of dice may helpful when thinking about polyhedra (and just fun to have in general). The observation is that the way the number of faces of the polyhedra relates to their number of vertices is counte r-intuitive: The cube (less faces) has more vertices than the octahedron, and the dodecahedron (less faces) has more vertices than the octahedron (i.e. would expect polyhedra w/ more faces to have more vertices?). More necessitates less. The observatio n of this phenomenon came from the fact that the cube is dual to the octahedron and that the dodecahedron is dual to the icosohedron. I assume that the phenomenon stems from the nature of the figures that the polyhedra are comprised of: squares have mor e sides (edges) than triangles, pentagons have more sides than triangles. Thus polyhedra created from squares/pentagons will have more vertices (have to connect more lines) than those created from triangles. How does this phenomenon of more necessitates less relate in higher dimensions (the fourth in particular)? It is a bit unclear and the chapter does not present all the relevant data to this discussion.
An important difference in the fourth dimension is that a duel figure is not created by connecting adjacent "centers of faces (polygons)" like in three dimensions but instead by connecting adjacent "centers of solids (polyhedra)" (pg. 98). A djacency is determined by sharing a similar vertex. Example: a hypercube has 16 vertices, each vertex being shared by four equally spaced cubes. Find center of cubes around a vertex and connect them into a regular three dimensional tetrahedron. Thus th e dual to the hypercube will be composed of 16 tetrahedron. How many vertices does this 16-cell have? Eight obviously, one for each "face" (i.e. cube) of the hypercube. So the phenomenon that the number of vertices of a polytope with more "faces" than another polytope will have less vertices.
By moving our idea of duality in three space to duality in four space we come upon an interesting case of perspective. The idea of "analogous" has two different meanings: a 3 cube is the analogue of the 2 cube because it is the "square drag ged parallel to itself" as presented in Flatland. By "analogy", the 4 cube is the analogue of the 3 cube because it is the three cube dragged parallel to itself. The assumption is that there is a progression of cubes (or simplexes for that matter) throu gh the dimensions and that they are all analogous to each other. This is correct. The ambiguity in the term "analogous" arrives when you try and say that the same parts of the cubes (i.e. edges, vertices) are analogous to each other in each dimen sion.
Examine how the respective cubes relate to each other in different dimensions: in 2D, a line bounds the square, while in 3D a face does the bounding, in 4D a solid, etc.. What then is the analogue of a side of a square? a square is bounded by 1D obj ects connected at a 0D object (the vertex). In three dimensions, a cube is bounded by 2D objects connected at a 1D object. In four dimensions, a 4 cube is bounded by 3D objects connected at a 2D object, etc.. Thus the "face" of the cubes (what a n-dime nsional being would see) goes (from 2D->4D) from a line to a plane to a solid. Thus the "vertex" of the cube (what the bounding objects are connected with) goes (from 2D->4D) from a point to a line segment to a square. (This pattern is the reason why Wi lliam Stringham (pg.94) probably had a difficult time convincing everyone on his ideas for constructing 4D polytopes even though his ideas were technically correct: they just weren't "analogous" Euclid's construction of 3D polyhedra).
So, when we talk about "duality" in three dimensions we are talking about a phenomenon, though created in similar ways, that has a different "value" than duality does in four dimensions. In three dimensions we connect centers of faces; in four we con nect centers of solids. The analogy would be in two dimensions to connect centers of sides (edges).
Back to the discussion presented first, how does the relationship between vertices and faces in three space relate to other relationships (to edges in three space, solids in four space)? This is to be put off to a later date, except for a little disc ussion:
Another interesting idea to think about is the similarity in how an icosohedron is built and a 600 cell is built. In three space, because I can see it, I can understand easily the two ways of building an icosohedron: i) align five equilateral triang les around a point and fold into 3 space to form a pentagonal pyramid; or ii) w/ a regular pentagon, pick a point in space and connect it to the corners of the pentagon so that the distance from the corner of the pentagon to the point in space equals the side length of the pentagon. I can see the pentagon that the triangles fold into to and can thus picture in my mind different qualities of the object. For a 600 cell though, matters are not as clear. The two methods for construction of 600 cell are: i ) align 5 tetrahedron around an edge and connect the faces into four space creating a ???; or ii) w/ an icosohedron, pick a point in four space and connect it to the vertices of the icosohedron so that the distance from the vertices to the point in space equals the edge length of the icosohedron. By method i we know that around any edge of the 600 cube there will be 5 tetrahedron. By method ii we know that around any vertex there will be 20 tetrahedron. How/why/what/where does this happen? At this po int it is difficult to intuit how these 4D shapes work and fit together.Michael Matthews Prof. Banchoff's comments