wendy wrote:All polytopes are honeycombs, and many honeycombs are polytopes.

One of the interesting things in hyperbolic geometry, is that you can have some 'tiling' whose edges are right-angles. An {8,3} made out of the octagons cut from an {8,4}, will, by the existance of {8,3,4}, give right-angles between the faces. It's sort of strange, but these polytopes, viewed at any distance, would be indistinguishable from a "real" polytope. In hyperbolic geometry, it is possible to look through the face of an icosahedron, and see all 19 other faces, in the same way as you see the opposite faces of a pyramid, looking through the base.

Wendy here states that for polytopes, they can be represented as a tiling of some curved shape (for orbiform ones it is usually a hypersphere) and so is a honeycomb; represented as a tiling of euclidean space and so is a honeycomb; and represented as a tiling of hyperbolic space (it can be paracompact, having euclidean honeycomb elements or hypercompact, having hyperbolic honeycomb elements.) and so is a honeycomb.

Now for honeycombs in general, some are not polytopes because not all of them have flat elements (note that edges in hyperbolic space look curved although they can be topologically straight.)

Your analogy provides how a n-sphere can be scaled up infinitely so that its surface looks like a hyperplane. There are theories about our universe being a 4D glome that may be finite (spherical universe) or infinite (flat universe). The funny thing is that if you go far enough in a line on a spherical universe, you will soon arrive at where you started. This is because you are within the surface of the glome and it curves in a great circle (the same thing in a glome that creates swirlprisms).

Related to that, 3D euclidean honeycombs are simply polychora that have 180 degree dichoral angles, meaning that the cells are arranged in a straight layout like the squares in a square tiling. This can be proven because the vertex figures of both 3D euclidean honeycombs and polychora are polyhedra.

Now back to your example, a 2D flatlander has a hard time visualizing a regular spherical tetrahedron. The best it can see is if the tetrahedron's vertices are specifically marked differently from the edges which are made transparent. It can then see four vertices and the edges. If the edges are opaque, the flatlander can see at most two of the vertices. This also applies to a 3D being within a glomeric pentachoron; the best it can see is a triangle outcropping along with the five vertices; and if the faces are opaque it can only see three vertices.

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Ilazhra! (Truth to everyone!)