This is probably already known, but I discovered this today and thought it was neat anyway: the total number of surtopes in an n-hypercube is precisely 3n.
PWrong wrote:[...]
00 - vertex
01 - side
02 - vertex
10 - side
11 - face
12 - side
20 - vertex
21 - side
22 - vertex
So there are 4 vertices, 4 sides and 1 face.
quickfur wrote:This is very useful, since I am currently working on a program that deals with polytope representation
bo198214 wrote:quickfur wrote:This is very useful, since I am currently working on a program that deals with polytope representation
Tell more!
PS: "Surtope" is what usually is called facet?
bo198214 wrote:Yes, I know (a bit) about lattices
The mathematicians word for surtope is face. That means every polytope on the surface no matter what dimension (including edges and vertices). Thatswhy the term 'face lattice'. The faces of maximal dimension are called facets all other faces are called proper faces.
bo198214 wrote:An allegory:
I know that the english language is in many cases not optimal (for one example, you mostly can not conclude the pronunciation), Esperanto fixes many of the drawbacks of the english language (and probably you even can conclude meaning from the stems in Esperanto).
Nonetheless nearly nobody speaks Esperanto, and so do I.
wendy wrote:[...]
Lots of this sort of thing goes on. The latest one i heard is, because cell is taken as 3-edge (ie surchoron), what everyone else calls a cell (eg a square on a gaming-board, or any solid tile in a tiling), becomes a new, and equally confusing "cellule" (from <= cellulation),
bo198214 wrote:Mathematics is anyway not the right place to enjoy language.
I personally dont like the word ring. Because its misleading to mean only structures like Z/nZ, i.e. if one sometimes add or multiplies numbers than one returns to smaller numbers.
Hey, the words in mathematics are not designed to make any sense, they are simply place holder for definitions with sometimes some vague relation to usual meanings.
bo198214 wrote:First we can only find a meaning-close word if the concept is part of common life. For example ring, group, field is arbitrary, because these concepts are uncommon to the normal human experience.
But then face and facet is quite close, while surtope is not even a common word. And as non-linguist I can not profit from reduction to stems because with latrix and hedrix, is simply dont know what the stems are, nor what they mean. My first association of surtope was a polytope that surrounds this polytope in some way, i.e. the construction of a new bigger polytope based on the original one. So the reconstruction of meaning does not work for me and then Id rather stick to the common mathematical parlance, if I anyway have to lookup the meaning.
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