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Questions tagged [polytopes]

In elementary geometry, a polytope is a geometric object with flat sides, which may exist in any general number of dimensions $n$ as an $n$-dimensional polytope or $n$-polytope.

In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions $n$ as an $n$-dimensional polytope or $n$-polytope. Reference: Wikipedia.

For example a two-dimensional polygon is a $2$-polytope and a three-dimensional polyhedron is a $3$-polytope.

An important category of polytopes is the category of regular polytopes. These are the polytopes whose symmetry group acts transitively on its vertices, edges, faces, etc. In $2$ dimensions, these are the regular polygons, in $3$ dimensions, these are the Platonic solids and the Kepler-Poinsot polyhedra, in $4$ dimensions, these are one of six convex figures, or one of ten non-convex ones, and in higher dimensions, these include only analogs of tetrahedra, cubes, and octahedra.

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Sections of an abstract polytope

An abstract polytope is a certain kind of partially-ordered set. Its elements or "faces" are ranked by "dimension" and also partially ordered via a pairwise "incidence" relation between elements of adjacent ranks. For some abstract polytope $P$, of…
Guy Inchbald
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Is the Intersection of an Integral and half-integral polytopes half-integral?

Given an integral polytope $\{x \in \mathbb{R}^n | A_1x \leq b , x\geq 0_n \}$ where the extreme points are integral, and another half-integral polytope $\{x \in \mathbb{R}^n | A_2x \leq b , x\geq 0_n\}$, what are some techniques or theorems to show…
AspiringMat
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describing a Simplicial complex of a regular polytope through the action of the automorphism group

This is from page 41 of "Abstract regular polytopes" Suppose that we have a group $\Gamma = \langle\sigma_1,\cdots,\sigma_n\rangle$ generated by involutions and denote $\Gamma_J = \langle\sigma_j\ \vert\ j\notin J\rangle$ for…
user405156
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Simplicial polytope in $\mathbb{R}^n$ with $n+2$ vertices

I am interested in simplicial polytopes of dimension $n$ with exactly $n+2$ vertices. Is there a nice characterization of those? For $n=2$ there is of course only the quadrilateral but what about in arbitrary dimensions? Are the $f$-vectors known?
Hans
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Finding a vector normal to a face of a polytope

Suppose I have an $n-1$ dimensional facet of an $n$ dimensional polytope, where the facet is expressed by a set of points. For a given $n-2$ face of the facet, how can I find a vector that is both: normal to the face as well as being in the same…
John
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Number of Half-Unit Pentachora in a Unit Hexacosichoron

As the title summarizes, I am unable to find out how many 5-cells {3,3,3} (pentachora) with a circumscribed diameter (d) of 1/2 can fit into a 600-cell {3,3,5} (hexacosichoron) with d=1, {3,3,3,6}. The method is analogous to fitting 6 triangles {3}…
Primeity
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Efficiently checking if a polytope has more than $N$ lattice points in its relative interior.

Let $P$ be a polytope in $\Bbb Q^d$. I am curious if there is an efficient way to check if $P$ has more than one integral point in its relative interior. More generally, is there an efficient way to check if $P$ has more than $N$ integral points in…
Brian Fitzpatrick
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Vertices coordinates for the regular 4-polytopes in 4d

Hi there I'm looking for the coordinates for each point of the 4 regular polytopes in 4d (we can use side=1 for simplicity). I have found them for the tesseract. But I don't know how to obtain the rest.
Jestern Alberto Novello
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Name for regular polytope vertex embedding

What I mean by that is using some vertices of a regular polytope to construct another, such as a tetrahedron in a hexahedron. I've thought about this topic, but I do not know its name, if it has any.
inakilbss
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Is there a specific name for polytopes that only have 0s and 1s as coordinates?

I only know of the name 0/1-polytope, but whenever I search for it on Google, I get articles written by the same author. So I think there is another name for it.
Julian
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About the union $P\cup Q$ of two polytopes $P, Q$

How can the union $P\cup Q$ of two polytopes $P, Q$ also be a polytope?
X. Gao
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Extending homeomorphism on the boundary to the interior

Let $A$ be a square and $B$ be a triangle on the plane. They are homeomorphic. Given $\phi:\partial A\rightarrow \partial B$ homeomorphism, does there exist a map $\phi':A\rightarrow B$ homeomorphism such that restriction of $\phi'$ on $\partial A$…
k99731
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