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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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It this an image of a rotating polytope in $\mathbb R^4$?

https://www.facebook.com/hoyle.anderson/videos/10155639583220910/ Can anyone say what this actually is?
Michael Hardy
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Polytopes: proving completeness of set of facets

Let $P$ be a $d$-dimensional convex polytope. $P$ is contained in $[0,1]^d$ and all vertices have only integral coefficients. Given a set of facets of $P$, how to check that this set is maximal. i.e. that it is the set of all facets of $P$? [update]…
stefan
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What is special about the 11-cell and 57-cell?

Reading about the 11-cell and 57-cell I find two facts implied often: They are particularly notable among the abstract regular 4-polytopes. They are related to each other. I'll establish why I think they are notable: Both polytopes are notable…
Sriotchilism O'Zaic
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Descartes-like formula in 4D

In 3D, Descartes' formula states that the total angular defect of a convex polyhedron must always be equal to $4\pi$. In particular, this implies that if we have a bound on the "bluntness" of a polyhedron's vertices, we can also obtain a bound on…
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For the n^th dimension, where n is ≥5, there exists only 3 regular convex polytopes proof

A lot of books stated that is true, however, no proof is given. I was wondering if anyone could help me and show me a proof? Statement: For the n-th dimension, where n is ≥5, there exists only 3 regular convex polytopes Thank you
Jack Wother
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What are higher degree cantellations?

Compare higher degree rectifications. What are higher degree cantellations? I didn't find any definition on internet, at least no operational description. You can find constructions by coordinates for "bicantellated 7-cube" or "tricantellated…
Hao Chen
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Is there a lower bound on the number of facets of a full-dimensional convex polytope

As the title say's: Imagine a full-dimensional convex polytope. Is there a lower bound (or even exact formula) for the minimum number of facets the polytope has?
user1742364
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Is there a notion of dual for a spherical polytope?

I am aware of the notion of polar dual for a flat convex polytope (by a flat convex polytope, I mean the convex hull of finitely many points in $\mathbb{R}^d$). Suppose you have instead a spherical polytope. Is there a notion of duality for a…
Malkoun
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Number of elements of regular 4D polytopes

Is there a simple method for determining the number of vertices, edges, faces, and cells of the 6 regular convex 4D polytopes? For the 3D platonic solids, it can be shown using combinatorial logic that for a polyhedron with Schläfli symbol {p,q} pF…
txyz
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competing definitions for polytope dual

Along with a polytope P one has the notion of its dual which is officially defined via the inner product. However, in three dimensions at least, the dual is often pictured simply by placing a point in each face of P and then taking the convex hull.…
user2052
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How to know if polytope in V-representation is full dimensional?

Let $v_i$, for $i=1,...,V$ be the vertices of a convex polytope in $R^n$. That is, each $v_i$ is a point in $n$-dimensions. How can I determine if the polytope is full dimensional?
a06e
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Given only dimensions for N vectors, is it possible to construct a polytope?

We are given a set of magnitude of vectors. These vectors are free to move and rotate in space. How can we verify that whether this set of given vectors form a polytope?
soumyaghosh
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Definition of a cone in Ziegler's "Lectures on Polytopes"

Ziegler in his lectures defines a cone as follows: a nonempty set of vectors $C \subset \mathbb{R}^d$ that with any finite set of vectors also contains all their linear combinations with nonnegative coefficients. So is this saying that $C$ is just…
Hoji
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Polytopes characterization in $\mathbb R^n$

Given $P = \{x \in\mathbb R^n \mid a_1x_1 + \ldots + a_nx_n = \text{constant}\}$, $(a_1, \ldots , a_n) \ne 0$. Can $P$ be a polytope? I think that with $N = 1$, $P$ is a point. Can a point in $\mathbb R^1$ be a polytope? Thank you all!
Fabio Carello
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Is there 4-polytope with rhombic dodecahedron cells?

I'm not as mathy as I wish I were, so wrt the advice stackexchange just presented me with: to "explain what I don't know," I'm afraid that's a rather long list, stretching deep into the "unknown unknowns" territory where be dragons. But I do know…
Travis Well
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