Questions tagged [polyhedra]
For questions related to polyhedra and their properties.
1,251 questions
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Relationship between number of linearly independent rows/columns and rank of the active constraints matrix in non standard and standard polyhedra?
There is a dissymmetry between non standard and standard polyhedra I don't understand.
In non standard problems, a basic solution is defined by choosing $n$ Linearly Independent constraints (rows) ...
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Fourier-Motzkin Elimination for Single Vertex Computation
Given a convex polytope $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \},ドル I want to know whether we can use Fourier-Motzkin elimination (or an adaptation therefore) to compute one vertex of $P$ (or to ...
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How many regular rhombic polyhedra exist?
Here are my definitions for regular, semi-regular, and irregular polyhedra:
A regular polyhedron is a convex, non-intersecting 3-dimensional shape made with polygon faces connected at edges and ...
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What are the holosnubs of the regular polyhedra?
According to polytope wiki, if P is a regular polyhedron, and H is the holosnub of P, then H will be a uniform polyhedron, if H is not degenerate or a polyhedron compound.
The stella octangula is the ...
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Why aren't there infinite star polytopes?
Surely we can assemble, from some schlafli symbol $\{p/q,n/m\}$ some arbitrary regular polytope? I understand that some of these constructions are infeasible, but surely not all of them are? Could we ...
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Understanding changes in a polyhedron structure when there are perturbations on the restrictions vector
I have the polyhedron
$$ P := \left\{ {\bf x} \in \mathbb{R}^{\binom{n}{k}} : {\bf A} {\bf x} = {\bf b} , \hspace{0.3em} {\bf 0} \leq {\bf x} \leq {\bf 1} \right\} $$
where the matrix ${\bf A} \in \...
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Identifying the dodecahedra in the third stellation of the pentagonal icositetrahedron
According to this blog, the third stellation of the pentagonal icositetrahedron (which is the dual of the snub cube) is a compound of two irregular dodecahedra. They look to me like they could be ...
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Equivalent definitions of Basic Solutions in Linear Programming problems.
I'm reading Introduction to Linear Optimization by D. Bertsimas and J. Tsitsiklis. In Chapter 2, Section 2, the authors provide the following definition for a basic solution:
"Consider a ...
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Wythoffian constructions and polyhedra expansion
I'm trying to understand to Wythoffian constructions. In particular, how to show that when additional mirror is activated, all the previous faces remain (translated and dilated) and are separated by ...
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symmetry of a polytope after mapping one facet
When doing homework on algebra, on the symmetries of regular polygons and regular polyhedra, I observed, that mapping vertices of a single edge in regular polygon to another or mapping vertices of a ...
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Is the "icosahedron with one inverted pentagonal pyramid" unique in this sense?
What are all polyhedra with the following properties?
The polyhedral surface is a subset of $\mathbb{R}^3$ and is homeomorphic to $S^2$; there are no self-intersections. All faces are congruent ...
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Volume of polytope spanned by vertex pairs $(A_i,A_i')$ is invariant under translation of diagonals $A_iA_i'$
2D case
Let a convex quadrilateral $Q= \operatorname{conv}\{A_1,A_1',A_2,A_2'\}$ have vertex pairs
$$(A_1,A_1'),\quad (A_2,A_2'),$$
and define the "diagonal vectors"
$$d_i = \overrightarrow{A_iA_i'} = ...
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Is the dual of any polyhedron a planar graph?
Here all edges in polyhedrons are straight. One edge can only be associated with two faces. For every polyhedron, there exists another polyhedron in which faces and polyhedron vertices occupy ...
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Dehn Invariants of all Platonic, Archimedean, and Catalan Solids. [closed]
I am trying to find the Dehn Invariants of the set of Platonic Solids, Archimedean Solids, and the duals of the Archimedean Solids, the Catalan Solids. We will use 's' to denote the side length of ...
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If all of a polyhedron's faces are concave, how many must it have?
Suppose I have a (non-self-intersecting, nondegenerate) polyhedron in $\mathbb{R}^3$ none of whose faces are convex or meet one another at 180o angles. Such polyhedra turn out to exist, such as this ...