Questions tagged [polyhedra]
For questions related to polyhedra and their properties.
1,252 questions
0 votes
0 answers
31 views
Quantitative bounds for Lipschitz neighborhood retractions of polyhedra in $\mathbb{R}^d$?
Let $P \subset \mathbb{R}^d$ be a (compact) polyhedron, i.e., the geometric realization of a finite simplicial complex linearly embedded in $\mathbb{R}^d$. In the case of interest, $P$ is a subcomplex ...
- 3,396
0 votes
0 answers
18 views
Fourier-Motzkin Elimination for Single Vertex Computation
Given a convex polytope $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$, I want to know whether we can use Fourier-Motzkin elimination (or an adaptation therefore) to compute one vertex of $P$ (or to ...
- 1
0 votes
0 answers
34 views
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 ...
- 115
3 votes
2 answers
96 views
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 ...
- 33
0 votes
1 answer
51 views
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 ...
2 votes
0 answers
35 views
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 \...
- 33
1 vote
1 answer
54 views
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 ...
- 11
1 vote
1 answer
45 views
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 ...
1 vote
1 answer
78 views
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 ...
- 322
1 vote
1 answer
43 views
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 ...
- 150
0 votes
2 answers
126 views
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 ...
- 1,532
2 votes
0 answers
63 views
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'} = ...
- 695
0 votes
0 answers
69 views
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 ...
- 1,757
3 votes
0 answers
42 views
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 ...
- 113
24 votes
3 answers
681 views
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 ...
- 19.1k