# Tagged Questions

Questions related to polyhedra and their properties.

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### efficiency of different whole-number-mass-to-a-power in balancing a regular triangle/tetrahedron

I saw this qustion: http://puzzling.stackexchange.com/questions/186/whats-the-fewest-weights-you-need-to-balance-any-weight-from-1-to-40-pounds Suppose you want to create a set of weights so ...
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### Maximum area of triangle inside a convex polygon

Prove that within any convex polygon of area $A$, there exists a triangle with area at least $cA$, where $c=\tfrac{3}{8}$. Are there any better constants $c$? I'm not sure how to approach this ...
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### General algorithm to cap an n-dimensional convex polyhedra

I am looking for a way to cap an $n$-dimensional ($n$ > 3) polyhedra, that is to say: Given an $n$ dimensional set of vertices and faces (including hyperplane equation), and an $n$ dimensional ...
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### permutohedron vs permutahedron

Why are there two spellings for the terms denoting the sets $$\mathrm{Conv}\left(\left\{(\sigma(1),\ldots,\sigma(n))\,\middle|\, \sigma\in S_n\right\}\right)\qquad(n\in\mathbb{N}^+)\,,$$ namely, ...
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### Maximum number of vertices in intersection of triangle with box

Suppose we have a triangle and a box in 3D. The intersection of the triangle with the (solid) box will be a polygon with some number of vertices (possibly zero). The vertex count will vary according ...
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### Integrating Over a Product of (Non-Separable) Piecewise Functions (Hyper-Solid Angle of a Convex Polyhedral Cone)

My problem is as follows: given a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ where $n$ is some integer of order 10 and $f$ is defined by a product of (non-separable) linear piecewise functions, ...
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### Is a snub disphenoid more oblate or prolate?

Figuring out which deltahedra are oblate/prolate (of coarse the platonic solids are spherical) was pretty easy for all of them except for the snub disphenoid... Anyone know what the radia of the ...
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### Drawing a Truncated Octahedron

I'm trying to draw a truncated octahedron in MATLAB. This is also known as a permutahedron so my strategy is to link up all the vertices via adjacent transpositions of permutations in $S_4$. What I ...
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### How many faces does the resulting polyhedron have?

Take a regular tetrahedron of edge one. Also take a square-based pyramid, whose edges are all one (therefore the side faces are equilateral triangles of same size as the faces of the tetrahedron). ...
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### Coloring the pentagonal hexecontahedron

So, I'd like to color the pentagonal hexecontahedron in a way that is satisfying aesthetically and mathematically. For me this equates to, in order of priority - 1. No same-colored faces can share an ...
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### Tetrahedron inside a sphere

What's the largest regular tetrahedron (having side length $x$) you can fit inside a sphere with a unit radius?
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### Convex 4-polytopes requiring 6 or more colors

Projected into 3-D space, a convex 4-polytope looks like a collection of convex polyhedra. If any two convex cells sharing a face have different colors, how many colors are required? In the paper ...
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### Width of a cone

Let $V=\{v_k\}$ be a collection of vectors of $\Bbb{R}^n$, and define their cone to be the set of all their non-negative linear combinations: $$C(V):=\Big\{ \sum_k a_k\,v_k; \; a_k\ge 0 \Big\}\;.$$ ...
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### Polyhedron, understanding face vs facet.

I've the two following definitions, for which I was trying to understand the difference. For a given polyhedron $P$ a face $F$ is both $P$ itself or the intersection of $F$ with $P$. A facet is ...
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### Relationship between circumscribed sphere radius and edge length of a dodecahedron? [duplicate]

Hello and I'm a secondary student doing a math exploration, but I'm currently stuck with this problem... Can anyone kind enough to show me the derivation of the relationship between the circumscribed ...
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### Maximal unit lengths in 3D with $n$ points.

Given $n$ points in 3D space (V), what is the maximal number of unit distance lengths (E) between those points? Here are a few possibilities. Some of them are chromatic spindles. ...
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### Finding a convex decomposition of a point in a polytope

Suppose I'm given the set of vertices, $\{v_i \}$, of a convex polytope. Suppose that I'm also given a point $p$ in terms of its coordinates, and I'm promised that $p \in \mbox{conv} \{v_i \}$. How ...
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### Understanding definition of “dimension” of a subset of $\mathbb{R}^n$

In a book of combinatorial optimization the following definition is stated: A polyhedron in $\mathbb{R}^n$ is a set of type $P = \left\{x \in \mathbb{R}^n \;:\; Ax \leq b \right\}$ for some matrix ...
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### Configuration of five or more mutually equidistant points in space.

How is it proved that there is no configuration of five or more mutually equidistant points in $R^3$? Is it done by induction? I'm stuck. Help would be appreciated. Well, surely equilateral ...
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