4
votes
2answers
84 views

Why are polyhedra related to the prime numbers 2, 3 and 5, but not to the prime number 7?

Just take a quick glance at all the numbers in these Wikipedia pages on polyhedra: http://en.wikipedia.org/wiki/Platonic_solid http://en.wikipedia.org/wiki/Archimedean_solid ...
13
votes
3answers
941 views

Making a convex polyhedron with two sheets of paper

Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that ...
2
votes
0answers
157 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
0
votes
1answer
114 views

Problem in understanding a proof there are five Platonic solids.

Thanks to several comments by Gerry Myerson, it is now clear that I wasn't clear, up to a state where I seriously confused myself. In a renewed attempt: Recently, I've been thinking about Platonic ...
12
votes
2answers
277 views

Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
0
votes
0answers
31 views

Finding the min of $L/V^2$ where $L$ be the product of six edges and $V$ be the volume of a tetrahedron.

I'm interested in making and solving new mathematical questions. I made the following question: Let $L$ be the product of the lengths of the six edges of a tetrahedron $K$. Also let $V$ be the volume ...
2
votes
0answers
64 views

About the relation between a tetrahedra and spheres moving in a tetrahedra

I found the following question in a book: There exists a regular triangle $OAB$ which has edge-length $2$. Let $H, I, J$ be a foot of the perpendicular line drawn from a point $P$ in $OAB$ to the ...
4
votes
3answers
144 views

Nested Tetrahedrons

Find 2 tetrahedrons $ABCD$ and $EFGH$ such that $EFGH$ lies completely inside $ABCD$. The sum of edge lengths of $EFGH$ is strictly greater than the sum of edge lengths of $ABCD$. I am completely ...
2
votes
1answer
69 views

The structure of realization spaces of polyhedral graphs

Given a polyhedral graph with $v$ vertices, $e$ edges and $f$ faces, each possible realization of the graph as a geometric (convex) polyhedron corresponds to a point in $\mathbb{R}^{3v}$, ...
6
votes
2answers
103 views

Regular Polyhedrons

In $\mathbb{R}^3$, there are five regular polyhedrons (up to similarity), and can be parametrized by number of vertices, edges and faces. What is the number of regular polyhedrons in $\mathbb{R}^n$, ...
2
votes
1answer
188 views

convex polyhedron edge property

I have a convex polyhedron (with integral nodes). I only calculate in euclidian spaces. Let N be the set of nodes, c the center (arithmetic mean) of the polyhedron. I now want to determine if a line ...
1
vote
0answers
45 views

Orthocentric tetrahedron homothety [duplicate]

Possible Duplicate: Tetrahedron parallel planes feet of altitudes I was wondering if you could help me with the following problem: Let H be the orthocentre of a tetrahedron ABCS incribed ...
0
votes
0answers
128 views

Tetrahedron parallel planes feet of altitudes

Could somebody help me prove that the plane KLM, where K, L, M are feet of altitudes of an orthocentric tetrahedron ABCS inscribed in a sphere is parallel to the base ABC of this pyramid? I would ...
8
votes
1answer
288 views

Chebyshev center = center of mass?

I would like to know for which convex polyhedra $P$ in $\mathbb{R}^3$, is the center of the largest sphere enclosed in $P$ (a.k.a. the Chebyshev center, or the incenter) the same as the center of ...