Questions related to polyhedra and their properties.

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4
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3answers
16k views

Height of a tetrahedron

How do I calculate: The height of a regular tetrahedron, side length 1. Just to be completely clear, by height I mean if you placed the shape on a table, how high up would the highest point be from ...
6
votes
3answers
2k views

Angle between lines joining tetrahedron center to vertices

What are the angles formed at the center of a tetrahedron if you draw lines to the vertices? I'm trying to make these: I need to know what angles to bend the metal.
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 ...
7
votes
1answer
181 views

How many $n$-colorings up to rotation using exactly 2 of each color are there on a $2n$-polyhedron?

I'm a high-school student and I stumbled across a YouTube video explaining how Rubik's cubes work. A Rubik's cube has 6 colors, one for each side, but I started thinking about ways to $n$-color the ...
9
votes
1answer
176 views

Is the Euler characteristic $\chi =2$ for the prism with a hole?

I keep getting $\chi=2$ for the solid in the picture. It's a prism with a hole joining two opposite sides. I remember reading that $\chi=0$ for such solids. Help me find my error. I'd appreciate if ...
4
votes
1answer
179 views

Maximal volume for given surface area of an $n$-hedron

Is there a term for a polyhedron with $n$ faces (or, similarly, $n$ vertices) that maximises the enclosed volume for a given surface area (equivalently, minimises the surface area for a given volume)? ...
0
votes
2answers
480 views

Calculate Spherical Distance between points

I have googled this and not come up with an answer yet, but basically, I'm trying to find out the distance between each point or vertice on a sphere (all points are evenly spaced). I already know this ...
0
votes
2answers
1k views

What are the vertices of a regular tetrahedron embeded in a sphere of radius R

Imagine you had a sphere of radius R centered at the origin. What are the coordinates of the vertices of the regular tetrahedron which is circumscribed by the sphere? One of the vertices of the ...
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 ...
21
votes
2answers
313 views
+100

What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...
20
votes
3answers
8k views

Why are there 12 pentagons and 20 hexagons on a soccer ball?

Edge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron. Is there some insight into why the alternation of pentagons and hexagons yields an approximated ...
8
votes
4answers
8k views

Calculating the probability of a coin falling on its side

A classical example that's given for probability exercises is coin flipping. Generally it is accepted that there are two possible outcomes which are heads or tails. However, it is possible in the real ...
13
votes
3answers
943 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 ...
11
votes
2answers
316 views

How to prove there are exactly eight convex deltahedra?

A deltahedron is a polyhedron whose faces are equilateral triangles. It is well-known that there are exactly eight convex deltahedra, and it is easy to find out that this was first proved by ...
4
votes
0answers
51 views

Mappings preserving convex polyhedra

It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra. Can you give a characterization of the class of mappings that preserve convex polyhedra?
4
votes
1answer
298 views

Tetrahedron volume

How to calculate volume of tetrahedron given lengths of all it's edges?
7
votes
1answer
453 views

Name of this convex polyhedron?

Does anyone recognize / know the name of the convex polyhedron depicted below as the intersection of a Cuboctahedron and a Rhombicdodecahedron? Please note you have to interpret this picture and ...
4
votes
3answers
228 views

Geodesics on a polyhedron

Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the "equator" of the octahedron eventually will. But for what ...
2
votes
1answer
111 views

What is the minimal isoperimetric ratio of a polyhedron with $5$ vertices?

I'm asking and answering this question to provide a partial answer to this question and a comment on this answer at MO. The isoperimetric ratio $\mu$ of a solid is the ratio $A^3/V^2$, where $A$ is ...
2
votes
1answer
189 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 ...
2
votes
1answer
144 views

The set of distances between three points chosen with uniform probability on a finite interval

I pick three numbers $(x_1, x_2, x_3)$, where the value each $x_i$ is a real number selected with uniform probability on the interval $[J, K]$. I then plot a point in three-dimensions where {$x, y, ...
2
votes
1answer
135 views

Convex polyhedron with five, six, or seven vertices at distinct corners of a cube

What are the names of the convex polyhedron with five, six, or seven vertices, where all vertices lie at distinct corners of a cube? I'm particularly interested in the five vertex case.
1
vote
0answers
50 views

Stereographic projection of the icosahedron and snub cube?

Using a steoreographic projection, the three equations associated with the icosahedron with unit circumradius, inradius, and midradius (respectively) are, $$f=z^{20} - 228z^{15} + 494z^{10} + 228z^5 ...
1
vote
4answers
163 views

Inequality for each $a, b, c, d$ being each area of four faces of a tetrahedron

We know 'triangle inequality'. I'm interested in the generalization of this inequality. Here is my question. Question: How can we represent a necessary and sufficient condition for each positive ...
0
votes
2answers
147 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...