Questions related to polyhedra and their properties.

learn more… | top users | synonyms

2
votes
1answer
28 views

polytope with 12 vertices and 48 edges

It seems like you can construct a polytope with 12 vertices, where each vertex connects to all the other vertices except 3. So there must be a totalt of 48 edges. (each of the 12 vertcies connects to ...
2
votes
0answers
45 views

What honeycomb has the highest volume to edge length ratio?

This question is analagous to the Kelvin Problem where the solution, the Weaire-Phelan Structure, has the highest volume to surface area ratio; however, the cell volume is compared to edge length ...
3
votes
1answer
34 views

What does an Euler characteristic of a topological space greater than 2 topologically mean?

Recently I've found a polyhedron with Euler characteristic $\chi=9$. This is made from the Octahemioctahedron with adding the intersections of the hexagon faces as vertices. It has $V=13, E=36, ...
4
votes
2answers
139 views

Why is Octahemioctahedron topologically a torus?

I'm afraid, that I have a very bad space vision, because I don't see, that Octahemioctahedron is topologically a torus. Could somebody explain it for me, why is it? Scene 2. @aes: Finally, ...
3
votes
1answer
104 views

Largest of the smallest angles of incidence from arbitrary point to tetrahedron vertex/centroid line

Picture a regular tetrahedron where each vertex has a line through the centroid and a plane normal to it. I need to show that the range of the smallest angles of incidence from an arbitrary point to ...
1
vote
0answers
75 views

The skeleton of Eulerian polyhedra

There is (at least) two kind of validity domain of Euler's $v−e+f=2$ polyhedron formula. One is the "Eulerian" polyhedra, i.e. simply connected polyhedra with simply connected faces (see here). The ...
1
vote
0answers
35 views

How to find out the circumscribed radius of a snub dodecahedron?

A snub dodecahedron is produced by partially rotating all 12 regular pentagonal faces of a small rhombocosidodecahedron. A snub dodecahedron has 80 congruent equilateral triangular faces, 12 congruent ...
1
vote
1answer
40 views

Is there any simply connected polyhedron with a not simply connected face?

According to Wikipedia, For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2. Is it really necessary to specify here, that ...
3
votes
1answer
65 views

How does the existence of Platonic graphs imply the existence of Platonic solids?

I will use the following definitions Platonic graph: A 3-connected planar graph with faces bounded by the same number of edges and vertices having the same number of incident edges. (remark: ...
1
vote
1answer
51 views

computing Lefschetz number

We have a fixed point theorem which says that : Let $X$ be a compact polyhedron, $f:X\rightarrow X$ be a continuous map. If $L(f)\neq 0$ then $f$ must have a fixed point. (Lefschetz number is ...
5
votes
1answer
64 views

Is the adjective 'regular' necessary in the definition of Platonic solids?

The definition I mean can be found in the tag Wiki of Platonic solid tag and also in Wikipedia: Definition 1: A Platonic solid is a regular, convex polyhedron with congruent faces of ...
0
votes
0answers
50 views

Convex polyhedra with edges of equal length

This is again a natural category of polyhedra without having an own name. Is it possible, that their graphs are the same as the graphs of polyhedra with faces of regular polygons? My question is ...
4
votes
2answers
79 views

Dihedral angles of a pentakis dodecahedron

I'm new to the world of mathematical descriptions of polyhedra, and I'm wondering if, for a Pentakis Dodecahedron, the dihedral angles are uniform at each vertex. The visualization of the P.D. on the ...
2
votes
0answers
29 views

What is the name of convex polyhedra with congruent faces of regular polygons?

The definition of platonic solids is the following (see Wikipedia): In Euclidean geometry, a Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same ...
0
votes
2answers
28 views

Is there a term for an “unbounded simplex”?

Is there a general term for regions like $\{(x,y):x>y\}$ and $\{(x,y,z): x>y>z\}$, i.e., regions which are simplexes with one open?
0
votes
1answer
18 views

Conway polyhedra notation calculator?

I recently read about Conway polyhedra notation, and I want to experiment with it. Are there any programs that take the notation, and output a representation of the shape?
4
votes
1answer
79 views

3D Dodecahedron model: Construction question.

The image below is a 2D construction that, when cut-out and folded appropriately (hopefully it is intuitively clear how to cut and fold), forms a 3D dodecahedron. It works great: I've successfully ...
2
votes
2answers
31 views

possible polyhedra from euler's formula

I'm not very clear with the euler's formula, and I couldn't find it anywhere. I'm sorry if it is a double post. F + V - E = 2 Is the euler's formula. If the equation balances, is it polyhedra all ...
1
vote
1answer
38 views

surface area of 'cylinder' with the top cut at an angle

I don't know what the name for this shape is, so in essence it is a cylinder, radius at base $r$, which has had a wedge of the top cut off at an angle so that rather than a circle the upper face is an ...
2
votes
0answers
45 views

Analytic-geometrical properties of dodecahedron

Consider the following projection of a dodecahedron: An equilateral triangle can be projected to make points $A, B, C, D, E, F$ intersect with it's edges. What would be the mathematical proof (if ...
0
votes
1answer
26 views

Something after after duals?

So if I have a polyhedron, x. Is it possible for a shape to be inscribed inside x (y). And inside y is z, but inside z is x. Is there a shape that could be x? If so what is it? Also, x, y, and z are ...
1
vote
1answer
16 views

Operations on polyhedra

Is there an operation on polyhedra that add $1$ vertex and $1$ facet (thus, due to Euler's formula, add two edges)? Here, a polyhedron is the convex hull of a finite number of non coplanar points in ...
4
votes
0answers
25 views

Computer realisation of automorphism group of polyhedron

I have a polyhedron (I am especially interested in the case of Platonic solids) and the graph corresponding to its skeleton. I also have some data associated with this graph (e.g. different ...
0
votes
0answers
26 views

Why is the component sum of a polymatroid analogous to size of a set and what is its relation to independence of a matroid?

I was learning about sub modularity and matroids in the following microsoft tutorial and they introduce the concept of polymatroid in parallel to matroids. In that talk they also introduce the concept ...
1
vote
1answer
18 views

Motions that Leave a Prism Invariant

Identify the group of all motions that leave a right prism invariant. It seems like the normal dihedral group, $D_n$, with respect to the base would be a subgroup of this group. That is to say that ...
3
votes
0answers
32 views

3D models of the unfoldings of the hypercube?

There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the tesseract, into 3D.1 These unfoldings (or "nets") are analogous to the 11 unfoldings of the 3D cube into the plane.2 ...
1
vote
1answer
32 views

Labelling the Vertices of a Polyhedron

It is possible to label the vertices of a cube with strings of 3 binary digits in such a way that two vertices are adjacent if and only if the correspondent strings differ one from the other in ...
0
votes
0answers
31 views

Ball-of-wacks combinations

The six-color version of the ball-of-wacks consists of thirty rhomboidal pieces, which can be combined to form a rhombic triacontahedron. There are six colors, each with five pieces. One challenge ...
1
vote
0answers
21 views

Which convex polytopes do not have stellations?

Some polytopes, like the cube and the tetrahedron, do not have stellations (see http://mathworld.wolfram.com/Stellation.html). Has anyone characterized all convex polytopes (generally or in small ...
1
vote
1answer
56 views

“Octahedron” made from two pyramids of different heights.

I wonder how to name such shape: It's commonly used by e.g. 3ds max to visualize the bone in animation system. It consist of two pyramids with the exact same square base. It would be a ...
0
votes
0answers
30 views

Computing a lower bound for the minimal componentwise distance of vertices of polyhedra

Let $A$ be a matrix in $\mathbb{R}^{m \times n}$ and let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a polytope. I want to compute a lower bound on the minimal componentwise distance of two ...
6
votes
2answers
603 views

Why can't a perpetual motion polyhedron exist?

I've been thinking about polyhedrons, when placed on a table on a certain face, will tip over and keep tipping over infinitely. I'm trying to prove mathematically that such a polyhedron doesn't exist. ...
0
votes
0answers
25 views

Distance from polyhedron to bounding ellipsoid

How can I find the distance from the edges of the polyhedron to the boundary of the ellipsoid? The ellipsoid is parameterized by: $$E = [ x : (x - x_c)^T H (x - x_c) \le m^2 ] $$ And it covers the ...
4
votes
1answer
43 views

Icosahedron and inscribed cube

We can inscribe a cube in dodecahedron (see this), where $12$ faces of dodecahedron give the $12$ edges of the cube. Can we inscribe cube in icosahedron?
1
vote
0answers
30 views

How to count the number of integer points in a polyhedral?

Let $k, l$ be positive integers and $k_1, l_1$ non-negative integers. I have a polyhedral give by the following inequalities: $$ A = \{(r_1,r_3,r_4,r_5): r_1+3r_4\leq l_1, -k_1 \leq r_5-r_3 \leq 0, ...
0
votes
0answers
8 views

How to write a polyhedra formula explicitly?

Let $m$ be a positive integer and $$ A_m = \{r=(r_1,r_2,r_3,r_4) \in \mathbb{Z}_{+}^4: r_4 \leq r_2, 2r_1+3r_2+3r_3 \leq m \}. $$ Let $$ch_m = \sum_{r \in A} ch((m-r_1-3r_2-3r_3)\omega_1 + (r_2 + r_3 ...
1
vote
0answers
176 views

Symmetries of Archimedean Solids

There are five platonic solids, and 13/15 (which is correct?) Archimedean Solids. The finite groups of isometries of Euclidean $3$-space are the finite subgroups of $SO(3,\mathbb{R})$ or ...
0
votes
1answer
23 views

Equality of polyhedra

Is a minimal representation for a polyhedron unique? And if so can we use this to prove that two polyhedra are equal (or maybe the same is a better definition).
0
votes
1answer
50 views

Is there any regular polyhedron that is not of Euler characteristic 2

Is there any regular polyhedron that 1. consist of congruent regular polygons as its faces 2. each vertex has same number of adjacent edges but nonetheless not of characteristic 2? (say, torus or ...
3
votes
2answers
116 views

Is Dodecahedron tesselation somehow possible?

In this video (at 3:25) there is an animation of planets inside a dodecahedron matrix (or any data-structure that best fit this 3d mosaic). I tried reproducing it with 12 sided dices, or in Blender, ...
1
vote
1answer
47 views

Proving the upper bound of edges in a convex polyhedron

The question is the following: Suppose Every face of a convex polyhedron has at least $5$ vertices and every vertex has degree $3$. Prove that if the number of vertices is $n$, then the number of ...
2
votes
1answer
149 views

Five Cubes in Dodecahedron

I will demonstrate why the group of rotational symmetries of a Dodecahedron is $A_5$. For that, we have to find five nice objects, on which the group of symmetries acts. One object is "Cubes" ...
1
vote
1answer
66 views

Is there a name for an irregular polyhedron of fourteen sides?

I was given a popup calendar that is cardboard and has 14 sides. The top and bottom are equilateral hexagons and each side of the top hexagon is attached to a trapezoid that widens away from the top ...
0
votes
0answers
71 views

The maximum pentahedron in a sphere

Suppose that there exists a pentahedron in a sphere whose radius is $1$ and that each vertex of the pentahedron can exist on the surface of the sphere. Question : What is the max of the volume of ...
1
vote
5answers
317 views

Relation between edgelengths in a tetrahedron with two right angles and three equal edges

I have got a problem I can't solve myself. I had an attempt, but it's wrong. I was told to draw a grid of this tetrahedron and then it's easier to find a solution (I tried it, but I don't see ...
2
votes
1answer
57 views

On the Formula: $V-E+F=2$

I am looking back at the history of the formula "$V-E+F=2$" for any polyhedron in Euclidean 3-space ($V=$number of vertices; $E=$ number of edges, $F=$ number of faces). Cauchy gave following proof of ...
1
vote
1answer
24 views

P - bounded polyhedron, L - linear map. Show that L(P) is a bounded polyhedron

Let $P = \{x\in \mathbb{R}^n \ | \ Ax\leq b\}$ be a bounded polyhedron. Let $L:\mathbb{R}^n \rightarrow \mathbb{R}^k$ be a linear map. Show that $L(P):=\{L(x)\ | \ x\in P\}$ is a bounded polyhedron. ...
5
votes
0answers
77 views

From Icosahedron to Pentagonal hexecontahedron (Floret Tessellation)

Inspired by this post: Floret Tessellation of a Sphere I tried to transform myself an icosahedron into its simplest Floret tessellation. But I am having trouble when applying the 'method' given in the ...
0
votes
0answers
14 views

Which functions preserve convexity of the cells of a polyhedral tessellation of $\mathbb{R}^n$?

Let $\mathcal{X}_1, \ldots, \mathcal{X}_m \subseteq \mathbb{R}^n$ be disjoint sets with $\bigcup_{i=1}^m \mathcal{X}_i = \mathbb{R}^n$. Furthermore, let each $\mathcal{X}_i$, $i = 1, \ldots, m$, be an ...
1
vote
5answers
153 views

Deriving Euler's Polyhedron Equation for middle school/high school students

I've been perusing the 20 proofs, but they all seem to involve graph theory and complexities... How would you derive this simple formula for high school or middle school students? Specifically: ...