Questions related to polyhedra and their properties.

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1answer
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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: ...
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0answers
20 views

Specific polygons in \R^{3} [on hold]

Find all (convex) polyhedra P in $\mathbb{R}^{3}$ with following property: for every two vertices $v, u \in P$: $[u,v] \in \partial P$ . Here $[u, v] = \{w \; \vert \; w = tv + (1-t)u, \; t \in ...
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1answer
44 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
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1answer
55 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 ...
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3answers
177 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 ...
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0answers
48 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 ...
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2answers
61 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 ...
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0answers
22 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 ...
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2answers
24 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?
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1answer
16 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?
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4answers
235 views

Is the rhombic dodecahedron the only isohedral polyhedron that tiles 3-space (other than the cube)?

Is the rhombic dodecahedron the only face-transitive (or isohedral, i.e. all faces are the same) polyhedron that seamlessly tiles 3-dimensional Euclidean space (other than the cube)? I'm looking ...
2
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1answer
93 views

Minkowski sum of two polytopes via the halfspace representation

If i have two polytopes denoted by $P_1, P_2 \subset \mathbb{R}^d$, suppose their halfspace representations are respectively $H_1x \leq K_1$ and $H_2x \leq K_2$. Now, considering their Minkowski sum, ...
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2answers
24 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 ...
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1answer
19 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 ...
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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 ...
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1answer
14 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 ...
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0answers
20 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 ...
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1answer
173 views

Biggest Little Polyhedron

The Biggest Little Polygon problem asks for the polygon with greatest area where the largest diameter is 1. Let's add a dimension and find the largest volumes. What is the biggest little polyhedron ...
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0answers
23 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 ...
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4answers
22k 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 ...
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1answer
17 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 ...
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0answers
25 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 ...
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1answer
497 views

Space-filling polyhedra (or honeycomb) survey?

Is there a survey anywhere of space-filling polyhedra? MathWorld's article, space-filling polyhedron, mentions about 400 being seen in pre-1981 books and papers. Wikipedia mentions 28 convex uniform ...
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1answer
29 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 ...
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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 ...
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0answers
20 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 ...
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1answer
42 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 ...
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2answers
33 views

Prove that a convex $d$-polytope has at least $d+1$ facets

This seems trivial but I can't come up with a formal proof. I think there should be a way to do this inductively but I can't figure out how$\ldots$ Any help much appreciated
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1answer
421 views

Find the Polyhedron Enclosed by Multiple Faces

I have two faces ($S_1$ and $S_2$), each bounded by a series of Vertex $V$. These two faces may or may not intersect. In addition to that there are two vertical faces $S_3$ and $S_4$, that connect ...
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2answers
365 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 ...
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0answers
28 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 ...
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2answers
201 views

Are there formulae to determine close-packing polyhedra?

Is there a formula to determine which polyhedra will tessellate in 3D without any spaces?
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2answers
632 views

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 ...
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2answers
599 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
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0answers
24 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 ...
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0answers
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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, ...
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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 ...
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3answers
104 views

Does a polyhedron with 7 hexagons and 20 pentagons exist?

A beautiful polyhedron with 20 hexagons and 60 pentagons can be seen here: http://robertlovespi.wordpress.com/2013/11/03/a-polyhedron-with-80-faces/ . Euler formula and the corresponding Diophantine ...
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0answers
173 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 ...
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1answer
20 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).
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6answers
133 views

there is any relation between $\pi$, $\sqrt{2}$ or a generic polygon?

I'm a programmer, I'm always looking for new formulas and new way of computing things, to satisfy my curiosity I would like to know if there are any formulas, or I should say equalities, that make use ...
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2answers
105 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, ...
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1answer
45 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 ...
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5answers
314 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 ...
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1answer
46 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 ...
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1answer
133 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" ...
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1answer
60 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 ...
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0answers
69 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 ...
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1answer
55 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 ...
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1answer
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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. ...