Questions related to polyhedra and their properties.

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6
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2answers
588 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. ...
2
votes
1answer
408 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 ...
0
votes
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 ...
1
vote
0answers
27 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
7 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
3answers
99 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 ...
1
vote
0answers
144 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
16 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
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6answers
117 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 ...
5
votes
1answer
472 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 ...
3
votes
2answers
88 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, ...
0
votes
1answer
32 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 ...
1
vote
5answers
309 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 ...
1
vote
1answer
41 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
66 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
48 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
55 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 ...
2
votes
1answer
48 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
21 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. ...
3
votes
4answers
226 views

What shape is this?

im doing a question that involves a shape with 8 faces, 10 vertices and 16 edges. Can anyone enlighten me as to what this shape is called? Many Thanks
3
votes
3answers
159 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 ...
5
votes
0answers
43 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
8 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
108 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: ...
8
votes
4answers
137 views

How many $A_5$ are there inside $A_6$?

I am reading a paper that says: There are $12$ versions of $A_5$ in $A_6$: $1) $ the permutations that leave one thing unmoved. $2)$ the permutations of the six pairs of antipodal ...
7
votes
1answer
243 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)? ...
1
vote
1answer
21 views

What is a good simple definition or characterization of 'polyhedral space'?

I would like to give a definition of polyhedral space in $\mathbb{R}^n$ that is easy to understand by people that has some Maths knowledge but are neither expert in Calculus nor any other Mathematics ...
3
votes
1answer
83 views

Show that Icosahedral group does not have normal subgroup of order 5

Here is my work so far: Let $G$ denote the group of orientation-preserving isometries of Icosahedron. I have shown that $G$ acts transitively on $G/G_s$ where $G_s$ is the isotropy group, namely ...
1
vote
4answers
58 views

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). ...
8
votes
2answers
606 views

Is the Euler characteristc defined wrong? If not, why not?

Ever since learning that $$\chi(S_0\# S_1) = \chi(S_0)+\chi(S_1)-2$$ (where $\chi$ denote the Euler characteristic), I've wondered whether $\chi$ isn't "defined wrong." If we let $\chi' = 2-\chi,$ ...
0
votes
1answer
30 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
1
vote
0answers
40 views

Determining the set of a linear transformation of elements in a polyhedron

I have a set defined by linear inequalities of the form: $X = \{x : Ax \le b\}$. For any $x \in X$, I write $y = Gx$ where $G$ is a matrix (the dimension of $y$ is less than the dimension of $x$). ...
1
vote
2answers
75 views

Why does induction procedure of Euler characteristic fail for non-convex polyhedra? What am I missing?

Euler characteristic of convex polyhedra is always $V-E+F=2$. Induction procedure reduces edges and vertices until we are down to one vertex whose $V-E+F=2$ and hence you are done. The same ...
10
votes
1answer
137 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
0
votes
1answer
26 views

Proof: Minkowski sum polytope implies A and B polytopes

Suppose $A$ and $B$ are convex sets and their Minkowski sum $A+B$ is a polytope. How do you prove that $A$ and $B$ are polytopes as well?
0
votes
0answers
17 views

Maximization over linear surjective mapping of polyhedron

I am reading this paper and confused about the derivation of equation (11) (page 3, bottom of column 2). I will rephrase it in this question. Let $\mathcal{P}_r = \{ x \in \mathbb{R}^n : P_r x \leq ...
3
votes
1answer
26 views

Have face-transitive polyhedra been completely classified?

I've been researching polyhedral dice, and everywhere I look it states that face-transitive (and therefore fair) dice include polyhedra of various families, like the platonic solids, catalan solids, ...
2
votes
0answers
20 views

Does every polyhedral graph have a path cover with non-empty paths?

I'm looking to prove or disprove the following conjecture: Every polyhedral graph has a path cover with vertex disjoint, non-zero (length $\ge 1$) paths. Any pointers to literature are appreciated. ...
2
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0answers
40 views

On the Name of the Amplituhedron

Shouldn't the 'amplituhedron' really be called an 'amplitutope' since it's really a polytope and not strictly a polyhedron?
0
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2answers
35 views

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 ...
0
votes
0answers
27 views

How is the Uniqueness of Equilateral Tetrahedra Proved? [duplicate]

Equilateral tetrahedrons all have this property: For any two of its vertices exists a third vertex, which forms an equilateral triangle with these 2 vertices. (It doesn't necessarily have to be a ...
3
votes
1answer
50 views

Volume of overlap between two convex polyhedra

I have two convex polyhedra represented by triangle meshes. I can easily determine if they are in contact or not, but when they are in contact then I would like to determine the volume of their ...
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0answers
72 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, ...
4
votes
3answers
175 views

Can a tetrahedron lying completely inside another tetrahedron have a larger sum of edge lengths?

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 ...
1
vote
0answers
21 views

Folding a Given Net into a Polyhedron Automatically!

There are some applications to fold predefined nets into the polyhedra, e.g. "Poly" or this applet. Is there any application which automatically folds any net generated by the user, if possible?
1
vote
1answer
32 views

Find volume of parallelepiped where all faces are rhombuses with edge $a$ and angle $60^\circ$.

Find volume of parallelepiped where all faces are rhombuses with edge $a$ and angle $60^\circ$. I first noticed that $\dfrac{h}a=\sin60^\circ$, so I easly found that height of the rhombus is ...
2
votes
0answers
26 views

Create Platonic solids from the coxeter group (vertexes & edges & faces)

How can one define vertexes, edges and faces from the Coxeter group? For example, for all platonic solids? I would like to create a general function that takes the Coxeter diagram as input, and gives ...
2
votes
0answers
31 views

Discrete Analogue of the Poincaré Conjecture and Simple Connectedness

I apologize if this question is badly worded or obvious, but I have no formal topology background. I have put some effort into trying to find something, but nothing turned up, perhaps due to my lack ...
0
votes
1answer
70 views

What are the formulas for the number of vertices, edges, faces, cells, 4-faces, …, $n$-faces, of convex regular polytopes in $n \geq 5$ dimensions?

I know that in dimension $n \geq 5$ there are only 3 kind of convex regular polytopes in each dimension: the $n$-simplex, the $n$-cube and the $n$-orthoplex. I would like to know if there are ...
1
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0answers
110 views

Show that every extreme point in Q is either an extreme point of P or a convex combination of two adjacent extreme points of P

P is a bounded polyhedron in $\mathbb{R}^n$, $a$ a vector in $\mathbb{R}^n$, and $b$ some scalar. Define $$Q = {x \in P | a'x = b}$$. Show that every extreme point in Q is either an extreme point of P ...