Questions related to polyhedra and their properties.

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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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37 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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Polyhedron with an expontial number of extreme points under a map

I'm looking for an $\mathcal{H}$-polyhedron $P\subseteq \mathbb{R}^n$ and a map $C\colon\mathbb{R}^n \to \mathbb{R}^2$ such that the number of vertices of $C\cdot P$ is exponential in $n$.
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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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47 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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301 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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44 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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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. ...
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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 ...
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6 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 ...
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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: ...
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1answer
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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 ...
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4answers
133 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 ...
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2answers
604 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 $\chi' = 2-\chi,$ ...
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4answers
56 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). ...
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39 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 $x$). ...
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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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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 ...
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1answer
19 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?
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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 ...
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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, ...
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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 ...
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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. ...
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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?
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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 ...
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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 ...
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68 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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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?
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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 ...
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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 ...
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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 ...
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108 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 ...
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Vertices of Polyhedral

Suppose there are matrix $A\in\mathbb{R}^{n \times m}$ and vector $b\in\mathbb{R}^n$. Consider a non-empty polyhedron $P = \{Ax \leq b\} $. Then, there exists a vector $\bar{x}\in P $ such that ...
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How many arrangements (correct and incorrect) possible with cube puzzle pieces in a particular unfolded form?

I am trying to do a computer program for arranging cube puzzle pieces in an unfolded form. A cube can be unfolded into 11 forms or nets. I have chosen a single net for now for simplicity purposes and ...
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Explanation regarding Kaleido index for polyhedra

I can't seem to find any information about the Kaleido index number used in geometry (see 'K# at http://en.wikipedia.org/wiki/List_of_Wenninger_polyhedron_model'). I found an abstract called "Uniform ...
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Linear programming - Textbook recommendations

Next term, I will attend a course on linear programming. Due to the assignments, we will have to write many thorough proofs. I anticipate that we will be supposed to cope with in-depth background ...
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Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
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61 views

Visualizing Platonic Solid group symmetries

How do you visualize the rotation symmetries, to classify a icosahedron for example as Ih, H3, [5,3], (*532)
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Drawing a Truncated Octahedron

I'm trying to draw a truncated octahedron in MATLAB. This is also known as a permutahedron so my strategy is to link up all the vertices via adjacent transpositions of permutations in $S_4$. What I ...
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For an app teaching about polyhedra, what are some core characteristics to include?

For fun: I'm building a 3d app that teaches about polyhedra. What should I include? The obvious didactic elements for each polyhedron would be: Fundamental polygon's Vertices 
Edges
 Faces
 (and ...
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Showing that group of orientation preserving isometries of Icosahedron is a simple group

Let $G$ denote the group of orientation preserving isometries of Icosahedron. To prove the claim, I have shown that $\nexists \ N \ \triangleleft \ G$ such that $|N|=5.$ $\nexists \ N \ ...
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How to locate sphere resting in a corner of a polyhedron?

I am working on an interface to a computational solid geometry program. I would like this program to be able to fillet corners (although fillet sometimes seems to refer to an internally smoothed ...
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76 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 ...
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Is there a nice characterization of circumscribed hexahedra?

A convex quadrilateral that contains a circle tangent to its sides is called a tangential or circumscribed quadrilateral. There is a very nice characterization of tangential quadrilaterals known as ...
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Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
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2answers
70 views

Maximum number of vertices in intersection of triangle with box

Suppose we have a triangle and a box in 3D. The intersection of the triangle with the (solid) box will be a polygon with some number of vertices (possibly zero). The vertex count will vary according ...
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LP: An algorithm to decide whether a polyhedron is a subst of another polyhedron

I've encountered the following question which I am unable to solve: $$ P = \{\vec x | A\vec x \geq \vec a\} \\ Q = \{\vec x | B\vec x \geq \vec b\}\\ P, Q \subseteq R^n $$ Find an algorithm to ...
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a general definition of the volume of a high dimensional polytope

I would like to find a general definition of the volume for a full dimensional polytope in $R^n$. Could anyone give me a hint please! Thank a lot
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Why is the 24-cell (also called Icositetrachoron or Hyperdiamond) the unique regular convex polychoron which has no direct three-dimensional analog?

The 24-cell is self-dual, and is the unique regular convex polychoron which has no direct three-dimensional analog. http://mathworld.wolfram.com/24-Cell.html I don't understand why that is ...
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Time to roll through subterranean chords

Imagine a spherical airless body. It is small enough that central pressure allows a tunnel to be built from north pole to the south pole. I jump in the tunnel at the north pole and fall to the south ...