Questions related to polyhedra and their properties.

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In linear programming, how to check whether a convex polyhedron is contained in another

Suppose we have two convex polyhedra $P_1=\{x\in \mathbb{R}^n \mid A_1 x \geq b_1 \}$ and $P_2=\{x\in \mathbb{R}^n \mid A_2 x \geq b_2 \}$ Is there a way to check whether $P_1 \subseteq P_2$? I was ...
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1answer
47 views

What is the depth of water above the prism?

I have been practising for a math competition and came across the following question: A fishtank with base $100\,\rm cm$by $200\,\rm cm$ and depth $100\,\rm cm$ contains water to a depth of ...
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1answer
15 views

Subvector and related subspace

This might be easier than I think, but I got stuck. Assume a vector $y=[y_1,\ldots,y_n]\in Y$, where $Y$ is a convex polyhedron. Assume a $k$-dimensional subvector of $y$, namely ...
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1answer
31 views

Geometric Representation of Quasidihedral Groups

I am going back through Dummit/Foote studying for a prelim and came across the 'quasidihedral' or 'semi-dihedral', group of order $2^n$, with presentation $\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs ...
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55 views

Isomorphism of divisors

Consider the cartier divisor group $CDiv_{T_{N}}(X_{\Sigma})$ defined on the fan $X_{\Sigma}$. I am having trouble proving the following assertion that there is a natural isomorphism ...
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2answers
122 views

Convex polyhedron and its Gauß-curvature

I am trying to prove: A convex polyhedron has positive Gauß-Curvature at every vertex. What we know: Gauß-Curvature at every vertex is given by $K(p) = 2\pi - \sum\limits_{\text{angle } ...
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1answer
56 views

Finding generators of toric ideals

Consider the affine toric variety $V \subset k^{5}$ parametrized by $$\Phi(s,t,u) = (s^{4},t^{4},u^{4},s^{8}u,t^{12}u^{3}) \in k^{5}$$ where k is an algebraically closed field of characteristic 2. ...
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1answer
73 views

$3\mathrm D$ Projection Of $4\mathrm D$ Polyhedron

Can someone identify this shape? I think it is a $3\mathrm D$ projection of $4\mathrm D$ polyhedron. The body in the center seems to be a truncated octahedron, so as the body in the middle. The ...
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How to determine the outward normal vector of a face of a hexahedron if the orientation (CW or CCW) of vertices of the face is unknown

Consider a single quadrilaterally-faced hexahedron. If given the co-ordinates of the vertices, $\mathbf{v}_i$, of a face in counter-clockwise orientation, I can compute the corresponding unit outward ...
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1answer
38 views

How to prove a point in a set is an extreme point of the set ?

Def: an extreme point of a set $K$ is the point that cannot be expresssed as a convex combination of other points in $K$. Apart from the definition, what else arguments can we use to prove that a ...
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Dinamically generate Goldberg polyhedra G(m,n)

In these pages the autor provided a lot of info about some Goldberg polyhedra (http://en.wikipedia.org/wiki/Goldberg_polyhedron): http://dmccooey.com/polyhedra/DualGeodesicIcosahedra.html ...
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1answer
39 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 ...
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0answers
53 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 ...
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1answer
38 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, ...
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2answers
141 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, ...
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1answer
110 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 ...
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80 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 ...
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43 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 ...
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1answer
44 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 ...
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1answer
74 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: ...
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1answer
59 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 ...
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1answer
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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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53 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
107 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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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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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?
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21 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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1answer
87 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 ...
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3answers
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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
56 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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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 ...
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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
18 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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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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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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1answer
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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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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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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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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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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
60 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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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
611 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. ...
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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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1answer
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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?
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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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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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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
26 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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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 ...