Questions related to polyhedra and their properties.

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How can I determine the radius of a dodecahedron?

I am making a dodecahedron that needs to fit inside of a sphere. The sphere has a diameter of 56mm. What is largest possible measurement of one segment of a pentagon side of a dodecahedron that would ...
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2answers
245 views

Problem in understanding a proof there are five Platonic solids.

Thanks to several comments by Gerry Myerson, it is now clear that I wasn't clear, up to a state where I seriously confused myself. In a renewed attempt: Recently, I've been thinking about Platonic ...
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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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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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7answers
158 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
55 views

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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3answers
4k views

Tetrahedron inside a sphere

What's the largest regular tetrahedron (having side length $x$) you can fit inside a sphere with a unit radius?
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5answers
24k views

Height of a tetrahedron

How do I calculate the height of a regular tetrahedron having 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 ...
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1answer
84 views

Three theorems for Polyhedra, Polytopes, and Cones

Are there readable proofs of the following theorems? A polytope (bounded polyhedron) is the convex hull of a finite set of points. A polyhedral cone is generated by a finite set of vectors. That is ...
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1answer
397 views

What property of certain regular polygons allows them to be faces of the Platonic Solids?

It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes. What property of those regular ...
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2answers
232 views

Is there a forumla to find out if n faces can be made into a 'regular polyhedron'?

I'm not too sure about the exact terminology since Wikipedia is throwing me all over the place. I'm looking for a formula to find out if for n faces a 'regular polyhedron' can exist. In case that's ...
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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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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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0answers
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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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3answers
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insphere/circumsphere ratio of a polyhedron the same as its dual polyhedron?

Is the $r/R$ ratio for any polyhedron always the same as the $r/R$ ratio of the dual of that polyhedron? Given any polyhedron, we can find the biggest sphere that fits inside it (its insphere) and ...
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1answer
154 views

What is the minimal isoperimetric ratio of a polyhedron with $5$ vertices?

I'm asking and answering this question to provide a partial answer to this question and a comment on this answer at MO. The isoperimetric ratio $\mu$ of a solid is the ratio $A^3/V^2$, where $A$ is ...
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1answer
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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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3answers
125 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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1answer
85 views

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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1answer
130 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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0answers
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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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2answers
449 views

Deforming a truncated icosahedron into its circumscribing sphere

Imagine that I have a truncated icosahedron consisting of 60 identical vertices, each of degree $deg(v) = 3$, and fixed edge length $L$. I'd like to assign some constant curvature or bending angle ...
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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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1answer
108 views

volume of polyhedra which correspond to icosahedral fullerenes

There is, I believe, a sequence of polyhedra whose shape approaches that of the icosahedron (they all have twelve pentagonal faces and the rest hexagons), and starts: regular dodecahedron (C$_{12}$, ...
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4answers
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Why are there 12 pentagons and 20 hexagons on a soccer ball?

Edge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron. Is there some insight into why the alternation of pentagons and hexagons yields an approximated ...
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0answers
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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
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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
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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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1answer
109 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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0answers
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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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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
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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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0answers
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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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0answers
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Are there 3D tilings of a 3D projective hyperplane or 3-sphere?

I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings (honeycombs) of a 3D projective ...
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1answer
87 views

How to find that Dehn invariant of a dodecahedron?

What is the Dehn invariant of a regular dodecahedron with center (0,0,0), and radius 1?
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1answer
49 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?
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0answers
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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
354 views

Relative side lengths of dual dodecahedron and icosahedron

If the side length of a dodecahedron=1, then what is the side length of its dual icosahedron whose vertices occupy the same space as the mid-points of the faces of the dodecahedron. I've read that ...
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3answers
368 views

Minimum and maximum diagonal of a dodecahedron

Let $s$ be the edge length of a regular dodecahedron. As a function of $s$, what is the dodecahedron's minimum and maximum diagonal (i.e. cross-section)?
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Cleverest construction of a dodecahedron / icosahedron?

One can show, as an elementary application of Euler's formula, that there are at most five regular convex polytopes in 3-space. The tetrahedra, cube, and octohedra all admit very intuitive ...
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2answers
478 views

How to find the maximum diagonal length inside a dodecahedron?

I am trying to find the maximum length of a diagonal inside a dodecahedron with a side length of $2.319914107\times10^{89}$ meters. I am not sure if any other information than that is needed, if it ...
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1answer
81 views

Formula for greatest cross section of regular dodecahedron

Is there a formula for the area of greatest cross section of a regular dodecahedron? For example, this can be viewed as finding a hole big enough for it to fit into.
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1answer
80 views

Demi dodecahedron - what is it?

What is a demi dodecahedron? I have not been able to find the geometry of a demi dodecahedron. From latin, what makes a polygon a demi dodecahedron? Can a demi dodecahedron possibly contain 9 faces? ...
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1answer
105 views

Symmetries of a dodecahedron

Suppose we want to measure order of the group symmetries of a dodecahedron, and we know that If $G$ is a group and $S$ is a set on which $G$ acts and $s\in S$, then Order of G=(Order of stabiliser of ...
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1answer
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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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2answers
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Coloring dodecahedron

I found some months ago that there are the Polya's enumeration theorem to compute number of colorings of dodecahedron. I got interested to find how to show by using only Burnside's lemma that there ...