Questions related to polyhedra and their properties.

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2
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3answers
34 views

trying to grasp disphenoid tetrahedral honeycomb, what are the dihedral angles?

What are the dihedral angles in a disphenoid with four identical triangles, each having one edge of length $2$ and two edges of length $\sqrt{3}$? Tried to look it up, but couldn't find it...
3
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1answer
37 views

The Rhombohedron

I am trying to model a rhombohedron (using Blender) as a first pass to building Dürer's solid so I am trying to calculate the (x,y,z) values for a given side length 'a' and angle 'theta' (starting ...
0
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0answers
11 views

Stellating the Octahedron

I am trying to create a very primitive animation/demonstration that shows the stellation of an octahedron to yield the stella octangula. Unfortunately, it seems that the mental image I have for ...
3
votes
2answers
37 views

Faster Algorithms for Convex Hulls

I was interested in the following: Given two polyhedra $P_1, P_2$ specified in the form: $$ P_1 = {x : A_1x \le b_1 } $$ $$ P_2 = {x : A_2x \le b_2 } $$ Whereas $ x \in R^n$ and $b_1, b_2$ are ...
-1
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0answers
23 views

Calculate the inradius of a cell in hyperbolic {p,q,r} tiling?

As written in the title, I need to calculate the inradius of a cell in hyperbolic tiling with Schlafli symbol {p,q,r}. You can link to a document which have the formula or write the formula here. That ...
0
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2answers
26 views

How do you find redundant constraints for a feasible region?

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm only trying to find the non-redundant constraints that define a feasible region (i.e. I ...
0
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2answers
26 views

How many face we could make regular convex polyhedron

I want to tile the sphere as many face as possible. And I want every face be the same size and shape. Is it possible to generate more than 100 or 1000 faces of regular convex polyhedron?
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0answers
18 views

Show that the set $\{y_1a_1+y_2a_2: -1\le y_1,y_2\le 1\}$ is a polyhedron

Show that the set is a polyhedron and express it in the form: $S = \{Ax\leq b, Fx = g\}$, $S=\{y_1a_1+y_2a_2 | -1\leq y_1\leq 1, -1\leq y_2\leq 1\}$ where $a_1,a_2\in\mathbb{R}^n$ My attempt: A ...
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1answer
17 views

Polyhedral cone as conic hull of a finite set

I am reading notes on optimization and it was claimed that all polyhedral cones in $K\subseteq \mathbb{R}^n$ can be written Cone(R) where $R\subseteq \mathbb{R}^n$ is a finite set. That is, if K is a ...
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0answers
6 views

Inner construction of polyhedral set: cone part uniquely determined

The "inner construction" of a polyhedral set is: given $V,R\subseteq \mathbb{R}^n$, $|V|,|R|\in \mathbb{Z}^+$ (nonempty, finite), put $S:=Co(V)+Cone(R)$. It was claimed that Cone(R) is uniquely ...
3
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2answers
122 views

What are the rules for a Tetartoid pentagon?

The tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry. It has twelve identical ...
3
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2answers
1k views

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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votes
3answers
3k views

Angle between lines joining tetrahedron center to vertices

What are the angles formed at the center of a tetrahedron if you draw lines to the vertices? I'm trying to make these: I need to know what angles to bend the metal.
0
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1answer
39 views

Regular polyhedra surface area [closed]

When I looked at an image of a regular octahedron, I found that its surface was composed of triangles. Then I looked at a regular icosahedron and this was the same case! Then I saw a regular ...
1
vote
1answer
49 views

Small Stellated Dodecahedron, generating triangle vertices

I have been trying to draw a small stellated dodecahedron (would post an image if I had enough rep) using OpenGL, and would like to generate the vertices programmatically. I'm looking for a way to map ...
1
vote
0answers
18 views

For polytopes, does union and linear transformation commute?

Given two (convex) polytopes $P_1$ and $P_2$ and a linear transformation $T$, is it true that: $$T(P_1 \cup P_2) = TP_1 \cup TP_2$$ What if $P_1$ and $P_2$ are not convex?
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0answers
35 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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1answer
40 views

The maximum of several affine functions is a polyhedral function

A function $f: \mathbb{R}^n \mapsto (-\infty,\infty]$ is polyhedral if its epigraph is a polyhedral, i.e. $$\text{epi}f=\{(x,t)\in \mathbb{R}^{n+1} | \ \ C\left( \begin{matrix} x\\ t \end{matrix} ...
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0answers
57 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 ...
2
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3answers
63 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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4answers
492 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). ...
3
votes
5answers
292 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 ...
1
vote
1answer
57 views

Cubohemioctahedron

Am I missing something here? Do I see shapes differently than everyone else? A Cubohemioctahedron is cited to have a Euler characteristic of negative 2. This is because most texts say its 10 faces ...
1
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1answer
26 views

Linear Independence Proof using Rotation

I am attempting to prove that three axes of rotation are linearly independent and so form a basis for $\mathbb{R}^3$. I assert that given the fact that rotations about each axis involve no rotation ...
0
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0answers
31 views

The definition of face - in regard to polyhedral fans

Question: How is face defined rigorously for bullet point 2 below. Definition: A convex polyhedral fan, $F$, of polyhedral cones, all living in the same vector space, requires two things: If ...
1
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1answer
22 views

In an icosahedron subdivided n times, how can I find the coordinates of adjacent centroids?

I think it would be helpful to refer to this image when trying to follow my description: http://i.imgur.com/nRXQo3W.jpg (taken from http://experilous.com/1/blog/post/procedural-planet-generation). ...
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0answers
23 views

Polyhedral surface with infinitely many triangulations with same combinatorics

Is there an example of a polyhedral surface that has infinitely many triangulations with the same combinatorics?
2
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1answer
365 views

Relative side lengths of dual dodecahedron and icosahedron

If the side length of a dodecahedron is $1$, 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 ...
1
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2answers
261 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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2answers
132 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 } ...
0
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7answers
173 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 ...
3
votes
2answers
66 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
52 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 ...
0
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1answer
17 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 ...
7
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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
25k 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 ...
2
votes
1answer
99 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 ...
15
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1answer
404 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 ...
2
votes
2answers
248 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 ...
9
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1answer
89 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 ...
4
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1answer
35 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 ...
4
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0answers
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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3answers
997 views

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 ...
5
votes
1answer
170 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 ...
4
votes
1answer
63 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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votes
3answers
136 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
93 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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vote
1answer
157 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
15 views

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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vote
2answers
466 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 ...