Questions related to polyhedra and their properties.

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4
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2answers
106 views

Is projection of a convex polyhedron on a plane a convex polygon?

If we have a convex polyhedron with vertices $\mathbf{V}$ and project it on a plane $\mathbf{P}$, is this procedure equivalent to projecting points in $\mathbf{V}$ on the plane $\mathbf{P}$ and then ...
0
votes
0answers
20 views

Volume of Dodecahedron

A dodecahedron undergoes the transformation $$T(x,y,z) \to (3x,3y,3z)$$ What is the ratio of the new dodecahedron to the volume of the old dodecahedron? A) $3$ B) $3\sqrt3$ C) $9$ D) $27$ My ...
2
votes
1answer
172 views

Consistent formula for Regular Polyhedral Volumes

Back in high school I re-discovered the formula for regular polygonal areas like so: $$A = nx^2\frac{\cot(\pi/n)}{4}$$ Where $A$ was the area of the regular polygon, $n$ was the number of sides and ...
4
votes
1answer
81 views

Elegant proof of icosohedron property

This problem was question A1 on the 2013 Putnam contest. Is there a better way to solve this problem than just using pigeonhole principle? Specifically, is there a group theoretic way to interpret ...
1
vote
1answer
42 views

What do you call a convex polyhedron whose symmetry group is transitive on the facets?

I'd like to know a name/source for the following concept: Let $P$ be a convex polyhedron in $\mathbb{R}^3$. Let $G$ be its symmetry group, and let $F$ be the collection of (top-dimensional) faces of ...
0
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0answers
43 views

Geometrically, what is the difference between a “flat face” and a “non-flat” face?

I was curious when I was checking sites like MathisFun, and I came across a pretty unclear system that defines a "flat face" and as a "non-curving" face of a shape; a polyhedron. However, I have to ...
1
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0answers
49 views

Stereographic projection of the icosahedron and snub cube?

Using a steoreographic projection, the three equations associated with the icosahedron with unit circumradius, inradius, and midradius (respectively) are, $$f=z^{20} - 228z^{15} + 494z^{10} + 228z^5 ...
1
vote
1answer
147 views

Coordinates of the Vertices of a Goldberg Polyhedron

I'd like to be able to generate visualizations of the pentagon Goldberg Polyhedra from scratch (i.e. I'm looking for the math, not a software library or package to do this). I can generate truncated ...
0
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0answers
35 views

Farkas lemma corollary and duality

Which is the application of the Farkas lemma corollary on duality? Let $$ Ax \leq b $$ be a system of inequalities, and let $$ P = \{y^TA = 0, y^Tb \leq -1, y \geq 0\} $$ be a polyhedron. May I say ...
0
votes
1answer
65 views

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite ...
6
votes
1answer
99 views

Can I specify the edge lengths of a simplicial polyhedron?

Let $X$ be a convex polyhedron in $\mathbb{R}^3$ whose faces are all triangles, and let $\ell$ be a function which assigns a positive real number to each edge of $X$. We say that $\ell$ is realizable ...
2
votes
2answers
64 views

Generalization of sum of angles to polyhedra?

The sum of interior angles of a polygon is (n-2)*180. Is there a similar statement for the sum of the solid angles of a polyhedra? Is there any non-trivial relationship, $f(\alpha,\beta,...)=0$, ...
1
vote
0answers
41 views

Dodecahedron: How do we get the distance between 2 opposite faces?

I am deciphering a CSS code that Ana Tudor Maria has done. http://codepen.io/thebabydino/pen/qIfbL In her example, she has a formula that calculates the distance between 2 opposite faces. I have no ...
1
vote
1answer
55 views

Proving that a polynomial about the volume of a tetrahedron is irreducible

We know that the volume of a tetrahedron $ABCD$ can be represented as ...
2
votes
0answers
157 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
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0answers
41 views

Net for both cube and regular tetrahedron

At how to fold it by Joseph O'Rourke, there is a net given that can be folded into a cube or irregular tetrahedron. Is there a net that can be folded into either a cube or regular tetrahedron?
0
votes
1answer
110 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 ...
1
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1answer
122 views

Finding the vertices of a rhombic dodecahedron

I'm trying to figure out a straightforward way to find the vertex (x,y,z) coords for a rhombic dodecahedron. Besides starting with a rhombus and rotating it around at the proper angles, I have no ...
2
votes
1answer
109 views

Dimension of polyhedron defined by inequalities and rank of implied equalities

I'm looking at "Optimization Over Integers" by Bertsimas and Weismantel and I have a question about one of the examples in the book. I'm getting a conflicting answer and I'm not sure what I'm ...
12
votes
2answers
275 views

Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
1
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0answers
69 views

Indexing Goldberg (0,n) polyhedron faces

I would to know how to uniquely identify a face of a Goldberg (0,n) polyhedron: http://en.wikipedia.org/wiki/Goldberg_polyhedron#Icosahedral_G.280.2Cn.29_polyhedra It's possible to uniquely assign ...
1
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1answer
28 views

Give examples of polytopes $\Delta$ in $\mathbb{AR}^3$ such that

With Sym $\Delta$ of the set $\Delta$ consisting of all isometries of $\mathbb{AR}^n$ that map $\Delta$ onto $\Delta$, Sym $\Delta$ acts transitively on the set of vertices of $\Delta$ but is ...
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1answer
81 views

Instrinsic definition of concave and convex polyhedron

Is it possible to distinguish a concave polyhedron from a convex one by mesurements made only on its surface, without a reference to the 3d space around it?
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0answers
28 views

Standard for intrinsic polyhedron definition using angular deficit?

Is there a standard definition of a given polyhedron using only intrinsic properties (those which can be measured by a 2d being living on its surface) and particularly angular deficit at a vertex ...
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0answers
33 views

How to fit a cuboid into a polyhedra?

I have multiple points which create a solid (polyhedra). And now I want to place a cuboid inside this solid in a way that it uses the maximum amout of space inside. Are there any solutions for this ...
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0answers
44 views

Convex polyhedron is union of simplices

Given a convex polyhedron $P$, how can we prove that every point $x \in P$ is in some simplex whose vertices are vertices of $P$? One proof is to inductively build a triangulation of $P$. If $P$ is ...
0
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0answers
31 views

Finding the min of $L/V^2$ where $L$ be the product of six edges and $V$ be the volume of a tetrahedron.

I'm interested in making and solving new mathematical questions. I made the following question: Let $L$ be the product of the lengths of the six edges of a tetrahedron $K$. Also let $V$ be the volume ...
2
votes
0answers
63 views

About the relation between a tetrahedra and spheres moving in a tetrahedra

I found the following question in a book: There exists a regular triangle $OAB$ which has edge-length $2$. Let $H, I, J$ be a foot of the perpendicular line drawn from a point $P$ in $OAB$ to the ...
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vote
4answers
161 views

Inequality for each $a, b, c, d$ being each area of four faces of a tetrahedron

We know 'triangle inequality'. I'm interested in the generalization of this inequality. Here is my question. Question: How can we represent a necessary and sufficient condition for each positive ...
0
votes
0answers
76 views

Buckminster Fuller's dymaxion map

I'm looking for the projection equations of the dymaxion map, and more generally, the equations of the map projection of a sphere onto any geodesic dome, the mapping onto a icosahedron (the dymaxion ...
3
votes
2answers
90 views

Forming a polyhedron from concave polygonal faces.

A polyhedron is a convex, three dimensional region bounded by a finite number of polygonal faces. So is it possible that some of those polygonal faces be concave ? Can concave polygons be used in the ...
0
votes
1answer
276 views

How do I determine the Tait-Bryan angles (yaw, pitch, and roll) of polyhedron faces to its center?

I'm modeling a pentagonal hexecontrahedron by placing faces and then rotating them. I've determined the center of each face by using the Cartesian coordinates of the vertices of its dual polyhedron ...
1
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1answer
57 views

Integral Polyhedra: Integer on each face

The general topic is unimodular matrices and integral polyhedra. I am really new to this field and I am studying for an exam in an advanced operations research course. In this case we are always ...
1
vote
1answer
81 views

Bounded polyhedra closed under rotation, intersection and complement

Are there any known types of bounded polyhedra, which exist in all Euclidean dimensions, and are are closed under intersection, rotation and relative complement? In other words, I am looking for a set ...
3
votes
0answers
44 views

Regular apeirohedra?

Have been toying with structures that I think are best describe as unbounded regular polyhedra. More specifically I arrived at non-convex polyhedra that are unbounded in one direction: Alternate ...
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0answers
30 views

$X$ is a point in a bounded polyhedron $\ \in R^n $ with $n-1$ active constraints

Lets take a vector $d$ which is orthogonal to the active constraint. Since the polyhedron is bounded: We'll move to a point $x+\alpha*d$ where we will activate another constraint let's name it j. ...
2
votes
1answer
370 views

coordinates of icosahedron vertices with variable radius

I was looking on the wikipedia page about icosahedrons and it says that for edge length $a$ the radius of the circumscribed sphere around the icosahedron is given by $r = a \times ...
0
votes
1answer
107 views

Computing bounding box of polytope (system of linear inequalities)

Given a N real valued variables and a set of linear inequality constraints, I would like to find a minimal bounding box which encapsulates the convex polytope defined by these constraints. I think ...
1
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1answer
50 views

Mathematical word for geometrical object?

Is there a mathematical word to designate the concept of a geometrical object like: square cube tesseract N-dimensional cube circle sphere hypersphere regular and non-regular polygons regular and ...
6
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0answers
93 views

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 ...
2
votes
0answers
33 views

The dimension of birkoff polytope

Let $P_m$ be a subset for R^mxm be the polytope given by: $x_i,_j \ge 0$ $x_i,_1 + ... + x_i,_m \le 1$ $x_1,_j + ... + x_m,_j \le 1$ $\sum_{1 \le i,j \le m } \ x_i,_j \ge m-1$ Contruct a ...
1
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1answer
43 views

Duals of Deltahedra

What are the names of the duals of the Snub Disphenoid and the Triaugmented Triangular Prism? I built models of the eight convex deltahedra and their duals using spherical magnets as vertices, and ...
5
votes
0answers
77 views

Is Paley-13 graph a unit distance graph in 3D space?

The 13-node Paley graph has vertices 1 to 13 that are connected by an edge when their difference is one of the values $(1,3,4,9,10,12)$ Can this graph be put into 3D space so that all edges have ...
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0answers
52 views

How to define polyhedra?

Wikipedia does not provide a concise definition of "polyhedron" in $\mathbb R^n$. What is the "best" - in whatever sense - definition of this class of objects? I am interested in a definition where ...
3
votes
1answer
422 views

Euler's formula for triangle mesh

Can anyone explain to me these two facts which I don't get from Euler's formula for triangle meshes? First, Euler's formula reads $V - E + F = 2(1-g)$ where $V$ is vertices number, $E$ edges number, ...
4
votes
2answers
178 views

Has anyone discovered a convex space-filling 15-faced polyhedron?

I've been looking for extensive surveys regarding space-filling polyhedra, but have only come across Michael Goldbergs "Convex polyhedral space-fillers of more than twelve faces" from 1979, stating ...
3
votes
1answer
92 views

Calculate polyhedra vertices based on faces

I have some origami polyhedra which I know the type of faces it has and how they are connected (such as this torus) and I want to calculate the co-ordinates of the vertices to use as an input to ...
2
votes
1answer
203 views

Number of edge colorings in a tetrahedron with three colors. Is my solution correct?

I've tried to count rotationally distinct edge colorings (both proper and improper) in a regular tetrahedron with three colors. Could you take a look if it's correct? First the improper colorings. ...
1
vote
1answer
89 views

Is this graph coloring problem solved correctly?

On this Wikipedia page about Burnside's lemma, it is calculated that there are 57 rotationally distinct colorings of the faces of a cube with three colors. I'm confused by the way it is done. They ...
0
votes
1answer
254 views

Goldberg polyhedra coordinates

I would 3D-print some Goldberg Polyhedra importing in Sketchup, the coordinates provided on these links: 72 faces (2,1) - (coordinates) 132 faces (3,1) - (coordinates) 192 faces (3,2) - ...