Questions related to polyhedra and their properties.

learn more… | top users | synonyms

2
votes
0answers
46 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 ...
1
vote
0answers
62 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 ...
2
votes
0answers
67 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 ...
1
vote
4answers
199 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 ...
3
votes
2answers
112 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
359 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
vote
1answer
93 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
95 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
49 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 ...
3
votes
2answers
1k views

Angles between two vertices on a dodecahedron

Say $20$ points are placed across a spherical planet, and they are all spaced evenly, forming the vertices of a dodecahedron. I would like to calculate the distances between the points, but that ...
1
vote
0answers
42 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
741 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
137 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
vote
1answer
52 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 ...
10
votes
1answer
194 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
36 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
vote
1answer
48 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
103 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 ...
1
vote
0answers
67 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
1k 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, ...
5
votes
2answers
296 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 ...
4
votes
1answer
117 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
325 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. ...
2
votes
1answer
100 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
350 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) - ...
7
votes
1answer
365 views

How many $n$-colorings up to rotation using exactly 2 of each color are there on a $2n$-polyhedron?

I'm a high-school student and I stumbled across a YouTube video explaining how Rubik's cubes work. A Rubik's cube has 6 colors, one for each side, but I started thinking about ways to $n$-color the ...
4
votes
3answers
189 views

Can a tetrahedron lying completely inside another tetrahedron have a larger sum of edge lengths?

Find 2 tetrahedrons $ABCD$ and $EFGH$ such that $EFGH$ lies completely inside $ABCD$. The sum of edge lengths of $EFGH$ is strictly greater than the sum of edge lengths of $ABCD$. I am completely ...
2
votes
0answers
52 views

Convex cone as sum of simplices?

In 3D a pyramid with a square base can be decomposed into the sum of two tetrahedra, i.e. two 3-simplices. I am dealing with a homogeneous N-dimensional system of inequalities and my solution is a ...
2
votes
0answers
186 views

$\{x:Ax\leq 0\}$ contains a subset of type $\{x:A'x=0, ax\leq 0\}$

If $C:=\{x:Ax\leq 0\}\neq\{x:Ax=0\}$, an independent set of rows of $A$ can be chosen, one denoted by $a$ and the others put as rows into a matrix $A'$, such that $\{x:A'x=0,ax\leq 0\}\subseteq C$. ...
1
vote
1answer
26 views

Estimate the number of integral solutions inside a convex polyhedron

How can I compute an estimate of the number of integral solutions (points) inside a bounded convex polyhedron with dimension $d$? I'm interested more in an efficient way to estimate the number of ...
1
vote
1answer
211 views

How tell if a polyhedron contains a lattice point

So given a polyhedron $Ax \le b$ Is there an Algorithm or formula to determine whether said polyhedron contains a lattice point (integer point) I was thinking a couple things: brute force ...
4
votes
2answers
464 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 ...
2
votes
1answer
77 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? ...
5
votes
1answer
146 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 ...
9
votes
1answer
274 views

Maximal volume for given surface area of an $n$-hedron

Is there a term for a polyhedron with $n$ faces (or, similarly, $n$ vertices) that maximises the enclosed volume for a given surface area (equivalently, minimises the surface area for a given volume)? ...
2
votes
1answer
118 views

Which polyhedra have an even number of faces touching each vertex?

A two-coloring of the faces of a polyhedron is always possible when an even number of faces meet at each vertex. http://www.georgehart.com/virtual-polyhedra/colorings.html Is there a name for ...
2
votes
0answers
71 views

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 ...
5
votes
1answer
128 views

Is There a Formalization of Cauchy's $F - E+V = 2$ proof?

Can anyone provide, or direct me to a formalized version of Cauchy's proof that for any convex polyhedron with $F$ faces, $E$ edges and $V$ vertices that $F - E + V = 2$. I am willing to accept the ...
4
votes
0answers
68 views

Mappings preserving convex polyhedra

It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra. Can you give a characterization of the class of mappings that preserve convex polyhedra?
0
votes
2answers
535 views

Calculate Spherical Distance between points

I have googled this and not come up with an answer yet, but basically, I'm trying to find out the distance between each point or vertice on a sphere (all points are evenly spaced). I already know this ...
0
votes
3answers
181 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
1
vote
0answers
41 views

Smallest amount of planes to enclose a closed space in extended projective geometry $\mathbb R^3_{\pm\infty}$

The smallest amount of planes to enclose a polyhedron is 4 in the euclidean $\mathbb R^3$ where it encloses a tetrahedron. What is the smallest amount of planes to enclose a closed space in extended ...
1
vote
1answer
218 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 ...
4
votes
1answer
104 views

Are the face–centroid pyramids of a convex congruent-faced polyhedron congruent?

Let a convex polyhedron $P$ be given, all of whose faces are congruent. Consider any pyramid formed by a face of $P$ as its base and the centroid of $P$ as its vertex. Allowing congruence to admit ...
4
votes
3answers
309 views

Geodesics on a polyhedron

Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the "equator" of the octahedron eventually will. But for what ...
1
vote
1answer
66 views

Part from “Regular polytopes” which I don't understand

This is a paragraph from "Regular polytopes" by Coxeter that I don't understand. Although it is not always possible to include all the vertices of a polyhedron in a single chain of edges, it ...
2
votes
1answer
80 views

The structure of realization spaces of polyhedral graphs

Given a polyhedral graph with $v$ vertices, $e$ edges and $f$ faces, each possible realization of the graph as a geometric (convex) polyhedron corresponds to a point in $\mathbb{R}^{3v}$, ...
8
votes
2answers
846 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 ...
1
vote
1answer
245 views

Does Voronoi tessellation in 3D always produce convex polyhedrons?

I'm more or less certain that the Voronoi tessellations (using Euclidean distance measure) produce convex polygons/polyhedrons. Is there a way to prove this mathematically? Or am I wrong? I am ...