The polyhedra tag has no usage guidance.

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### Number of polyhedra with N faces?

A. Up to isomorphism, how many polyhedra with N faces are there? Assume each face can be a triangle, square, pentagon, hexagon, etc... Furthermore each edge can be resized to any nonzero positive ...

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### Unit-Distance Polyhedra

What polyhedra are known to have two vertices adjacent if and only if they are of distance $d$ apart, for fixed $d$? For example, regular Platonic solids satisfy this condition, so I am looking for ...

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### Testing whether two vertices are neighbours

I face the following problem: I am given a high-dimensional, convex, bounded polyhedron in both vertex description: $X = \mathrm{conv} \, \{ v_1, \ldots, v_K \}$ and halfspace description: $X = \{ x \...

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### Distinguishing (possibly lower dimensional) $1$-skeleton of a regular graph inscribed in a sphere

Consider you have two (possibly same) convex $1$-skeleton of a regular graph $A$ and $B$ in $m$-dimensions inscribed in a sphere with possibly exponential number of vertices in $n$-dimension with ...

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### Harborth conjecture and polyhedra

Harborth conjecture state that every planar graph can be drawn on a plane only using staight line segments of rational or integral edge length.
( There is a good mathoverflow page for this conjecture, ...

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### Convex polyhedra, combinatorial types and Symmetry

Steinitz theorem says that combinatorial types of convex polyhedra is identified with 3-connected planar graph, called by a polyhedral graph.
A symmetry of polyhedral graph means that a vertex ...

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### Term for meshes bounding a non-degenerate tetrahedral mesh in 3D?

Is there a term in the literature referring to triangle meshes that are the closed boundary of some non-self-intersecting, non-degenerate tetrahedral mesh embedded in $R^3$?
This class of triangle ...

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### Checking integer points of given infinity norm on intersection

If we have convex region $\mathscr R$ given as intersection of a convex polytope $\mathscr P$ and an ellisoid $\mathscr E$ in $\ell^2$ norm is there an efficient way to test if there is an integer ...

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### Labeling edges of an icosahedron with sum constraints

The question is inspired by this previous MO question. There it was shown that it's possible to label the edges of a cube by the numbers $\{1,2,\ldots,6,8,9, \ldots, 13\}$ in such a way that:
Three ...

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### Self-avoiding/reflecting geodesics on a convex surface

Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$.
For me $S$ is a convex polyhedron,
but I am happy to view $S$ as a smooth body with positive Gaussian curvature
at each point, or ...

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### symmetric group of regular polyhedrons

Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,
$\Delta^2$:
$\Delta^3$:
Let $...

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### Which -icial sets produce the “standard” representations of symmetric groups?

Suppose you have a system of cell complexes (say, even convex polyhedra) $(P_n)_{n\geqslant0}$ which occur as faces of each other and are used to define the corresponding notion of "$P_*$-set". So ...

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### Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?

Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$,
$$f(v) = A[v]_+ + B[-v]_+$$
surjective? Here $[.]_+$ is an elementwise ...

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### Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints
$$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq \vec{...

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### Convex polyhedron and its Gauß-curvature [closed]

I have asked this question on MathSE and no one could give me an answer. So I'll post my question here.
What I am trying to prove:
A convex polyhedron has positive Gauß-Curvature at every vertex.
...

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### What's that shape? Inferring a 3D shape from random shadows

Let $P$ be a bounded, simply connected region of $\mathbb{R}^3$.
$P$ could be a polyhedron, or a smooth shape, or an arbitrary shape;
I'll assume below that $P$ is a (non-degenerate, perhaps non-...

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### Convex Polyhedra Scissors Congruence Problem

I am currently writing a geometry paper "Rectifications of Convex Polyhedra" and I am confused to have discovered what appears to be a remarkable discrete geometric fact:
Conjecture: Let $P$ be a ...

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### Construct polygon/polyhedron containing all points not externally visible w.r.t given polygon/polyhedron?

Is there an algorithm to construct a polyhedron containing all points in space for which there exists no ray to infinite not intersecting a given polyhedron?
In 2D, we could consider polygons. For ...

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### Digital topology, animal problem, 2-sphere and torus

I have the following question relating digital topology, surfaces, particularly $S^2$ and torus.
Can a body $B$ constructed with cubes (without cavities or tunnels) and with frontier homeomorphic to ...

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### Computionally efficient vertex enumeration for (convex) polytopes

Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...

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### A Jordan Separation Theorem for Polyhedral Surfaces

Let me begin by defining what a polyhedral surface is.
A path-connected subset $ P $ of $ \mathbb{R}^{3} $ is called a polyhedral surface iff it is the union of a finite collection $ \mathcal{C} $ of ...

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### The intersection of two $l_1$ balls

Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ with respect to the $l_1$ norm that have different radii and different centers. Is there an upper bound for the number of vertices that $B_1\cap B_2$...

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### Polyhedra containing hexagones only [closed]

It is well-known that Euler's theorem gives raison d'être to polyhedra containing exactly 12 pentagons if they are connected by 3 in a vertex. The number of hexagons may be arbitrary (in fact >1). In ...

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### Which (polytopal) fans/polytopes are secondary?

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.
The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional ...

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### Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Recently Mark McClure constructed and displayed
the 261 unfoldings of the hypercube (tesseract)
in response to the question,
"3D models of the unfoldings of the hypercube?":
The first 9 unfoldings ...

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### regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations

While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably ...

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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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### Examples of Polyhedra with Large Shadows

Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...

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### The limit of edge-midpoint convex polyhedra

Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$,
replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$.
Continuing this process, we obtain a ...

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### How to approximate volumes of m-dimensional manifolds with m-dimensional polyhedra

The areas of a sequence of polyhedra approaching a surface need not approach the area of the surface, but there are theorems guaranteeing that this be so. (T. Rado, On the Problem of Plateau, Chapter ...

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### Higher dimensional generalization of: Any quadrilateral tiles the plane?

Any (non-self-intersecting) quadrilateral tiles the plane.
(MathWorld image.)
Q. What is the strongest known generalization of this statement to higher dimensions?
I.e., $\mathbb{...

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### $\mathcal{H}$-polyhedron under a linear map

Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$.
Moreover, let $M \colon \mathbb{R}^n \to \...

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### Build a topological polytope with a specified CW-structure

I am a topologist and not quite familiar with the tools for building a polytope. I would like to build some topological polytope which is an somewhere in between permutohedron and associahedron which ...

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### Can anyone suggest a text on polyhedral theory?

Can anyone suggest a text on polyhedral theory? Particularly on increasing the number of faces under projections. 0,1 polytopes

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### Put P inside Q! polygons/polyhedra

We have two Polygons/Polyhedra P and Q.
Does there exist a polynomial time algorithm to decide if we can put P (using translation and rotation) inside Q or not?
First think about the case of which P ...

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### Cut and Fold Polyhedron!

I have two convex polyhedra such that their sums of side areas are equal. It is true that I can cut one of them and flatten it on the plane, then fold the flattened polygon to reach the other ...

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### Polyhedron not circumscribed about a sphere

Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
My teacher ...

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### Light inside a polyhedron

I have two questions the same as Mostafa's Question:
Visibility of vertices in polyhedra
Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ ...

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### Visibility of vertices in polyhedra

Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...

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### Surface area of convex hull [duplicate]

Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?

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### Enumerating Tri-vertex transitive polyhedra n > 3 faces

How many unique vertex transitive polyhedra exist where each vertex has 3 incident edges
for polyhedra with n (= # faces) > 3 ?

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### What was the Question that led Euler to his Investigations on Polyhedra?

The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and that story is a must in every introduction to graph theory.
...

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### Regular cross-sections of a dodecahedron; analogous sections of 4-polytopes

One can intersect a dodecahedron with a plane and
obtain an equilateral triangle, a square, a regular pentagon,
a regular hexagon, and a regular decagon:
&...

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### Making a convex polyhedron with two sheets of paper

Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that $S$...

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### Seeking criteria for “threadable” pairs of centrosymmetric polyhedra

Let $A$ and $B$ be origin-centered centrosymmetric polyhedra in $\mathbb{R}^3$:
"for every point $(x, y, z)$ [...] there is an indistinguishable point $(-x, -y, -z)$."
Say that $A$ and $B$ are ...

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### Convex polyhedra jammed in $k$ disjoint holes

For a given convex polyhedron $P \subset \mathbb{R}^3$,
I was imagining finding the optimal "fixing" of $P$ in holes (or jamming them in "mud"),
which led to the following question.
First, scale $P$ ...

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### Did ancient mathematicians know Euler's characteristic for convex polyhedra?

The formula $V-E+F=2$ is so simple that I can't believe that it was really Euler (or perhaps Descartes) who first observed it (I mean the formula itself in some generality, not necessarily a valid ...

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### Polyhedra with minimal edge length

Given a fixed volume and fixed surface area I would like to construct polyhedra that minimize the total length of the edges. This seems like a straight-forward problem to solve by brute force for ...

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### “Derived” polyhedra and polytopes

The notion of derived polygon is natural and leads to remarkable convergence.
Start with a polygon, and replace it by locating a point on every edge
a fraction $\alpha$ between the two endpoints. For ...

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### Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...