The polyhedra tag has no usage guidance.

**24**

votes

**2**answers

1k views

### Can the unsolvability of quintics be seen in the geometry of the icosahedron?

Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...

**22**

votes

**3**answers

964 views

### 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 ...

**16**

votes

**1**answer

324 views

### 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., ...

**15**

votes

**1**answer

374 views

### Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...

**34**

votes

**1**answer

2k views

### Pach's “Animals”: What if the genus is positive?

Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...

**29**

votes

**3**answers

1k views

### About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$.
...

**14**

votes

**2**answers

939 views

### Is the tensor product of polyhedra a polyhedron?

Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...

**6**

votes

**0**answers

146 views

### 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 ...

**16**

votes

**2**answers

594 views

### “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 ...

**8**

votes

**3**answers

837 views

### On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1.
For example, it's easy to prove that ...

**9**

votes

**1**answer

315 views

### Complexity of the union of randomly rotated unit cubes

It is a remarkable fact that the union of congrent cubes
has only at most near-quadratic combinatorial complexity,
$O^*(n^2)$ for $n$ cubes, known to be almost tight.
This contrasts with the union of ...

**6**

votes

**1**answer

157 views

### Embedding of flat surfaces

Let $S$ be a orientable compact surface with a flat euclidean structure with conical singularities (cf. [T] for instance). Let also $\mathcal P$ be a polyhedral euclidean decomposition of $S$ (with ...

**14**

votes

**1**answer

480 views

### 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
...

**8**

votes

**1**answer

473 views

### 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 ...

**6**

votes

**2**answers

203 views

### 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 ...

**6**

votes

**0**answers

350 views

### Maximum volume convex body coverable by a unit square

Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected)
pieces (where $k=1$ means just the square). Your task is to construct the largest volume
convex body ...

**2**

votes

**0**answers

95 views

### 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 ...

**7**

votes

**1**answer

166 views

### Convex deltahedra in higher dimensions

There are eight convex polyhedra whose faces are equilateral triangles, so-called
deltahedra:
(Image from here)
Q. Have the equivalent higher-dimensional ...

**4**

votes

**2**answers

170 views

### 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 ...