Questions tagged [polyhedra]
The polyhedra tag has no usage guidance.
43
questions
26
votes
2
answers
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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
...
37
votes
2
answers
2k
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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 ...
27
votes
3
answers
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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 ...
10
votes
3
answers
2k
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 ...
96
votes
4
answers
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A curious relation between angles and lengths of edges of a tetrahedron
Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...
28
votes
5
answers
2k
views
Visibility of vertices in polyhedra
Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
26
votes
4
answers
894
views
Why do some uniform polyhedra have a "conjugate" partner?
While browsing through a list of uniform polynohedra, I noticed that the square of the circumradius $R_m$ of the small snub icosicosidodecahedron ($U_{32}$) with unit edge lengths is,
$$R_{32}^2 =\...
25
votes
3
answers
966
views
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 faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...
24
votes
1
answer
2k
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Building a genus-$n$ torus from cubes
I wonder if this has been studied:
What is the fewest number of unit cubes
from which one can build an $n$-toroid?
The cubes must be glued face-to-face,
and the boundary of the resulting object ...
19
votes
2
answers
2k
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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 ...
18
votes
2
answers
953
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 ...
18
votes
1
answer
654
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., $\mathbb{R}^d$ ...
11
votes
1
answer
518
views
How to correctly state Cauchy's rigidity theorem?
Cauchy's rigidity theorem is usually cites briefly as
Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent.
As a more formal generalization to general ...
5
votes
1
answer
311
views
Are the polyhedral cones the only examples of cones that remains closed when they are added to vector subspaces?
Let $C \subset \mathbb{R}^{n}$ be a closed convex cone. If one wants to know whether the linear map $T:\mathbb{R}^{n} \to\mathbb{R}^m$ sends the closed set $C$ to another closed one, $T(C)$, it is ...
44
votes
1
answer
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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 ...
34
votes
4
answers
2k
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 $\frac{S^\prime}{S}$.
...
24
votes
1
answer
1k
views
Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?
A six year old question,
Which unfoldings of the hypercube tile $3$-space?, has just been answered by
Moritz Firsching:
All $261$ unfoldings tile space!
So now we know:
For $d=2$, the unfolding of ...
20
votes
4
answers
906
views
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 ...
14
votes
0
answers
477
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 ...
13
votes
0
answers
464
views
What are the known convex polyhedra with congruent faces?
Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
12
votes
12
answers
1k
views
Database of integer edge lengths that can form tetrahedrons
Is there a collection of lists of six integer edge lengths that form a tetrahedron? Is there a computer program for generating such lists? I need to find approximately thirty such tetrahedral ...
10
votes
1
answer
609
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 ...
10
votes
1
answer
541
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 $B_1\cap B_2$...
10
votes
1
answer
423
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 ...
10
votes
1
answer
3k
views
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 ...
9
votes
1
answer
276
views
Thinnest covering of the plane by regular pentagons
Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons?
By covering I mean every point of the plane is covered.
By thinnest I mean the proportion of the plane covered ...
9
votes
2
answers
313
views
Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?
A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...
9
votes
0
answers
530
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 ...
9
votes
1
answer
407
views
The $32$-deg polynomial for the tetrahedron inscribed in the icosahedron?
This MO answer discusses this table involving the maximal side lengths of the five Platonic solids $T,C,O,D,I$ inscribed in the other solids,
This table is also found in Moritz Firsching's paper. I ...
7
votes
1
answer
294
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 ...
7
votes
2
answers
377
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 ...
7
votes
3
answers
401
views
Average caliper diameter (mean width) of a polyhedron
Define the caliper diameter of a polyhedron as follows:
Let $P_1$ and $P_2$ be two planes both of which are parallel to the x axis such that the perpendicular distance between $P_1$ and $P_2$ is the ...
6
votes
1
answer
252
views
Can a dodecahedron be deformed into a great stellated dodecahedron?
Can a convex regular dodecahedron be deformed into a great stellated dodecahedron while keeping all pentagons planar and all edges of nonzero length the whole time?
5
votes
3
answers
655
views
Alexandrov's generalization of Cauchy's rigidity theorem
Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions.
The relevant statement in the article is not linked to any source. The sources at the ...
5
votes
1
answer
237
views
Convex polyhedra with non-congruent faces
Question: Are there convex polyhedra wherein all faces are convex polygons with same area and perimeter and no two faces are mutually congruent?
Remarks: If the answer to above is "no", then,...
4
votes
2
answers
382
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 ...
4
votes
1
answer
123
views
Any visualization software for the intrinsic metric of a convex polyhedron?
I'd like to find a visual simulation of what it would be like to 'live' in a polyhedron with the intrinsic, piecewise-Euclidean length metric. Of course, to make it easier to visualize, I'd prefer to ...
3
votes
2
answers
164
views
Bounding distance to a polyhedron
I need to estimate the Euclidean distance from a point $x\in {\mathbb R}^m$ to a polyhedron $P\subset {\mathbb R}^m$ in terms of distances from $x$ to the tangent hyperplanes which define $P$. By a ...
2
votes
0
answers
90
views
Dodecahedron deformation II
(Follow-up to this question)
Can a dodecahedron be deformed into a great stellated dodecahedron while maintaining the number of dimensions each element occupies?
2
votes
1
answer
894
views
Does the Lebesgue Differentiation Theorem hold for regular polytopes?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
2
votes
1
answer
207
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Constructing a Polyhedron given areas of its faces
I want to visualize a set of data as a polyhedron in 3d space. Imagine set A includes areas of such polyhedron's faces. I assume the first step is to check if there exist a polyehdron by making sure ...
2
votes
3
answers
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non-convex Polytope definition
I have a simple question. I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$,
we can define a convex polytope in the following way:
$$P:= \Big\{ \sum_{u\in S} \mu_u u \,\Big| \, \...
1
vote
1
answer
524
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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.
...