All Questions
Tagged with euclidean-geometry polyhedra
15
questions
8
votes
1
answer
430
views
If we know the combinatorics of a polyhedron, and all but one of its dihedral angles, does that uniquely determine the remaining dihedral angle?
If we know the combinatorics of a polyhedron, and all but one of its dihedral angles, does that uniquely determine the remaining dihedral angle?
I’m happy to assume the polyhedron is simply connected, ...
- 2,866
15
votes
1
answer
477
views
Dividing a polyhedron into two similar copies
The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
9
votes
1
answer
164
views
Is it possible for the dihedral angles of a polyhedron to all grow simultaneously?
(Originally on MSE.)
Suppose $P$ and $Q$ are combinatorially equivalent non-self-intersecting polyhedra in $\mathbb{R}^3$, with $f$ a map from edges of $P$ to edges of $Q$ under said combinatorial ...
- 2,498
3
votes
1
answer
130
views
Bounding distance to an intersection of polyhedra
Let $P$ and $Q$ be polyhedra in ${\mathbb R}^m$ with a non-empty intersection. I believe there should exist a constant $C_{PQ}>0$ such that for any point $x\in {\mathbb R}^m$ the following ...
- 1,459
3
votes
2
answers
188
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 ...
- 1,459
4
votes
3
answers
972
views
Is there a pyramid with all four faces being right triangles? [closed]
If such a pyramid exists, could someone provide the coordinates of its vertices?
5
votes
1
answer
306
views
Embedding an icosahedron
A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group.
Let me start with the example of a ...
- 9,215
6
votes
0
answers
249
views
What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?
Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...
- 723
2
votes
1
answer
126
views
Distance between two polyhedra that takes incidence structure into account
Suppose that we have two polyhedra $P_1$ and $P_2$ in $\mathbb{R}^3$. I would like to define such a metric $\rho(P_1, P_2)$ that depends on several factors, but currently I don't know how to do it ...
- 123
96
votes
4
answers
5k
views
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}, \...
- 4,316
7
votes
3
answers
408
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 ...
- 101
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 $\frac{S^\prime}{S}$.
...
- 4,747
20
votes
1
answer
882
views
Tetrahedra passing through a hole
Assume a plane $P\subset\mathbb R^3$ has a hole $H$, and that the hole is topologically a compact disc. Being so, $P\setminus H$ does not separate the space. A regular tetrahedron $\sigma^3$ (of edge-...
- 4,747
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 ...
- 4,747
3
votes
1
answer
2k
views
Finding a minimum bounding sphere for a frustum
I have a frustum (truncated pyramid defined by six planes) and I need to compute a bounding sphere for this frustum that's as small as possible.
I can choose the centre of the sphere to be right in ...
- 31