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Results tagged with convex-polytopes
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user 39521
Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming
24
votes
Accepted
Rational inscribed realization of the regular dodecahedron
An example
Yes, here is a list of rational coordinates lying on the unit sphere, the convex hull of which is combinatorially equivalent to a regular dodecahedron. This polyhedron is invariant under r …
- 12.4k
5
votes
Accepted
Edges of the contact polytope of the Leech lattice
Using the unimodular scaling of the Leech lattice, the length of each minimal vector is $\sqrt{4}$. Fixing a particular minimal vector $u$, the remaining minimal vectors $v$ are:
1 vector $v$ with $\ …
- 12.4k
8
votes
Accepted
Is a polytope that has in-spheres for faces of all dimensions already regular?
This is true in all dimensions, and can be proved by induction (on $d$) applied to the following (slightly stronger) hypothesis:
Theorem: If $P$ is a convex $d$-polytope with $k$-in-spheres for all $ …
- 12.4k
4
votes
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
The following comment in the question intrigued me:
In fact, it's possible to show that the linear symmetries of
$\mathbb{R}^6$ that preserve the Cayley-Menger determinant form the
Weyl group $D_6$, …
- 12.4k