6
votes
1answer
359 views

Higher dimensional Rubik's cube group

Since "cubes" with higher dimension than three exist I think it's natural to ask for higher dimensional Rubik's cubes. These so called hypercubes don't seem to have been described from a group ...
1
vote
0answers
74 views

Group actions on polytopes in indefinite integer lattices

Is anything at all known about polytopes in indefinite integer lattices? I'm interested in lattice automorphisms which preserve certain polytopes of "high regularity" (e.g. cones). As a first step, ...
0
votes
3answers
271 views

The symmetry group of $\mathbb Z^d$

Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm. I would like to write $\mathbb Z^d = G / H$, where $G$ is the ...
11
votes
2answers
706 views

covers of $Z^\infty$

Is it possible to cover $Z^\infty$ (the infinite direct sum of $Z$'s with the $l_1$-metric) by a finite set of collections of subsets $U^0,...,U^n$ such that each collection $U^i$ consists of ...
6
votes
3answers
673 views

Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance Part II

For some context see Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance As per Noah's answer and JBL's comment this was false as stated. However, I think the following ...
9
votes
1answer
418 views

Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance

Problem: Consider a random walk on the lattice $\mathbb{Z}^2$ where on each iteration a particle either stays at its current location or moves to a neighboring vertex with probability 1/5. We start ...