Results 1 to 5 of about 5 (5)

An aperiodic monotile that forces nonperiodicity through dendrites

Bulletin of the London Mathematical Society, Volume 52, Issue 5, Page 942-959, October 2020., 2020
Abstract We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar–Taylor monotile, but can be
Michael Mampusti, Michael F. Whittaker
wiley   +1 more source

A generalised Euler–Poincaré formula for associahedra

Bulletin of the London Mathematical Society, Volume 51, Issue 1, Page 181-192, February 2019., 2019
Abstract We derive a formula for the number of flip‐equivalence classes of tilings of an n‐gon by collections of tiles of shape dictated by an integer partition λ. The proof uses the Euler–Poincaré formula; and the formula itself generalises the Euler–Poincaré formula for associahedra.
Karin Baur, Paul P. Martin
wiley   +1 more source

Undecidability of the Spectral Gap

Forum of Mathematics, Pi, 2022
We construct families of translationally invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem.
Toby Cubitt, David Perez-Garcia, Michael M. Wolf   +2 more
doaj   +1 more source

Combinatorial and harmonic-analytic methods for integer tilings

Forum of Mathematics, Pi, 2022
A finite set of integers A tiles the integers by translations if $\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $A\oplus B=\mathbb {Z}_M$ for some $M\in \mathbb {N}$ and $B ...
Izabella Łaba, Itay Londner
doaj   +1 more source

Tilings, sub-tilings, and spectral sets on p-adic space

Open Mathematics
In this article, we provide a characterization of tilings, sub-tilings, and spectral sets on the pp-adic space Qpd{{\mathbb{Q}}}_{p}^{d}. Our methods are based on pp-adic distributions and pp-adic Fourier analysis. We also obtained a sufficient condition
Kadir Mamateli
doaj   +1 more source