Closed form for the top of the boat–star tiling spectrum
Let $\Gamma_\bigstar$ be the tile graph of the boat–star tiling: Penrose's pentagon tiling P1, generated by the six-tile substitution of Grünbaum and Shephard (Definition 3.1 of the source), with one vertex per tile and tiles adjacent when they share an edge. Let $\Sigma_\bigstar$ be the spectrum of its graph Laplacian $\Delta = D - A$. Its bottom, $\min\Sigma_\bigstar = 0$, follows from elementary arguments; its top is not as easy to compute.
Problem. Can $\max\Sigma_\bigstar$ be computed in closed form?
Source: Damanik, Embree, Fillman and Mei, Question 8.3 in “Discontinuities of the integrated density of states for Laplacians associated with Penrose and Ammann–Beenker tilings”, Experimental Mathematics 33 (2024); arXiv:2209.01443.
Imported from arxiv.org/abs/2209.01443.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DS, MP
- Keywords
- aperiodic order, graph Laplacian, Penrose tiling, spectral edges
- Literature checked
- Oct. 1, 2026 by clemens-admin
References
@article{damanik2024,
author = {Damanik, David and Embree, Mark and Fillman, Jake and Mei, May},
title = {Discontinuities of the integrated density of states for {L}aplacians associated with {P}enrose and {A}mmann--{B}eenker tilings},
journal = {Experimental Mathematics},
volume = {33},
number = {4},
pages = {588--610},
year = {2024},
doi = {10.1080/10586458.2023.2206589},
eprint = {2209.01443},
archivePrefix = {arXiv}
}
@book{grunbaum1987,
author = {Gr{\"u}nbaum, Branko and Shephard, G. C.},
title = {Tilings and Patterns},
publisher = {W. H. Freeman and Company},
address = {New York},
year = {1987}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00023, "Closed form for the top of the boat–star tiling spectrum", clopen.solutions.