Filling space with hypercubes of two sizes -- The pythagorean tiling in higher dimensions

• Published in 2022
• Added on 2022-05-13

We construct a unilateral lattice tiling of $\mathbb{R}^n$ into hypercubes of two differnet side lengths $p$ or $q$. This generalizes the Pythagorean tiling in $\mathbb{R}^2$. We also show that this tiling is unique up to symmetries, which proves a variation of a conjecture by B\"olcskei from 2001. For positive integers $p$ and $q$ this tiling also provides a tiling of $(\mathbb{Z}/(p^n+q^n)\mathbb{Z})^n$.

Links

• http://arxiv.org/abs/2204.11529v2
• http://arxiv.org/pdf/2204.11529v2

Other information

BibTeX entry

@article{FillingspacewithhypercubesoftwosizesThepythagoreantilinginhigherdimensions,
	key = {FillingspacewithhypercubesoftwosizesThepythagoreantilinginhigherdimensions},
	type = {article},
	title = {Filling space with hypercubes of two sizes -- The pythagorean tiling in  higher dimensions},
	author = {Jakob F{\"{u}}hrer},
	abstract = {We construct a unilateral lattice tiling of {\$}\mathbb{\{}R{\}}^n{\$} into hypercubes of
two differnet side lengths {\$}p{\$} or {\$}q{\$}. This generalizes the Pythagorean tiling
in {\$}\mathbb{\{}R{\}}^2{\$}. We also show that this tiling is unique up to symmetries,
which proves a variation of a conjecture by B\"olcskei from 2001. For positive
integers {\$}p{\$} and {\$}q{\$} this tiling also provides a tiling of
{\$}(\mathbb{\{}Z{\}}/(p^n+q^n)\mathbb{\{}Z{\}})^n{\$}.},
	comment = {},
	date_added = {2022-05-13},
	date_published = {2022-12-07},
	urls = {http://arxiv.org/abs/2204.11529v2,http://arxiv.org/pdf/2204.11529v2},
	collections = {easily-explained,fun-maths-facts,geometry},
	url = {http://arxiv.org/abs/2204.11529v2 http://arxiv.org/pdf/2204.11529v2},
	year = 2022,
	urldate = {2022-05-13},
	archivePrefix = {arXiv},
	eprint = {2204.11529},
	primaryClass = {math.CO}
}