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

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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 $(\mathbb{Z}/(p^n+q^n)\mathbb{Z})^n$.

## Other information

key
FillingspacewithhypercubesoftwosizesThepythagoreantilinginhigherdimensions
type
article
date_added
2022-05-13
date_published
2022-09-14

### 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 = {2022-09-14},
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}
}