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

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- key
- FillingspacewithhypercubesoftwosizesThepythagoreantilinginhigherdimensions
- type
- article
- date_added
- 2022-05-13
- date_published
- 2022-04-26

### 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 = {2022-04-26}, 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} }