Computer geometry: Rep-tiles with a hole

Christoph Bandt; Dmitry Mekhontsev

Home/Archive/arXiv

arXiv · 1811.03929

Computer geometry: Rep-tiles with a hole

Christoph Bandt·Dmitry Mekhontsev

View original source ↗Read source document ↗Export metadata ↓

Abstract

A cube is an 8-rep-tile: it is the union of eight smaller copies of itself. Is there a set with a hole which has this property? The computer found an interesting and complicated solution, which then could be simplified. We discuss some problems of computer-assisted research in geometry.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Christoph Bandt, Dmitry Mekhontsev. 2018-11-09. Computer geometry: Rep-tiles with a hole. https://arxiv.org/abs/1811.03929

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The disjoint disks property for Busemann $G$-spaces

We prove that every finite-dimensional Busemann $G$-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere $L=S(c,r)$, we show that every embedded arc contained in an exact distance level is a homotopical $Z_2$-set in $L$. It follows that $L$ has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for $L\times\mathbb R$, and a local avoidance argument at the center yields DDP for the ambient $G$-space. Since finite-dimensional Busemann $G$-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.

math.MG↗

arXiv2026

Every Compact Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Space

Let $(\mathcal{M},d_{\mathrm{GH}})$ denote the Gromov-Hausdorff space of isometry classes of nonempty compact metric spaces. We prove that every nonempty compact metric space is isometrically embeddable into $(\mathcal{M},d_{\mathrm{GH}})$. More precisely, for every $D>0$ and every nonempty compact metric space $K$ with $\operatorname{diam} K\le D$, we realize the space of all $1$-Lipschitz functions on $K$ with values in $[0,D]$ as a family of metrics on a fixed Cantor space. Under this realization, the Gromov-Hausdorff distance agrees exactly with the uniform distance between functions, and each resulting metric space has diameter at most $76D$. We also construct finite approximations for which the Gromov-Hausdorff distance is given by an exact formula, together with a uniform approximation estimate.

math.MG↗

arXiv2026

Measure contraction property on isometric leaves and monotone fibres

For finite measures with positive densities on convex Euclidean supports, we prove that $MCP(κ,N)$ passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.

math.MG↗