Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1206.2231

Triangle Tiling I: The tile is similar to ABC or has a right angle

Michael Beeson
View original source ↗Read source document ↗Export metadata ↓

Abstract

An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile'". The tile may or may not be similar to ABC . This paper is the first of several papers, which together seek a complete characterization of the triples (ABC,N,T) such that ABC can be N -tiled by T . In this paper, we consider the case in which the tile is similar to ABC, the case in which the tile is a right triangle, and the case when ABC is equilateral. We use (only) techniques from linear algebra and elementary field theory, as well as elementary geometry and trigonometry. Our results (in this paper) are as follows: When the tile is similar to ABC, we always have "quadratic tilings'" when N is a square. If the tile is similar to ABC and is not a right triangle, then N is a square. If N is a sum of two squares, N = e^2 + f^2, then a right triangle with legs e and f can be N -tiled by a tile similar to ABC ; these tilings are called "biquadratic". If the tile and ABC are 30-60-90 triangles, then N can also be three times a square. If T is similar to ABC, these are all the possible triples (ABC, T, N) . If the tile is a right triangle, of course it can tile a certain isosceles triangle when N is twice a square, and in some cases when N is six times a square. Equilateral triangles can be 3-tiled and 6-tiled and hence they can also be 3n^2 and 6n^2 tiled for any n . We also discovered a family of tilings when N is 3 times a square, which we call the "hexagonal tilings." These tilings exhaust all the possible triples (ABC, T, N) in case T is a right triangle or is similar to ABC . Other cases are treated in subsequent papers.

Explore related subjects

math.MG

Keep this discovery

BibTeXRIS

Michael Beeson. 2012-06-05. Triangle Tiling I: The tile is similar to ABC or has a right angle. https://arxiv.org/abs/1206.2231

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

KEEP EXPLORING

Related papers

arXiv2026

Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles

Using a convex pentagonal monotile belonging to the Type 5 family, we investigate the relationships among the hat tile, turtle tile, and Tile$(1, 1)$. By applying Sugimoto's Perspective and Amfirifma's Perspective, we obtain four types of concave polygons, AH-tile, BH-tile, AT-tile, and BT-tile, each having Heesch number 1 under the conditions considered. We show that these polygons correspond to clusters used to generate the non-periodic tilings $\mathscr{T}_h$ and $\mathscr{T}_s$. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to $\textit{ASPmr}\{\text{A-tile}, \text{B-tile}\}$.

math.MG↗
arXiv2026

The mean distance to a simple closed curve on the sphere

Kimberling's Problem 10 asks for a simple closed curve of prescribed length $L$ (in particular, $L=4\pi$) on the unit sphere minimizing the mean geodesic distance $\mathcal{J}$ from a point of the sphere to the curve. For a positive integer $n$, put $\vartheta_{n}=\pi/(2n)$ and $L_{n}=2\pi/\sin\vartheta_{n}$. We show that the minimum of $\mathcal{J}$ over rectifiable simple closed curves of length at most $L_{n}$ equals $\vartheta_{n}-\tan(\vartheta_{n}/2)$, that it is attained only by curves of length exactly $L_{n}$, and that the sphere-filling ropes $\beta^{n,k}$ of Gerlach and von der Mosel attain it. Kimberling's case is $n=3$: at $L=4\pi$ the minimum is $\pi/6+\sqrt{3}-2=0.255649\ldots$, attained by an explicit six-arc curve and by its mirror image. For $L\le2\pi$ we determine $J(L)$, the infimum of $\mathcal{J}$ over curves of length $L$, exactly: it equals $\pi/2-L/(2\pi)$, attained precisely by the circles of length $L$. At the lengths $L_{n}$ we do not classify all minimizers, but show that every one of them bisects the sphere into two disks of area $2\pi$ and inradius $\vartheta_{n}$ whose inward collars have the largest possible area at every depth. The great circle is the only minimizer for $n=1$, and the $\beta^{n,k}$ are, up to congruence, the only ones of thickness at least $\sin\vartheta_{n}$. For arbitrary $L$ the function $J$ is nonincreasing, and together with the above this brackets it between two explicit values.

math.MG↗
arXiv2026

The topology of Gromov--Hausdorff space

We prove that the Gromov--Hausdorff space is homeomorphic to the Hilbert space. This paper is divided into four parts. In Part I, we construct an assignment of a full-support probability measure to every nonempty compact metric space that respects isometries and is continuous for simultaneous Hausdorff convergence of the spaces and weak convergence of the measures. In Part II, we use these measures to construct finite-dimensional local models whose induced pseudometrics approximate the original distances uniformly and whose norms and point maps vary continuously up to orthogonal changes of coordinates. In Part III, we use the local models to prove that the Gromov--Hausdorff space is an absolute retract for all metrizable spaces. In Part IV, we establish a discrete approximation property and conclude that the space of isometry classes of nonempty compact metric spaces is homeomorphic to the real separable infinite-dimensional Hilbert space.

math.MG↗