2 Triangle Tiling III: The Triquadratic Tilings

International Journal of Computational Geometry & Applications, 2011

A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of sidelength n for every n in the set. From Ref. 8 we know that ℕ itself tiles the plane. From that and Ref. 9 we know that the set of even numbers tiles the plane while the set of odd numbers doesn't. In this paper we explore the nature of this property. We show, for example, that neither tiling nor non-tiling is preserved by superset. We show that a set with one or three odd numbers may tile the plane—but a set with two odd numbers can't. We find examples of both tiling and non-tiling sets that can be partitioned into tiling sets, non-tiling sets or a combination. We show that any set growing faster than the Fibonacci numbers cannot tile the plane.

2012

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 four 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.

Journal of Combinatorial Theory, Series A, 1988

We will always take S to be finite and call a set congruent to it a tile. We will denote the ith tile in a tiling by Si and its smallest and largest elements by xi and zi, where the order is by smallest element, and let A, denote Ui, j Sj. The set (0, 2, 3, 5) can easily be seen to tile Z but not N. On the other hand, there is the open Question. If S tiles N must it tile some finite interval [0, n]? In the following we will consider the case ISJ = 3. With appropriate changes in terminology, [ 11 and later [2] showed that every 3-set of R tiles R. From the latter proof it follows that every 3-set tiles Z. In [3] an algorithm is given which yields the stronger result that every 3-set tiles N. An elegant non-constructive argument in shows that this algorithm applied to S = (0, n, m + H} tiles an interval of length at most 3.2"+"-' and suggests that the correct length might be O(nm). In this paper we examine this algorithm and show that it gives fairly orderly tilings provided m > 3n. In this case we give an exact description of the resulting tilings and their lengths which are in fact just about 3nm. The patterns which arise also give alternate tilings for smaller values of m. This provides a constructive proof that every 3-set tiles a finite interval. Using another approach, [5] gives a tiling of an interval of length at most 6(n + m) and shows that it is shortest possible in certain cases.

2012

An $N$-tiling of triangle $ABC$ by triangle $T$ (the `tile') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(\alpha,\beta,\gamma)$, and sides $(a,b,c)$. This paper takes up the case when $3\alpha + 2\beta = \pi$. Then there are (as was already known) exactly five possible shapes of $ABC$: either $ABC$ is isosceles with base angles $\alpha$, $\beta$, or $\alpha+\beta$, or the angles of $ABC$ are $(2\alpha,\beta,\alpha+\beta)$, or the angles of $ABC$ are $(2\alpha, \alpha, 2\beta)$. In each of these cases, we have discovered, and here exhibit, a family of previously unknown tilings. These are tilings that, as far as we know, have never been seen before. We also discovered, in each of the cases, a Diophantine equation involving $N$ and the (necessarily rational) number $s = a/c$ that has solutions if there is a tiling using tile $T$ of some $ABC$ not similar to $T$. By means of these Diophantine equation...

Geometriae Dedicata, 1987

The paper discusses homeomorphic types of (periodic) filings of the plane in terms of their associated Delaney symbol. Such a symbol consists of a (finite) set ~ on which three involutions ao,al and tr2 act from the right such that troa ~ = a2ao and there are two maps mol ,m12 :~ ~ ~ satisfying certain compatibility conditions. It is shown how the barycentric subdivision of a tiling can be used to define its Delaney symbol and that the symbol characterizes the tiling up to (equivariant) homeomorphisms. Furthermore, it is shown how properties of the tiling can be recognized from corresponding properties of the symbol and how this technique can be used to enumerate various types of tilings with specific properties. If necessary, this enumeration can be done by appropriate computer programs. Among other results, we have been able to vindicate the results by Griinbaum et al., announced in [8]. Finally, some recursive enumeration formulas, based on the Delaney symbol technique,, are stated.

Journal of Combinatorial Theory, Series A, 1990

2012

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. We wish to understand possible tilings by completely characterizing the triples (ABC, T, N) such that ABC can be N-tiled by T. In particular, this understanding should enable us to specify for which N there exists a tile T and a triangle ABC that is N-tiled by T ; or given N , to determine which tiles and triangles can be used for N-tilings; or given ABC, to determine which tiles and N can be used to N-tile ABC. This is our fourth paper on this subject. In this paper, we take up the last remaining case: when ABC is not similar to T , and T has one angle equal to 120 • , and T is not isosceles (although ABC can be isosceles or even equilateral). Here is our result: If there is such an N-tiling, then the smallest angle of the tile is not a rational multiple of π. In total there are six tiles with vertices at the vertices of ABC. If the sides of the tile are (a, b, c), then there must be at least one edge relation of the form jb = ua + vc or ja = ub + vc, with j, u, and v all positive. The ratios a/c and b/c are rational, so that after rescaling we can assume the tile has integer sides, which by virtue of the law of cosines satisfy c 2 = a 2 + b 2 + ab. A simple unsolved specific case is when ABC is equilateral and (a, b, c) = (3, 5, 7). The techniques used in this paper, for the reduction to the integer-sides case, involve linear algebra, elementary field theory and algebraic number theory, as well as geometrical arguments. Quite different methods are required when the sides of the tile are all integers.

European Journal of Combinatorics

We solve a problem of R. Nandakumar by proving that there is no tiling of the plane with pairwise noncongruent triangles of equal area and equal perimeter. We also show that any tiling of a convex polygon with more than three sides with finitely many triangles contains a pair of triangles that share a full side. * EPFL, Lausanne and MIPT, Moscow.

Geometriae Dedicata, 1992

arXiv (Cornell University), 2016

In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert's Nullstellensatz we obtain a necessary condition for tiling n-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of n-space by translates of a cluster V of prime size then there is a lattice tiling by V as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster V is lattice if V generates n-space.