Some Generalizations of the Pinwheel Tiling

We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but can be forced through local matching rules. In a countable number of cases, the tiles appear in a finite number of orientations but an infinite number of sizes, all within a set range, while in an uncountable number of cases both the number of sizes and the number of orientations is infinite. Received April 9, 1996, and in revised form September 16, 1996.     

Circular Symmetry of Pinwheel Diffraction

A method is given for explicitly determining the autocorrelation of the pinwheel tiling by use of the substitution system generating the tiling. Using this a new proof of the circular symmetry of the diffraction of the pinwheel tiling is given. Communicated by Jean Bellissard Submitted: 8/10/2004 Accepted: 21/11/2005     

铺瓷砖问题的推广

铺瓷砖问题是数学模型中著名的问题,在研究了铺瓷砖问题的基础上,对其进行了推广,并给出了各种推广情形的结果.     

A Double Tiling of Triangles and Regular Hexagons

A tiling of triangles and regular hexagons, which wraps around a focal point and covers the plane twice, is investigated using both synthetic triangle geometry and complex numbers. Received February 12, 1999, and in revised form October 25, 1999. Online publication May 16, 2000.     

A Homeomorphism Invariant for Substitution Tiling Spaces

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of ech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like tiling spaces. We also introduce a module structure on cohomology which is very convenient as well as of intuitive value.     

Natural Tiling, Lattice Tiling and Lebesgue Measure of Integral Self-Affine Tiles

In the existing theory of self-affine tiles, one knows thatthe Lebesgue measure of any integral self-affine tile correspondingto a standard digit set must be a positive integer and everyintegral self-affine tile admits some lattice Zⁿ as a translationtiling set of Rⁿ. In this paper, we give algorithms to evaluatethe Lebesgue measure of any such integral self-affine tile Kand to determine all of the lattice tilings of Rⁿ by K. Moreover,we also propose and determine algorithmically another type oftranslation tiling of Rⁿ by K, which we call natural tiling.We also provide an algorithm to decide whether or not Lebesguemeasure of the set K (K+j), j–Zⁿ, is strictly positive.     

A Cube Tiling of Dimension Eight with No Facesharing

   Abstract. A cube tiling of eight-dimensional space in which no pair of cubes share a complete common seven-dimensional face is constructed. Together with a result of Perron, this shows that the first dimension in which such a tiling can exist is seven or eight.     

Isohedral Polyomino Tiling of the Plane

A polynomial time algorithm is given for deciding, for a given polyomino P , whether there exists an isohedral tiling of the Euclidean plane by isometric copies of P . The decidability question for general tilings by copies of a single polyomino, or even periodic tilings by copies of a single polyomino, remains open. Received June 23, 1997, and in revised form April 6, 1998.     

Growth Rates in the Quaquaversal Tiling

Conway and Radin's ``quaquaversal' tiling of R ³ is known to exhibit statistical rotational symmetry in the infinite volume limit. A finite patch, however, cannot be perfectly isotropic, and we compute the rates at which the anisotropy scales with size. In a sample of volume N , tiles appear in O(N ^1/6 ) distinct orientations. However, the orientations are not uniformly populated. A small (O(N ^1/84 ) ) set of these orientations account for the majority of the tiles. Furthermore, these orientations are not uniformly distributed on SO(3) . Sample averages of functions on SO(3) seem to approach their ergodic limits as N ^-1/336 . Since even macroscopic patches of a quaquaversal tiling maintain noticeable anisotropy, a hypothetical physical quasicrystal whose structure was similar to the quaquaversal tiling could be identified by anisotropic features of its electron diffraction pattern. Received October 19, 1998, and in revised form March 11, 1999.     

Tiling space with the aid of the holomorph

The notion of a factorization of a group is generalized and a method is presented for obtaining new factorizations from old ones. The results are applied to obtain new fillings of the lattice spaces Z, Z ⊕ Z and the cube.     

A Version of the Poincare-Birkhoff-Witt Theorem

In this note I shall prove that if L is a finite-dimensionalLie algebra over a field F of characteristic zero which is generatedas an algebra by a set of elements {e₁, e₂,...,e_k}, then theuniversal enveloping algebra U(L) of L is linearly generatedby monomials spanned by the elements {e_i} of an a priori boundedwidth. As an application, a criterion of Kostant for a leftideal of U(L) to be of finite codimension is proved by purelyalgebraic means.     

A Cube Tiling of Dimension Eight with No Facesharing

Abstract. A cube tiling of eight-dimensional space in which no pair of cubes share a complete common seven-dimensional face is constructed. Together with a result of Perron, this shows that the first dimension in which such a tiling can exist is seven or eight.     

正规Tiling的邻居系

该文借助Tiling的拓扑图和边邻居图, 给出了正规Tiling 中两个Tiles 拥有相同的邻居系的一个充分必要条件, 并利用此条件证明了正规Tiling 中不可能存在三个不同的Tiles 具有相同的邻居系, 从而在一定程度上回答了Grünbaum 的一个公开问题.     

Tiling the torus and other space forms

We consider graphs on two-dimensional space forms which are quotient graphs /F, where is an infinite, 3-connected, face, vertex, or edge transitive planar graph andF is a subgroup of Aut(), all of whose elements act freely on . The enumeration of quotient graphs with transitivity properties reduces to computing the normalizers in Aut() of the subgroupsF. Results include: all isogonal toriodal polyhedra belong to the two families found by Grünbaum and Shephard; there are no transitive graphs on the Möbius band; there is a graph on the Klein bottle whose automorphism group acts transitively on its faces, edges, and vertices.This paper is an expanded version of a lecture presented to the Conference on Combinatorial Geometry, Oberwolfach, Germany, September 1984.     

A Semigroup Version of the Isoperimetric Inequality

In this article we extend the heat semigroup version of the isoperimetric inequality in Rⁿ established by M. Ledoux. To this purpose, the notion of the perimeter of a Caccioppoli set introduced by E. De Giorgi yields the appropriate measure theoretical background. In the proofs we use properties of the heat semigroup as well as its explicit integral representation.     

A Refined Version of the Lang-Trotter Conjecture

Let E be an elliptic curve defined over the rational numbersand r a fixed integer. Using a probabilistic model consistentwith the Chebotarev density theorem for the division fieldsof E and the Sato–Tate distribution, Lang and Trotterconjectured an asymptotic formula for the number of primes upto x which have Frobenius trace equal to r, where r is a fixedinteger. However, as shown in this note, this asymptotic estimatecannot hold for all r in the interval with a uniform bound for the error term, becausean estimate of this kind would contradict the Chebotarev densitytheorem as well as the Sato–Tate conjecture. The purposeof this note is to refine the Lang–Trotter conjecture,by taking into account the "semicircular law," to an asymptoticformula that conjecturally holds for arbitrary integers r inthe interval , with auniform error term. We demonstrate consistency of our refinementwith the Chebotarev density theorem for a fixed division field,and with the Sato–Tate conjecture. We also present numericalevidence for the refined conjecture.     

On the Universal Tiling Conjecture in Dimension One

We show that the spectral-tile implication in the Fuglede conjecture in dimension 1 is equivalent to a Universal Tiling Conjecture and also to similar forms of the same implication for some simpler sets, such as unions of intervals with rational or integer endpoints.