David Feldman, James Propp, Sinai Robins
We use Fourier methods to prove that if $n > 1$ translates of sublattices of $Z^d$ tile $Z^d$, and all the sublattices are Cartesian products of arithmetic progressions, then two of the tiles must be translates of each other. This is a multi-dimensional generalization of the Mirsky-Newman Theorem.
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Our earlier article proved that if $n > 1$ translates of sublattices of $Z^d$ tile $Z^d$, and all the sublattices are Cartesian products of arithmetic progressions, then two of the tiles must be translates of each other. We re-prove this Theorem, this time using generating functions. We also show that for $d \geq 1$, not every finite tiling of $Z^d$ by lattices can be obtained from the trivial tiling by the process of repeatedly subdividing a tile into sub-tiles that are translates of one another.
When $\mathbb{Z}^d$ is represented as a finite disjoint union of translated integer sublattices, the translated sublattices must possess some special properties. Such a representation is called a \emph{lattice tiling}. We develop a theoretical framework, based on multiple residues and dual groups, to provide a set of necessary and sufficient conditions for such a lattice tiling to exist. We also investigate the question of when a lattice tiling must possess at least two translated sublattices which are translates of one another.
Tiling of Strip Lattices and Asymptotics
本文研究等边三角形条带格的铺砌，建立两类三角形组合铺砌方式数与 Pell 数的联系，并首次分析所有铺砌所需瓦片的总数及分类数量。
We consider the problem of counting and classifying domino tilings of a quadriculated torus. The counting problem for rectangles was studied by Kasteleyn and we use many of his ideas. Domino tilings of planar regions can be represented by height functions; for a torus given by a lattice L, these functions exhibit arithmetic L-quasiperiodicity. The additive constants determine the flux of the tiling, which can be interpreted as a vector in the dual lattice (2L)*. We give a characterization of the actual flux values, and of how corresponding tilings behave. We also consider domino tilings of the…
In a recent preprint, Lai and Rohatgi compute the generating functions of lozenge tilings of "quartered hexagons with dents" by applying the method of "graphical condensation". The purpose of this note is to exhibit how (a generalization of) Theorems 2.1 and 2.2 in Lai and Rohatgi's preprint can be achieved by the Lindström--Gessel--Viennot method of non--intersecting lattice paths and a certain determinant evaluation.
In a recent preprint, Lai worked out the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents" which differ only in width. Lai achieved this by using "graphical condensation" (i.e., application of a certain Pfaffian identity to the weighted enumeration of matchings). The purpose of this note is to exhibit how this can be done by the Lindström--Gessel--Viennot method for nonintersecting lattice paths in a quite simple way. Basically the same observation, but restricted to mere enumeration (i.e., all weights of lozenge tilings are equal to $1$), i…
In 2003, Ciucu presented a unified way to enumerate tilings of lattice regions by using a certain Reduction Theorem (Ciucu, Perfect Matchings and Perfect Powers, Journal of Algebraic Combinatorics, 2003). In this paper we continue this line of work by investigating new families of lattice regions whose tilings are enumerated by perfect powers or products of several perfect powers. We prove a multi-parameter generalization of Bo-Yin Yang's theorem on fortresses (B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, MA, 1991). On the square lattice with zigzag paths, we consider two particul…
In a recent preprint, Lai showed that the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents", which differ only in width, factors nicely, and the same is true for the quotient of generating functions of weighted lozenge tilings of two "quarter hexagons with lateral dents". Lai achieved this by using "graphical condensation" (i.e., application of a certain Pfaffian identity to the weighted enumeration of matchings). The purpose of this note is to exhibit how this can be done by the Lindström--Gessel--Viennot method for nonintersecting lattice …
The collection CL(T) of nonempty convex sublattices of a lattice T ordered by bi-domination is a lattice. We say that T has the fixed point property for convex sublattices (CLFPP for short) if every order preserving map f from T to CL(T) has a fixed point, that is x > f(x) for some x > T. We examine which lattices may have CLFPP. We introduce the selection property for convex sublattices (CLSP); we observe that a complete lattice with CLSP must have CLFPP, and that this property implies that CL(T) is complete. We show that for a lattice T, the fact that CL(T) is complete is equivalent to the f…
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.
We solve and generalize an open problem posted by James Propp (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999) on the number of tilings of quasi-hexagonal regions on the square lattice with every third diagonal drawn in. We also obtain a generalization of Douglas' Theorem on the number of tilings of a family of regions of the square lattice with every second diagonal drawn in.
We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not necessarily local) transformations called {\em flips}. This study allows us to formulate an exhaustive generation algorithm and a uniform random sampling algorithm. We finally extend these results to other types of tilings (calisson tilings, tilings with bicolored Wang tiles).
In this paper we show the combinatorial structure of $\mathbb{Z}^2$ modulo sublattices selfsimilar to $\mathbb{Z}^2$. The tool we use for dealing with this purpose is the notion of association scheme. We classify when the scheme defined by the lattice is imprimitive and characterize its decomposition in terms of the decomposition of the gaussian integer defining the lattice. This arise in the classification of different forms of tiling $\mathbb{Z}^2$ by lattices of this type.
On the Lattice Tilings by Lee Balls: Q-Polynomials, Group-Ring Equations, and a Decision Algorithm
本文针对Golomb-Welch猜想，发展了基于Q多项式和群环方程的两种方法，推导了李球格铺砌的充要条件与新必要条件，通过算法排除了绝大多数(n,r)对，仅余(35,3)和(805,3)。
This paper studies the dimer model on the dual graph of the square-octagon lattice, which can be viewed as the domino tilings with impurities in some sense. In particular, under a certain boundary condition, we give an exact formula representing the probability of finding an impurity at a given site in a uniformly random dimer configuration in terms of simple random walks on the square lattice.
On Boolean sublattices of finite partition lattices
研究有限全域\(U\)的划分格\(Part(U)\)的极大布尔子格，通过树的子树划分等方法构建，证明其含最小和最大元，刻画含\(0\)布尔子格，表明超图为超树时可扩展，得出高度至少为三的划分格有极大布尔子格的结论。
We first show that the tilings of a general domain form a lattice which we then undertake to decompose and generate without any redundance. To this end, we study extensively the relatively simple case of hexagons and their deformations. We show that general domains can be broken up into hexagon-like parts. Finally we give an algorithm to generate exactly once every element in the lattice of the tilings of a general domain.
Combinatorial connections in snake graphs: Tilings, lattice paths, and perfect matchings
本文引入三角蛇图，建立其路径、底层蛇图完美匹配与铺砌之间的双射，并据此给出直蛇图完美匹配数的 Hankel 行列式表达及 Fibonacci、Pell 序列的行列式来源。
Each straight infinite line defined by two vertices of a finite square point lattice contains (covers) these two points and a - possibly empty - subset of points that happen to be collinear to these. This work documents vertex subsets of minimum order such that the sum of the infinite straight lines associated with the edges of their complete subgraph covers the entire set of vertices (nodes). This is an abstraction to the problem of sending a light signal to all stations (receivers) in a square array with a minimum number of stations also equipped with transmitters to redirect the light to ot…
It is well-known that the question of whether a given finite region can be tiled with a given set of tiles is NP-complete. We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right tromino alone. In the process, we show that Monotone 1-in-3 Satisfiability is NP-complete for planar cubic graphs. In higher dimensions, we show NP-completeness for the domino and straight tromino for general regions on the cubic lattice, and for simply-connected regions on the four-dimensional hypercubic lattice.
On Lattice Tilings of Asymmetric Limited-Magnitude Balls $\cB(n,2,m,m-1)$
本文研究非对称限幅错误球的格铺砌存在性，针对 $t=2$、$\km=\kp-1$ 的情形推导出 $m$ 与 $n$ 的必要条件，并证明 $m=2,3$ 时不存在相应格铺砌。
Let $P$ be a partially ordered set. If the Boolean lattice $(2^{[n]},\subset)$ can be partitioned into copies of $P$ for some positive integer $n$, then $P$ must satisfy the following two trivial conditions: (1) the size of $P$ is a power of $2$, (2) $P$ has a unique maximal and minimal element. Resolving a conjecture of Lonc, it was shown by Gruslys, Leader and Tomon that these conditions are sufficient as well. In this paper, we show that if $P$ only satisfies condition (2), we can still almost partition $2^{[n]}$ into copies of $P$. We prove that if $P$ has a unique maximal and minimal elem…
Cao & Yuan obtained a Blichfeldt-type result for the vertex set of the edge-to-edge tiling of the plane by regular hexagons. Observing that every Archimedean tiling is the union of translates of a fixed lattice, we take a more general viewpoint and investigate basic questions for such point sets about the homogeneous and inhomogeneous problem in the Geometry of Numbers. The Archimedean tilings nicely exemplify our results.
We use the subgraph replacement method to investigate new properties of the tilings of regions on the square lattice with diagonals drawn in. In particular, we show that the centrally symmetric tilings of a generalization of the Aztec diamond are always enumerated by a simple product formula. This result generalizes the previous work of Ciucu (1997) and Yang (1992) about symmetric tilings of the Aztec diamond. We also use our method to prove a closed-form product formula for the number of centrally symmetric tilings of a quasi-hexagon.
The problem of determining which infinite lattices are (isomorphic to) sublattices of free lattices is in general unsolved and extremely difficult. In this note, we reduce the problem by proving that all locally finite sublattices of free lattices are countable by using a result from Baldwin, Berman, Glass and Hodges on free algebras.
Periodic Tilings of Cardinality Twice a Prime or Nine in Arbitrary Dimension
本文证明任意维度中基数为奇素数两倍或九的有限平移平铺块总存在周期平铺补集，结合互素伸缩、单位根消失与周期二染色方法，并给出可平铺性判定算法。
Conway and Lagarias observed that a triangular region T(m) in a hexagonal lattice admits signed tiling by three-in-line polyominoes (tribones) if and only if m=9d-1 or m=9d for some integer d. We apply the theory of Groebner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m=dn^2-1 or m=dn^2 for some integer d. Explicit description of the Groebner basis allows us to calculate the "Groebner discrete volume" of a lattice region by applying the division algorithm to its `Newton polynomial'. Among immediate consequences is a description…
In 1945-46, C. L. Siegel proved that an $n$-dimensional lattice $Λ$ of determinant ${\rm det}(Λ)$ has at most $m^{n^2}$ different sublattices of determinant $m\cdot {\rm det}(Λ)$. In 1997, the exact number of the different sublattices of index $m$ was determined by Baake. This paper presents a systematic treatment for counting the sublattices and deduces a formula for the number of the sublattice classes of determinant $m\cdot {\rm det}(Λ)$.
Grätzer asked in 1971 for a characterization of sublattices of Tamari lattices (associahedra). A natural candidate was coined by McKenzie in 1972 with the notion of a bounded homomorphic image of a free lattice---in short, bounded lattice. Urquhart proved in 1978 that every associahedron is bounded (thus so are its sublattices). Geyer conjectured in 1994 that every finite bounded lattice embeds into some associahedron. We disprove Geyer's conjecture, by introducing an infinite collection of lattice-theoretical identities that hold in every associahedron, but not in every finite bounded lattice…
A simple but elegant result of Rival states that every sublattice $L$ of a finite distributive lattice $\mathcal{P}$ can be constructed from $\mathcal{P}$ by removing a particular family $\mathcal{I}_L$ of its irreducible intervals. Applying this in the case that $\mathcal{P}$ is a product of a finite set $\mathcal{C}$ of chains, we get a one-to-one correspondence $L \mapsto \mathcal{D}_\mathcal{P}(L)$ between the sublattices of $\mathcal{P}$ and the preorders spanned by a canonical sublattice $\mathcal{C}^\infty$ of $\mathcal{P}$. We then show that $L$ is a tight sublattice of the product of …
In 1959, F.Galvin and B.Jonsson characterized distributive sublattices of free lattices in their paper. In this paper, I will create new proofs to a portion of Galvin and Jónsson's results. Based on these new proofs, I will explore possible generalizations of F.Galvin and B.Jónsson's work by defining \emph{spanning pairs} and proving partial results which may help with analysing finite width sublattices of free lattices; and by making some new observations on finitely generated lattices over semidistributive varieties. The work done in this paper may assist in attacking the following long-stan…
The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.
This article builds on Thurston's height functions. His tiling algorithm is reinterpreted using lattice theory and then generalized in order to generate any tiling of a hole-free region. Combined with a natural encoding of tilings by words, this allows the generation of a linear extension of the lattice of the tilings.
We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimizati…
We study necessary conditions for the existence of lattice tilings of $\R^n$ by quasi-crosses. We prove non-existence results, and focus in particular on the two smallest unclassified shapes, the $(3,1,n)$-quasi-cross and the $(3,2,n)$-quasi-cross. We show that for dimensions $n\leq 250$, apart from the known constructions, there are no lattice tilings of $\R^n$ by $(3,1,n)$-quasi-crosses except for ten remaining cases, and no lattice tilings of $\R^n$ by $(3,2,n)$-quasi-crosses except for eleven remaining cases.
For every natural number $n\geq 5$, we prove that the number of subuniverses of an $n$-element lattice is $2^n$, $13\cdot 2^{n-4}$, $23\cdot 2^{n-5}$, or less than $23\cdot 2^{n-5}$. By a subuniverse, we mean a sublattice or the emptyset. Also, we describe the $n$-element lattices with exactly $2^n$, $13\cdot 2^{n-4}$, or $23\cdot 2^{n-5}$ subuniverses.
The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to various zeta functions of orders in imaginary quadratic fields.
Limited magnitude error model has applications in flash memory. In this model, a perfect code is equivalent to a tiling of $\mathbb{Z}^n$ by limited magnitude error balls. In this paper, we give a complete classification of lattice tilings of $\mathbb{Z}^n$ by limited magnitude error balls $\mathcal{B}(n,2,1,1)$.
We state the formula for the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given sublattice and give a partial proof of the formula. We show that the proof can be reduced to finding upper bounds on the number of vertices in certain classes of polygons. To obtain these bounds, we establish inequalities relating the number of edges of a broken line and the coordinates of its endpoints within a suitable class of broken lines.
Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice $Λ_1$ and to a sublattice $Λ_2$ may differ. Here, we examine the coloring of $Λ_1$ induced by $Λ_2$ to identify how the coincidence indices with respect to $Λ_1$ and to $Λ_2$ are related. This leads to a generalization of the notion of color symmetries of lattices to what we call color coincidences of lattices. Examples involving the cubic and hypercubic lattices are given to illustrate these ideas.
We study whether an asymmetric limited-magnitude ball may tile $\mathbb{Z}^n$. This ball generalizes previously studied shapes: crosses, semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting codes in a channel which changes a transmitted integer vector in a bounded number of entries by limited-magnitude errors. A construction of lattice tilings based on perfect codes in the Hamming metric is given. Several non-existence results are proved, both for general tilings, and lattice tilings. A complete classification of lattice tilings for two certain cases is proved.
We characterize the polytopes in $\mathbb{R}^d$ (not necessarily convex or connected ones) which multi-tile the space by translations along a given lattice. We also give a necessary and sufficient condition for two polytopes in $\mathbb{R}^d$ to be equidecomposable by lattice translations.
Van der Corput's provides the sharp bound vol(C) \le m 2^d on the volume of a d-dimensional origin-symmetric convex body C that has 2m-1 points of the integer lattice in its interior. For m=1, a characterization of the equality case vol(C)= m 2^d is equivalent to the well-known problem of characterizing tilings by translations of a convex body. It is rather surprising that so far, for m \ge 2, no characterization of the equality case has been available, though a hint to the respective characterization problem can be found in the 1987 monograph of Gruber and Lekkerkerker. We give an explicit ch…
Similar sublattices of the root lattice $A_4$ are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form $m^2 + mn - n^2$. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra $\mathbb{H} (\mathbb{Q} (\sqrt{5}))$ and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian rin…
There exist as many index-$k$ sublattices of the hexagonal lattice up to isometry as there exist lattice triangles with normalized volume $k$ up to unimodular equivalence, which can be explained using orbifolds. In dimension 3, it was noted that the number of sublattices of the fcc and the bcc lattices and the number of lattice tetrahedra all seem to be the same. We provide a bijection between the sublattices of the coweight lattice $A^*_n$ and the $n$-dimensional lattice simplices. It explains, proves, and generalizes the observed coincidences to arbitrary dimension.
An assosiahedron $\mathcal{K}^n$, known also as Stasheff polytope, is a multifaceted combinatorial object, which, in particular, can be realized as a convex hull of certain points in $\mathbf{R}^{n}$, forming $(n-1)$-dimensional polytope. A permutahedron $\mathcal{P}^n$ is a polytope of dimension $(n-1)$ in $\mathbf{R}^{n}$ with vertices forming various permutations of $n$-element set. There exist well-known orderings of vertices of $\mathcal{P}^n$ and $\mathcal{K}^n$ that make these objects into lattices: the first known as permutation lattices, and the latter as Tamari lattices. We establish…
Domino tilings, nonintersecting lattice paths and subclasses of Koutschan-Krattenthaler-Schlosser determinants
本文用多米诺平铺和不相交格点路径对Koutschan-Krattenthaler-Schlosser行列式的两个子类给出组合解释，并利用holonomic Ansatz和模归约方法证明了剩余的两个计数猜想。
A basic property in a modular lattice is that any two flags generate a distributive sublattice. It is shown (Abels 1991, Herscovic 1998) that two flags in a semimodular lattice no longer generate such a good sublattice, whereas shortest galleries connecting them form a relatively good join-sublattice. In this note, we sharpen this investigation to establish an analogue of the two-flag generation theorem for a semimodular lattice. We consider the notion of a modular convex subset, which is a subset closed under the join and meet only for modular pairs, and show that the modular convex hull of t…
We prove the nonexistence of lattice tilings of $\mathbb{Z}^n$ by Lee spheres of radius $2$ for all dimensions $n\geq 3$. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals $2$ and $2n^2+2n+1$ is a prime. As a direct consequence, we also answer an open question in the degree-diameter problem of graph theory: the order of any abelian Cayley graph of diameter $2$ and degree larger than $5$ cannot meet the abelian Cayley Moore bound.
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R^d}$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z^2} \times G_0$ for some finite abelian $G_0$. Our methods rely on encoding a certain class of "$p$-adically structured functions" in terms of certain functional equations.
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 Bölcskei from 2001. For positive integers $p$ and $q$ this tiling also provides a tiling of $(\mathbb{Z}/(p^n+q^n)\mathbb{Z})^n$.
Common tiling functions with small support
该研究解决了格点公共铺砌函数支撑直径的公开差距问题，构造出满足条件的格点族，得到最优支撑直径上界并回答了相关问题。
Folding a sequence $S$ into a multidimensional box is a well-known method which is used as a multidimensional coding technique. The operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes and not just a box. The new definition of folding is based on a lattice tiling for the given shape $\cS$ and a direction in the $D$-dimensional integer grid. Necessary and sufficient conditions that a lattice tiling for $\cS$ combined with a direction define a folding of a sequence into $\cS$ are derived. The immediate and most impressive application is some new lo…
On lattice tilings of $\mathbb{Z}^n$ by limited magnitude error balls $\mathcal{B}(n,2,k_{1},k_{2})$ with $k_1>k_2$
本文研究有限幅度错误球对整数格的格铺砌，完全刻画了 $\mathcal{B}(n,2,3,0)$ 与 $k_1=k_2+1$ 情形，并证明合数条件下高维不存在铺砌。
Generalized Integer Partitions, Tilings of Zonotopes and Lattices
本文利用动力系统与序理论，证明广义整数分拆集合构成分配格，并刻画了二维 zonotopes 铺砌集合为分配格的不交并，深化了对翻转铺砌行为的理解。
We construct a countable bounded sublattice of the lattice of all subspaces of a vector space with two non-isomorphic maximal Boolean sublattice. We represent one of them as the range of a Banschewski function and we prove that this is not the case of the other. Hereby we solve a problem of F. Wehrung.
The $k$-tiling problem for a convex polytope $P$ is the problem of covering $\mathbb R^d$ with translates of $P$ using a discrete multiset $Λ$ of translation vectors, such that every point in $\mathbb R^d$ is covered exactly $k$ times, except possibly for the boundary of $P$ and its translates. A classical result in the study of tiling problems is a theorem of McMullen that a convex polytope $P$ that 1-tiles $\mathbb R^d$ with a discrete multiset $Λ$ can, in fact, 1-tile $\mathbb R^d$ with a lattice $\mathcal L$. A generalization of McMullen's theorem for $k$-tiling was conjectured by Gravin, …
Suppose $f\in L^1(\mathbb{R}^d)$, $Λ\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+Λ$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles, with a possibly different weight. As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity. Our second result is a new characterization of convex polygons that multi-tile the plane by translations. It also provides a very efficien…
MacMahon enumerated the plane partitions in an $a \times b \times c$ box. These are in bijection to lozenge tilings of a hexagon, to certain perfect matchings, and to families of non-intersecting lattice paths. In this work we consider more general regions, called triangular regions, and establish signed versions of the latter three bijections. Indeed, we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. A combinatorial argument involving a new method, called resolution of a puncture, then shows that the signs are in fact equivalent. …
The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial conditions, at all sizes, there appears what we call an exact Sierpinski structure: the volume is filled with periodic patterns, glued together along straight lines, with the topology of a triangular Sierpinski gasket. Various lattices (square, hexagonal, kagome,...), initial conditions, and toppling rules show Sierpinski structures which are apparently unrelated and…
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