Lattice Tilings Research Articles

Multiresolution lattice-based support generation for additive manufacturing

The generation of support for 3D models toward 3D printing is a highly challenging task that is of great need in many additive manufacturing processes. In this work, we explore the use of multiresolution geometric lattices to generate support with controlled contact locations. That is, with bounds on the maximal distance between adjacent local support points. A variety of end-user controls over the synthesized support are provided, such as the angular slopes in the model that are provided with support and/or controls on the dimensions and sizes of the support lattice tiles. These controls are augmented with the option of an automated optimization via a direct link to analysis. We demonstrate this proposed lattice approach for support synthesis on several 3D models of different types.

Characterization of the two-dimensional fivefold and sixfold lattice tiles

Characterization of the two-dimensional fivefold and sixfold lattice tiles

Lattice Tilings of a Plane into Polyominoes and Molecular Layers in Crystal Structures: Structural Class cm, Z = 2(m)

Lattice Tilings of a Plane into Polyominoes and Molecular Layers in Crystal Structures: Structural Class cm, Z = 2(m)

Dynamical evolution of entanglement in disordered oscillator systems

We study the non-equilibrium dynamics of a disordered quantum system consisting of harmonic oscillators in a [Formula: see text]-dimensional lattice. If the system is sufficiently localized, we show that, starting from a broad class of initial product states that are associated with a tiling (decomposition) of the [Formula: see text]-dimensional lattice, the dynamical evolution of entanglement follows an area law in all times. Moreover, the entanglement bound reveals a dependency on how the subsystems are located within the lattice in dimensions [Formula: see text]. In particular, the entanglement grows with the maximum degree of the dual graph associated with the lattice tiling.

A Remarkable Summation Formula, Lattice Tilings, and Fluctuations

We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to efficiently analyze the average behavior of certain common nonlinear dynamical systems, such as the angle-doubling map, modulo 1. In particular, one can use this information to analyze how the behavior of individual orbits deviates from the global average (called fluctuations). More generally, the formula is valid in , where expanding maps give rise to so-called number systems. To illustrate the usefulness in this setting, we compute the fluctuations of a certain map on the plane.

Filling space with hypercubes of two sizes - The pythagorean tiling in higher dimensions.

We construct a unilateral lattice tiling of Rn into hypercubes of two differnet side lengths p or q. This generalizes the Pythagorean tiling in R2. 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 (Z/(pn+qn)Z)n.

On tilings of asymmetric limited-magnitude balls

We study whether an asymmetric limited-magnitude ball may tile Zn. 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.

Lattice-Point Mutually Guided Ground-to-Aerial Feature Matching for Urban Scene Images

Ground-to-aerial feature matching bridges information from cross-view images, which enables optimized urban applications, e.g., pixel-level geolocating and complete urban 3-D reconstruction. However, urban ground and aerial images typically suffer from drastic changes in viewpoint, scale, and illumination, together with repetitive patterns. Thus, direct matching of local features between ground and aerial images is particularly difficult because of the low similarity of local descriptors and high ambiguity in true–false match discrimination. For this challenging task, we propose a novel lattice-point mutually guided matching (LPMG) method in this article. We specifically address two key issues: 1) reducing descriptor variance and 2) enhancing true–false match discriminability. The former is solved by recovering the geometry and appearance of the underlying image region in 3-D through automatic view rectification on ground and aerial images. The latter is circumvented by replacing the conventional mismatch removal with an LPMG strategy. In this strategy, the topology structure of repeated façade elements (i.e., lattice), and the high reliable point matching seeds, are first extracted from the rectified ground and aerial images. Then, the point matching seeds guide the self-similar lattice tiles from two views to be precisely aligned, thereby estimating an accurate transformation model from lattice tile correspondences. Finally, the estimated model powerfully supervises the differentiation of true and false matches from the entire putative match set. Extensive experiments conducted on several datasets show that our method can obtain a considerable number of nearly pure correct matches from urban ground and aerial images, significantly outperforming those existing methods.

Low-complexity eigenstates of a ν = 1/3 fractional quantum Hall system

We identify the ground-state of a truncated version of Haldane’s pseudo-potential Hamiltonian in the thin cylinder geometry as being composed of exponentially many fragmented matrix product states. These states are constructed by lattice tilings and their properties are discussed. We also report on a proof of a spectral gap, which implies the incompressibility of the underlying fractional quantum Hall liquid at maximal filling ν = 1/3. Low-energy excitations and an extensive number of many-body scars at positive energy density, but nevertheless low complexity, are also identified using the concept of tilings.

Some New Results on Splitter Sets

Splitter sets have been widely studied due to their applications in flash memories, and their close relations with lattice tilings and conflict avoiding codes. In this paper, we give necessary and sufficient conditions for the existence of nonsingular perfect splitter sets, ${B}[-{k}_{1},{k}_{2}]({p})$ sets, where $0\le {k}_{1}\leq {k}_{2}=4$ . Meanwhile, constructions of nonsingular perfect splitter sets are given. When perfect splitter sets do not exist, we present four new constructions of quasi-perfect splitter sets. Finally, we give a connection between nonsingular splitter sets and Cayley graphs, and as a byproduct, a general lower bound on the maximum size of nonsingular splitter sets is given.

No lattice tiling of [formula omitted] by Lee sphere of radius 2

We prove the nonexistence of lattice tilings of Zn by Lee spheres of radius 2 for all dimensions n≥3. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals 2 and 2n2+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.

On self-affine tiles whose boundary is a sphere

Let M M be a 3 × 3 3\times 3 integer matrix each of whose eigenvalues is greater than 1 1 in modulus and let D ⊂ Z 3 \mathcal {D}\subset \mathbb {Z}^3 be a set with | D | = | det M | |\mathcal {D}|=|\det M| , called a digit set. The set equation M T = T + D MT = T+\mathcal {D} uniquely defines a nonempty compact set T ⊂ R 3 T\subset \mathbb {R}^3 . If T T has positive Lebesgue measure it is called a 3 3 -dimensional self-affine tile. In the present paper we study topological properties of 3 3 -dimensional self-affine tiles with collinear digit set, i.e., with a digit set of the form D = { 0 , v , 2 v , … , ( | det M | − 1 ) v } \mathcal {D}=\{0,v,2v,\ldots , (|\det M|-1)v\} for some v ∈ Z 3 ∖ { 0 } v\in \mathbb {Z}^3\setminus \{0\} . We prove that the boundary of such a tile T T is homeomorphic to a 2 2 -sphere whenever its set of neighbors in a lattice tiling which is induced by T T in a natural way contains 14 14 elements. The combinatorics of this lattice tiling is then the same as the one of the bitruncated cubic honeycomb, a body-centered cubic lattice tiling by truncated octahedra. We give a characterization of 3 3 -dimensional self-affine tiles with collinear digit set having 14 14 neighbors in terms of the coefficients of the characteristic polynomial of M M . In our proofs we use results of R. H. Bing on the topological characterization of spheres.

On the non-existence of linear perfect Lee codes: The Zhang–Ge condition and a new polynomial criterion

The Golomb–Welch conjecture (1968) states that there are no e-perfect Lee codes in Zn for n≥3 and e≥2. This conjecture remains open even for linear codes. A recent result of Zhang and Ge establishes the non-existence of linear e-perfect Lee codes in Zn for infinitely many dimensions n, for e=3 and 4. In this paper we extend this result in two ways. First, using the non-existence criterion of Zhang and Ge together with a generalized version of Lucas’ theorem we extend the above result for almost all e (i.e. a subset of positive integers with density 1). Namely, if e contains a digit 1 in its base-3 representation which is not in the unit place (e.g. e=3,4) there are no linear e-perfect Lee codes in Zn for infinitely many dimensions n. Next, based on a family of polynomials (the Q-polynomials), we present a new criterion for the non-existence of certain lattice tilings. This criterion depends on a prime p and a tile B. For p=3 and B being a Lee ball we recover the criterion of Zhang and Ge.

Equality Case in van der Corput’s Inequality and Collisions in Multiple Lattice Tilings

Van der Corput’s provides the sharp bound $$\mathop {\mathrm {vol}}\nolimits (C) \le m 2^d$$ 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 $$\mathop {\mathrm {vol}}\nolimits (C)= m 2^d$$ 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 characterization of the equality case for all $$m \ge 2$$ . Our result reveals that, the equality case for $$m \ge 2$$ is more restrictive than for $$m=1$$ . We also present consequences of our characterization in the context of multiple lattice tilings.

On the nonexistence of lattice tilings of [formula omitted] by Lee spheres

In 1968, Golomb and Welch conjectured that Zn cannot be tiled by Lee spheres with a fixed radius r≥2 for dimension n≥3. This conjecture is equivalent to saying that there is no perfect Lee codes in Zn with radius r≥2 and dimension n≥3. Besides its own interest in discrete geometry and coding theory, this conjecture is also strongly related to the degree-diameter problems of abelian Cayley graphs. Although there are many papers on this topic, the Golomb–Welch conjecture is still far from being solved. In this paper, we introduce some new algebraic approaches to investigate the nonexistence of lattice tilings of Zn by Lee spheres, which is a special case of the Golomb–Welch conjecture. Using these new methods, we show the nonexistence of lattice tilings of Zn by Lee spheres of the same radius r=2 or 3 for infinitely many values of the dimension n. In particular, there does not exist lattice tilings of Zn by Lee spheres of radius 2 for all 3≤n≤100 except 8 cases.

Multiple Lattice Tilings in Euclidean Spaces

AbstractIn 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

Graph coloring and Graham’s greatest common divisor problem

In this paper we introduce and study two graph coloring problems and relate them to some deep number-theoretic problems. For a fixed positive integer k consider a graph Bk whose vertex set is the set of all positive integers with two vertices a,b joined by an edge whenever the two numbers a∕gcd(a,b) and b∕gcd(a,b) are both at most k. We conjecture that the chromatic number of every such graph Bk is equal to k. This would generalize the greatest common divisor problem of Graham from 1970; in graph-theoretic terminology it states that the clique number of Bk is equal to k. Our conjecture is connected to integer lattice tilings and partial Latin squares completions.

Topology of a class of [formula omitted]-crystallographicreplication tiles

We study the topological properties of a class of planar crystallographic replication tiles. Let M∈Z2×2 be an expanding matrix with characteristic polynomial x2+Ax+B (A,B∈Z, B≥2) and v∈Z2 such that (v,Mv) are linearly independent. Then the equation MT+B−12v=T∪(T+v)∪(T+2v)∪⋯∪(T+(B−2)v)∪(−T)defines a unique nonempty compact set T satisfying To¯=T. Moreover, T tiles the plane by the crystallographic group p2 generated by the π-rotation and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that T∪(−T) is homeomorphic to a closed disk if and only if 2|A|<B+3. However, this characterization does not hold anymore for T itself. In this paper, we completely characterize the tiles T of this class that are homeomorphic to a closed disk.

On the Number of Neighbors in Normal Tiling

The paper is devoted to the normal tiling whose tiles are uniformly bounded and general connected closed sets instead of being restricted to polytopes or convex sets. We estimate the number of neighbors of a tile in the normal tiling and develop various novel techniques to derive lower and upper bounds. The bounds for lattice tilings are shown to be optimal.

ОЦЕНКА ЧИСЛА Р2–РАЗБИЕНИЙ ПЛОСКОСТИ НА ПОЛИМИНО ЗАДАННОЙ ПЛОЩАДИ

We consider the problem about a number of \(p2\)--tilings of a plane by a given area polyominoes. A polyomino is a connected plane geometric figure formed by joining one or more unit squares edge to edge. At present, various combinatorial enumeration problems connected to the polyomino are actively studied. There are some interesting problems on enuneration of various classes of polyominoes and enumeration of tilings of finite regions or a whole plane by polyominoes. The tiling is called \(p2\)--tiling, if each tile can be mapped to any other tile by the translation or the central symmetry, and this transformation maps the whole tiling to itself. \(p2\)-tilings are special case of regular plane tilings. Let \(t(n)\) be a number of \(p2\)--tilings of a plane by a \(n\)-area polyomino such that the lattices of periods of these tilings are sublattices of \(\mathbb{Z}^2\). It is proved that following inequality is true: \( C_12^n \leq t(n)\leq C_2n^4(2.68)^n\). To prove the lower bound we use the exact construction of required tilings. The proof of the upper bound is based on the Conway criterion of the existence of \(p2\)--tilings of a plane. Also, the upper bound depends on the theory of self-avoiding walks on the square lattice. Earlier similar results were obtained by authors for the number of lattice tilings of a plane by a given area polyomino (it's more simple type of a plane tilings by polyomino), and for the number of lattice tilings of the plane by centrosimmetrical polyomino.