Computer-generated aesthetic patterns arewidely used as design materials in various fields. Themost common methods use fractals or dynamicalsystems as basic tools to create various patterns. Toenhance aesthetics and c...Computer-generated aesthetic patterns arewidely used as design materials in various fields. Themost common methods use fractals or dynamicalsystems as basic tools to create various patterns. Toenhance aesthetics and controllability, some researchershave introduced symmetric layouts along with thesetools. One popular strategy employs dynamical systemscompatible with symmetries that construct functionswith the desired symmetries. However, these aretypically confined to simple planar symmetries. Theother generates symmetrical patterns under theconstraints of tilings. Although it is slightly moreflexible, it is restricted to small ranges of tilingsand lacks textural variations. Thus, we proposed anew approach for generating aesthetic patterns bysymmetrizing quasi-regular patterns using general kuniformtilings. We adopted a unified strategy toconstruct invariant mappings for k-uniform tilings thatcan eliminate texture seams across the tiling edges.Furthermore, we constructed three types of symmetriesassociated with the patterns: dihedral, rotational, andreflection symmetries. The proposed method can beeasily implemented using GPU shaders and is highlyefficient and suitable for complicated tiling with regularpolygons. Experiments demonstrated the advantages of our method over state-of-the-art methods in terms offlexibility in controlling the generation of patterns withvarious parameters as well as the diversity of texturesand styles.展开更多
Let M be a 3×3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D?Z^(3)be a digit set containing|det M|elements.Then the unique nonempty compact set...Let M be a 3×3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D?Z^(3)be a digit set containing|det M|elements.Then the unique nonempty compact set T=T(M,D)defined by the set equation MT=T+D is called an integral self-affine tile if its interior is nonempty.If D is of the form D={0,v,...,(|det M|-1)v},we say that T has a collinear digit set.The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets.In particular,we prove that a large class of these tiles is homeomorphic to a closed 3-dimensional ball.Moreover,we show that in this case,T carries a natural CW complex structure that is defined in terms of the intersections of T with its neighbors in the lattice tiling{T+z:z∈Z^(3)}induced by T.This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.展开更多
In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in L^2（H^d） by ...In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in L^2（H^d） by using self-similar tilings for the acceptable dilations on the Heisenberg group.展开更多
This review summarizes the self-assembly of block molecules forming unconventional two-dimensional(2D)periodic nanopatterns.Especially,we emphasize the structural evolution from simple columnar phases to complex 2D ti...This review summarizes the self-assembly of block molecules forming unconventional two-dimensional(2D)periodic nanopatterns.Especially,we emphasize the structural evolution from simple columnar phases to complex 2D tiling morphologies in soft materials including block copolymers,liquid crystals,giant molecules,etc.Then,the state-of-the-art nanofabrication technologies for making sophisticated nanostructures with specific functions via combining both bottom-up assembly and top-down lithography-based methods are discussed,highlighting the use of directed self-assembly processes.Finally,we provide our perspective on this area.By further increasing the complexity of block molecules and the designability of lithography,low-dimensional ordered morphologies will be particularly promising for further application in nanotechnology.展开更多
Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investiga...Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.展开更多
We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is...We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.展开更多
In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by ...In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by two discrete families denoted by ε^m,m ≥ 2,m ∈ N,F^k,k ≥ 4,k ∈ N and two sporadic tilings denoted by G and H.展开更多
The authOrs define the scenery flow of the torus. The flow space is the union of all flat 2- dimensional tori of area 1 with a marked direction (or equivalently the union of all tori with a quadratic differential of n...The authOrs define the scenery flow of the torus. The flow space is the union of all flat 2- dimensional tori of area 1 with a marked direction (or equivalently the union of all tori with a quadratic differential of norm 1). This is a 5-dimensional space, and the flow acts by following individual points under an extremal deformation of the quadratic differential. The authors define associated horocycle and translation flows; the latter preserve each torus and are the horizontal and vertical flows of the corresponding quadratic differential. The scenery flow projects to the geodesic flow on the modular surface, and admits, for each orientation preserving hyperbolic toral automorphism, an invariant 3-dimensional subset on which it is the suspension flow of that map. The authors first give a simple algebraic definition in terms of the group of affine maps of the plane, and prove that the flow is Anosov. They give an explicit formula for the first-return map of the flow on convenient cross-sections. Then, in the main part of the paper, the authors give several different models for the flow and its cross-sections, in terms of : stacking and rescaling periodic tilings of the plane; symbolic dynamics: the natural extension of the recoding of Sturmian sequences, or the S-adic system generated by two substitutions; zooming and subdividing quasi-periodic tilings of the real line, or aperiodic quasicrystals of minimal complexity; the natural extension of two-dimensional continued fractions; induction on exchanges of three intervals; rescaling on pairs of transverse measure foliations on the torus, or the Teichmuller flow on the twice-punctured torus.展开更多
The study of the dihedral f-tilings of the sphere S2 whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in a previous work. In this paper we continue this classification presenting the st...The study of the dihedral f-tilings of the sphere S2 whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in a previous work. In this paper we continue this classification presenting the study of all dihedral spherical f-tilings by scalene triangles and isosceles trapezoids in some cases of adjacency.展开更多
基金supported by the Key R&D Programs of Zhejiang Province(Nos.2023C01224 and 2022C01220)the National Natural Science Foundation of China(No.61702458)+1 种基金Yun Zhang was partially supported by Zhejiang Province Public Welfare Technology Application Research(No.LGG22F020009)Key Lab of Film and TV Media Technology of Zhejiang Province(No.2020E10015).
文摘Computer-generated aesthetic patterns arewidely used as design materials in various fields. Themost common methods use fractals or dynamicalsystems as basic tools to create various patterns. Toenhance aesthetics and controllability, some researchershave introduced symmetric layouts along with thesetools. One popular strategy employs dynamical systemscompatible with symmetries that construct functionswith the desired symmetries. However, these aretypically confined to simple planar symmetries. Theother generates symmetrical patterns under theconstraints of tilings. Although it is slightly moreflexible, it is restricted to small ranges of tilingsand lacks textural variations. Thus, we proposed anew approach for generating aesthetic patterns bysymmetrizing quasi-regular patterns using general kuniformtilings. We adopted a unified strategy toconstruct invariant mappings for k-uniform tilings thatcan eliminate texture seams across the tiling edges.Furthermore, we constructed three types of symmetriesassociated with the patterns: dihedral, rotational, andreflection symmetries. The proposed method can beeasily implemented using GPU shaders and is highlyefficient and suitable for complicated tiling with regularpolygons. Experiments demonstrated the advantages of our method over state-of-the-art methods in terms offlexibility in controlling the generation of patterns withvarious parameters as well as the diversity of texturesand styles.
基金supported by a grant funded by the Austrian Science Fund and the Russian Science Foundation(Grant No.I 5554)supported by National Natural Science Foundation of China(Grant No.12101566)。
文摘Let M be a 3×3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D?Z^(3)be a digit set containing|det M|elements.Then the unique nonempty compact set T=T(M,D)defined by the set equation MT=T+D is called an integral self-affine tile if its interior is nonempty.If D is of the form D={0,v,...,(|det M|-1)v},we say that T has a collinear digit set.The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets.In particular,we prove that a large class of these tiles is homeomorphic to a closed 3-dimensional ball.Moreover,we show that in this case,T carries a natural CW complex structure that is defined in terms of the intersections of T with its neighbors in the lattice tiling{T+z:z∈Z^(3)}induced by T.This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.
基金Sponsored by the NSFC (10871003, 10701008, 10726064)the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007001040)
文摘In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in L^2（H^d） by using self-similar tilings for the acceptable dilations on the Heisenberg group.
基金financially supported by the National Natural Science Foundation of China (Nos. 21925102, 21991132, 92056118 and 22101010)the financial support from the National Key R&D Program of China (No. 2018YFB0703702)Beijing National Laboratory for Molecular Sciences (No. BNLMS-CXXM-202006)
文摘This review summarizes the self-assembly of block molecules forming unconventional two-dimensional(2D)periodic nanopatterns.Especially,we emphasize the structural evolution from simple columnar phases to complex 2D tiling morphologies in soft materials including block copolymers,liquid crystals,giant molecules,etc.Then,the state-of-the-art nanofabrication technologies for making sophisticated nanostructures with specific functions via combining both bottom-up assembly and top-down lithography-based methods are discussed,highlighting the use of directed self-assembly processes.Finally,we provide our perspective on this area.By further increasing the complexity of block molecules and the designability of lithography,low-dimensional ordered morphologies will be particularly promising for further application in nanotechnology.
文摘Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.
文摘We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.
基金Supported by FEDER funds through COMPETE Operational Programme Factors of Competitiveness(Programa Operacional Factores de Competitividade)Supported by FSE+3 种基金Supported by Portuguese funds through the Center for Researchand Development in Mathematics and Applications(University of Aveiro)the Portuguese Foundation for Science and Technology(FCT Fundao para a Ciência e a Tecnologia)project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690supported partially by an NSERC Canada Discovery Grant
文摘In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by two discrete families denoted by ε^m,m ≥ 2,m ∈ N,F^k,k ≥ 4,k ∈ N and two sporadic tilings denoted by G and H.
文摘The authOrs define the scenery flow of the torus. The flow space is the union of all flat 2- dimensional tori of area 1 with a marked direction (or equivalently the union of all tori with a quadratic differential of norm 1). This is a 5-dimensional space, and the flow acts by following individual points under an extremal deformation of the quadratic differential. The authors define associated horocycle and translation flows; the latter preserve each torus and are the horizontal and vertical flows of the corresponding quadratic differential. The scenery flow projects to the geodesic flow on the modular surface, and admits, for each orientation preserving hyperbolic toral automorphism, an invariant 3-dimensional subset on which it is the suspension flow of that map. The authors first give a simple algebraic definition in terms of the group of affine maps of the plane, and prove that the flow is Anosov. They give an explicit formula for the first-return map of the flow on convenient cross-sections. Then, in the main part of the paper, the authors give several different models for the flow and its cross-sections, in terms of : stacking and rescaling periodic tilings of the plane; symbolic dynamics: the natural extension of the recoding of Sturmian sequences, or the S-adic system generated by two substitutions; zooming and subdividing quasi-periodic tilings of the real line, or aperiodic quasicrystals of minimal complexity; the natural extension of two-dimensional continued fractions; induction on exchanges of three intervals; rescaling on pairs of transverse measure foliations on the torus, or the Teichmuller flow on the twice-punctured torus.
基金Research funded by the Portuguese Government through the FCT-Fundaao para a Ciencia e a Tecnologia-under the project PEst-OE/MAT/UI4080/2011
文摘The study of the dihedral f-tilings of the sphere S2 whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in a previous work. In this paper we continue this classification presenting the study of all dihedral spherical f-tilings by scalene triangles and isosceles trapezoids in some cases of adjacency.
