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"Tiling"
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Looping with Disney Pixar Finding Dory
A simple, low-level, unplugged introduction to looping designed for young readers not yet ready for coding on computers. Beloved characters Dory and Nemo, from the world-famous Disney movie Finding Dory, draw in readers new to coding concepts-- Provided by publisher.
Book
Symbolic Extensions of Amenable Group Actions and the Comparison Property
In topological dynamics, the
Of course, the statement is preceded by the
presentation of the concepts of an entropy structure and its superenvelopes, adapted from the case of
eBook
Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile
We construct an example of a group G=Z2×G0 for a finite abelian group G0, a subset E of G0, and two finite subsets F1,F2 of G, such that it is undecidable in ZFC whether Z2×E can be tiled by translations of F1,F2. In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles F1,F2, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in Z2). A similar construction also applies for G=Zd for sufficiently large d. If one allows the group G0 to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.
Journal Article
Improved Rendering of Car Paint Sparkle with Random Tiling Approach
Car paint suppliers and automobile manufacturers show great interest in virtually assessing and designing new automotive paints. Car paint shows a complex visual appearance where the sparkling of the effect pigments plays a big role. Previously bi-directional texture functions (BTF) were proposed to represent a car paint with sparkling. Rendering requires the interpolation of the recorded texture images, that are part of the BTF. Conventional approaches for blending the images use linear interpolation, which causes reduced contrast and intensity, and a static dynamic of the glittering. Here, we propose a new random tiling approach for texture image interpolation where each pixel is interpolated in a short, random interval independently from each other. Results show that sparkling contrast and intensity are preserved, and that the dynamic of the sparkling is more realistic.
Journal Article
Substitution Discrete Plane Tilings with 2n-Fold Rotational Symmetry for Odd n
We study substitution tilings that are also discrete plane tilings, that is, satisfy a relaxed version of cut-and-projection. We prove that the Sub Rosa substitution tilings with a 2n-fold rotational symmetry for odd n>5 defined by Kari and Rissanen are not discrete planes—and therefore not cut-and-project tilings either. We then define new Planar Rosa substitution tilings with a 2n-fold rotational symmetry for any odd n, and show that these satisfy the discrete plane condition. The tilings we consider are edge-to-edge rhombus tilings. We give an explicit construction for the 10-fold case, and provide a construction method for the general case of any odd n. Our methods are to lift the tilings and substitutions to Rn using the lift operator first defined by Levitov, and to study the planarity of substitution tilings in Rn using mainly linear algebra, properties of circulant matrices, and trigonometric sums. For the construction of the Planar Rosa substitutions we additionally use the Kenyon criterion and a result on De Bruijn multigrid dual tilings.
Journal Article
Semi-regular Tilings of the Hyperbolic Plane
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.
Journal Article
On the Structure of Ammann A2 Tilings
We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in the sense of Solomyak (Discret Comput Geom 20:265–279, 1998). By the same techniques we show that Ammann A2 tilings are not robust in the sense of Durand et al. (J Comput Syst Sci 78(3):731–764, 2012).
Journal Article