Nodal points can be artificially synthesized using glide-reflection symmetries at crystal interfaces. This property was first demonstrated for a square-lattice phononic crystal at the
X
point of the ...first Brillouin zone (wavenumber
k
= ±
π/a
with
a
the lattice constant), for a half-lattice-constant glide. Here we show that the nodal point can be moved to the
Γ
point (
k
= 0) considering quarter-lattice-constant glide-reflection symmetry. Applying a continuous grading along the x-axis is further shown to leave the band structure mostly unaffected. In particular, the topological interface waves survive in the case that glide-reflection symmetry is only locally valid around the graded interface. As a result, the glide dislocation can be compensated for over a distance of a few crystal rows, to recover an apparently periodic crystal.
Curved Glide-Reflection Symmetry Detection LEE, Seungkyu; YANXI LIU
IEEE transactions on pattern analysis and machine intelligence,
02/2012, Letnik:
34, Številka:
2
Journal Article
Recenzirano
We generalize the concept of bilateral reflection symmetry to curved glide-reflection symmetry in 2D euclidean space, such that classic reflection symmetry becomes one of its six special cases. We ...propose a local feature-based approach for curved glide-reflection symmetry detection from real, unsegmented 2D images. Furthermore, we apply curved glide-reflection axis detection for curved reflection surface detection in 3D images. Our method discovers, groups, and connects statistically dominant local glide-reflection axes in an Axis-Parameter-Space (APS) without preassumptions on the types of reflection symmetries. Quantitative evaluations and comparisons against state-of-the-art algorithms on a diverse 64-test-image set and 1,125 Swedish leaf-data images show a promising average detection rate of the proposed algorithm at 80 and 40 percent, respectively, and superior performance over existing reflection symmetry detection algorithms. Potential applications in computer vision, particularly biomedical imaging, include saliency detection from unsegmented images and quantification of deviations from normality. We make our 64-test-image set publicly available.
Creating Symmetry Farris, Frank A
2015, 2015., 20150602, 2015-06-02
eBook
This lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks-a sort of potato-stamp ...method-Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling art images where the beauty of nature meets the precision of mathematics.
Featuring more than 100 stunning color illustrations and requiring only a modest background in math,Creating Symmetrybegins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you'll learn how to create breathtaking art images of your own.
Fun, accessible, and challenging,Creating Symmetryfeatures numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.
It is known that translations, symmetries with respect to points, and the identity map are the only isometries in general (normed or) Minkowski planes. Inspired by this “difference” to the Euclidean ...situation, we introduce so-called left-reflections in lines for the case of strictly convex Minkowski planes, and we develop a little theory on their products, yielding also results on glide reflections. As natural consequences we obtain several new characterizations of special normed planes, such as Radon planes or the Euclidean plane. All properties of left-reflections in strictly convex normed planes derived here hold in an analogous manner for correspondingly defined right-reflections in smooth Minkowski planes.
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