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The tiles in the square tiling have only one shape, and it is common for other tilings to have only a finite number of shapes. These shapes are called prototiles, and a set of prototiles is said to admit a tiling or tile the plane if there is a tiling of the plane using only these shapes.
Following Grünbaum and Shephard (section 1.3), a tiling is said to be regular if the symmetry group of the tiling acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first ...
Special cases are right triangles (p q 2). Uniform solutions are constructed by a single generator point with 7 positions within the fundamental triangle, the 3 corners, along the 3 edges, and the triangle interior. All vertices exist at the generator, or a reflected copy of it. Edges exist between a generator point and its image across a mirror.
A compact binary circle packing with the most similarly sized circles possible. [7] It is also the densest possible packing of discs with this size ratio (ratio of 0.6375559772 with packing fraction (area density) of 0.910683). [8] There are also a range of problems which permit the sizes of the circles to be non-uniform.
An oval (from Latin ovum 'egg') is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( projective geometry , technical drawing , etc.) it is given a more precise definition, which may include either one or two axes of symmetry of an ellipse .
Notably, Jarkko Kari gave an aperiodic set of Wang tiles based on multiplications by 2 or 2/3 of real numbers encoded by lines of tiles (the encoding is related to Sturmian sequences made as the differences of consecutive elements of Beatty sequences), with the aperiodicity mainly relying on the fact that 2 n /3 m is never equal to 1 for any ...
Any ellipsoid is the image of the unit sphere under some affine transformation, and any plane is the image of some other plane under the same transformation. So, because affine transformations map circles to ellipses, the intersection of a plane with an ellipsoid is an ellipse or a single point, or is empty. [8] Obviously, spheroids contain ...
Ceramic tiles may be painted and glazed. Small mosaic tiles may be laid in various patterns. Floor tiles are typically set into mortar consisting of sand, Portland cement and often a latex additive. The spaces between the tiles are commonly filled with sanded or unsanded floor grout, but traditionally mortar was used.
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