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Aperiodic tiling with "Tile(1,1)". The tiles are colored according to their rotational orientation modulo 60 degrees. [1] ( Smith, Myers, Kaplan, and Goodman-Strauss) In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way.
An aperiodic tiling using a single shape and its reflection, discovered by David Smith. An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- periodic tilings.
Curved space often refers to a spatial geometry which is not "flat", where a flat space has zero curvature, as described by Euclidean geometry. [ 1] Curved spaces can generally be described by Riemannian geometry, though some simple cases can be described in other ways. Curved spaces play an essential role in general relativity, where gravity ...
Socolar–Taylor tile. The Socolar–Taylor tile is a single non-connected tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane (due to the Sierpinski's triangle -like tiling that occurs), with rotations and reflections of the tile allowed. [1] It is the first known example of a single ...
The 15th monohedral convex pentagonal type, discovered in 2015. In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon . A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle ...
The roughly spherical shape of Earth can be empirically evidenced by many different types of observation, ranging from ground level, flight, or orbit. The spherical shape causes a number of effects and phenomena that combined disprove flat Earth beliefs . These include the visibility of distant objects on Earth's surface; lunar eclipses ...
Flat module. In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module M over a ring R is flat if taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact ...
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