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The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two directions.
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. Such a shape is called an einstein, a word play on ein Stein, German for "one stone". [2]
An example of such a tiling is shown in the adjacent diagram (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic . [ 3 ]
The term can also be used for non-periodic or aperiodic signals, like chirps and pulses. [3] In electronics, the term is usually applied to time-varying voltages, currents, or electromagnetic fields. In acoustics, it is usually applied to steady periodic sounds — variations of pressure in air or other media.
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. The Penrose tilings are a well-known example of aperiodic tilings. [1] [2]
Amorphous materials will have some degree of short-range order at the atomic-length scale due to the nature of intermolecular chemical bonding. [ a ] Furthermore, in very small crystals , short-range order encompasses a large fraction of the atoms ; nevertheless, relaxation at the surface, along with interfacial effects, distorts the atomic ...
Destructible materials: fracture of brittle solids, cutting [29] of soft bodies, and tearing of cloth. The finite element method is especially suited to modelling fracture [15] as it includes a realistic model of the distribution of internal stresses in the material, which physically is what determines when fracture occurs, according to ...
Video games are computer- or microprocessor-controlled games. Computers can create virtual spaces for a wide variety of game types. Some video games simulate conventional game objects like cards or dice, while others can simulate environs either grounded in reality or fantastical in design, each with its own set of rules or goals.