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The term aperiodic has been used in a wide variety of ways in the mathematical literature on tilings (and in other mathematical fields as well, such as dynamical systems or graph theory, with altogether different meanings). With respect to tilings the term aperiodic was sometimes used synonymously with the term non-periodic.
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 tilings obtained from an aperiodic set of tiles are often called aperiodic tilings , though strictly speaking it is the tiles themselves that are ...
Despite initial skepticism, the discovery gained widespread acceptance, prompting the International Union of Crystallography to redefine the term "crystal." [ 11 ] The work ultimately earned Shechtman the 2011 Nobel Prize in Chemistry [ 12 ] and inspired significant advancements in materials science and mathematics.
The atomic spacing of amorphous materials (such as glass) varies substantially between different pairs of atoms, therefore diffraction cannot be used to accurately determine atomic spacing. In this case, the average bond length is a common way of expressing the distance between its atoms. [citation needed]
[1] [2] A Euclidean graph is uniformly discrete if there is a minimal distance between any two vertices. Periodic graphs are closely related to tessellations of space (or honeycombs) and the geometry of their symmetry groups , hence to geometric group theory , as well as to discrete geometry and the theory of polytopes , and similar areas.
A metric or distance function is a function d which takes pairs of points or objects to real numbers and satisfies the following rules: The distance between an object and itself is always zero. The distance between distinct objects is always positive. Distance is symmetric: the distance from x to y is always the same as the distance from y to x.
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.
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. [1] Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.