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A tiling that has no periods is non-periodic. A set of prototiles is said to be aperiodic if all of its tilings are non-periodic, and in this case its tilings are also called aperiodic tilings. [5] Penrose tilings are among the simplest known examples of aperiodic tilings of the plane by finite sets of prototiles. [3]
Any prototile satisfying Conway's criterion admits a periodic tiling of the plane—and does so using only 180-degree rotations. [1] The Conway criterion is a sufficient condition to prove that a prototile tiles the plane but not a necessary one. There are tiles that fail the criterion and still tile the plane. [3]
Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + 96 / 2 − 1 = 48 [1]. In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary.
In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. [1] It can also be related to the relativistic velocity addition formula. [2] [3]
A period on the periodic table is a row of chemical elements. All elements in a row have the same number of electron shells. Each next element in a period has one more proton and is less metallic than its predecessor. Arranged this way, elements in the same group (column) have similar chemical and physical properties, reflecting the periodic law.
In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the d = 2 , 3 {\displaystyle d=2,3} case) by Hermann Weyl for eigenvalues for the Laplace–Beltrami operator acting on functions that vanish at the boundary of a ...
The bond angle for water is 104.5°. Valence shell electron pair repulsion (VSEPR) theory (/ ˈ v ɛ s p ər, v ə ˈ s ɛ p ər / VESP-ər, [1]: 410 və-SEP-ər [2]) is a model used in chemistry to predict the geometry of individual molecules from the number of electron pairs surrounding their central atoms. [3]
The composition of the braids σ and τ is written as στ.. The set of all braids on four strands is denoted by .The above composition of braids is indeed a group operation. . The identity element is the braid consisting of four parallel horizontal strands, and the inverse of a braid consists of that braid which "undoes" whatever the first braid did, which is obtained by flipping a diagram ...