SUMMARY / RELATED TOPICS

Dimension

In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere; the inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics and time are different categories and refer to absolute space and time; that conception of the world is a four-dimensional space but not the one, found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not defined spatially and temporally, but rather are known relative to the motion of an observer.

Minkowski space first approximates the universe without gravity. 10 dimensions are used to describe superstring theory, 11 dimensions can describe supergravity and M-theory, the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects. High-dimensional spaces occur in mathematics and the sciences, they may be parameter spaces or configuration spaces such as in Hamiltonian mechanics. In mathematics, the dimension of an object is speaking, the number of degrees of freedom of a point that moves on this object. In other words, the dimension is the number of independent parameters or coordinates that are needed for defining the position of a point, constrained to be on the object. For example, the dimension of a point is zero; the dimension is an intrinsic property of an object, in the sense that it is independent of the dimension of the space in which the object is or can be embedded. For example, a curve, such as a circle is of dimension one, because the position of a point on a curve is determined by its signed distance along the curve to a fixed point on the curve.

This is independent from the fact that a curve cannot be embedded in a Euclidean space of dimension lower than two, unless it is a line. The dimension of Euclidean n-space En is n; when trying to generalize to other types of spaces, one is faced with the question "what makes En n-dimensional?" One answer is that to cover a fixed ball in En by small balls of radius ε, one needs on the order of ε−n such small balls. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, but there are other answers to that question. For example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces. A tesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions", mathematicians express this as: "The tesseract has dimension 4", or: "The dimension of the tesseract is 4" or: 4D.

Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schläfli's 1852 Theorie der vielfachen Kontinuität, Hamilton's discovery of the quaternions and John T. Graves' discovery of the octonions in 1843 marked the beginning of higher-dimensional geometry; the rest of this section examines some of the more important mathematical definitions of dimension. The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension. For the non-free case, this generalizes to the notion of the length of a module; the uniquely defined dimension of every connected topological manifold can be calculated.

A connected topological manifold is locally homeomorphic to Euclidean n-space, in which the number n is the manifold's dimension. For connected differentiable manifolds, the dimension is the dimension of the tangent vector space at any point. In geometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2 are elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to "work". This state of affairs was marked in the various cases of the Poincaré conjecture, where four different proof methods are applied; the dimension of a manifold depends on the base field with respect to which Euclidean space is defined. While analysis assumes a manifold to be over the real numbers, it is sometimes useful in the study of complex manifolds and algebraic varieties to work over the complex numbers instead. A complex number has an imaginary part y, where x and y are both real numbers.

Gymnopilus luteus

Gymnopilus luteus called the "Yellow Gymnopilus" is a distributed mushroom of the Eastern United States, it contains the hallucinogens psilocybin and psilocin. Mistaken for Gymnopilus junonius. Pileus: 5 — 10 cm, Convex or nearly flat with an incurved margin that overhangs the gills. Buff yellow to warm buff orange slightly darker towards the center, smooth, silky or finely floccose-fibrillose, sometimes floccose-sqaumulose toward the center, flesh firm, pale yellow. Staining orange-brownish or sometimes bluish-green where injured or on age. Gills: Adnexed, close, pale yellow, becoming rusty brown with age. Spore Print: Rusty brown. Stipe: 4 — 8 cm.5 — 1.5 cm thick, equal to enlarging below, firm, colored like the cap, developing yellowish-rusty stains when handled, finely hairy, partial veil forms a fragile submembraneous ring or fibrillose annular zone near the apex. Staining orange-brownish or sometimes bluish-green where injured or in age. Taste: Bitter. Odor: Pleasant. Microscopic features: Spores 6 — 9 x 4 — 5 µm minutely warty, dextrinoid, surface finely roughened, no germ pore.

Pleurocystidia present, cheilocystidia scarcely projecting beyond the basidia, variously shaped. Caulocystidia absent. Clamp connections present. Bruising: Green or light blue bruising at the base on on the pileus. Gymnopilus luteus is found growing solitary to gregariously or in small clusters on dead hardwood and conifers, June - November distributed in eastern North America. List of Gymnopilus species Stamets, Paul. Psilocybin Mushrooms of the World. Berkeley: Ten Speed Press. ISBN 0-9610798-0-0. Hesler, L. R.. North American species of Gymnopilus. New York: Hafner. 117 pp

Video Game Masters Tournament

The Video Game Masters Tournament was an event, created in 1983 by Twin Galaxies to generate world record high scores for the 1984 U. S. Edition of the Guinness Book of World Records, it was the most prestigious contest of that era and the only one that the Guinness book looked to for verified world records on video games at the time. This contest was conducted under the joint efforts of Twin Galaxies and the U. S. National Video Game Team in 1983, 1984 and 1985 and by the U. S. National Video Game Team alone in 1986 and 1987. During its first year, 1983, the event was called the North America Video Game Challenge, but was changed to the Video Game Masters Tournament by the second year. In order to administer the contest, Twin Galaxies appointed the U. S. National Video Game Team to the task of collecting and verifying scores from the eight cities that participated in the 1983 edition of this event; the cities were Washington. The 1984 event was conducted in eight cities while the 1985 event was conducted in 35 cities.

The 1985 Video Game Masters Tournament was conducted as a fundraiser to raise donations for CARE. The fundraiser set aside half of all entry fees for the CARE/Twin Galaxies African Relief Fund