Violet is the color at the end of the visible spectrum of light between blue and the invisible ultraviolet. Violet color has a dominant wavelength of 380–450 nanometers. Light with a shorter wavelength than violet but longer than X-rays and gamma rays is called ultraviolet. In the color wheel used by painters, it is located between blue and purple. On the screens of computer monitors and television sets, a color which looks similar to violet is made, with the RGB color model, by mixing red and blue light, with the blue twice as bright as the red; this is not true violet, for it does not match the color of a single wavelength shorter than that of blue light. The color's name is derived from the violet flower. Violet and purple look similar, but violet is a spectral color, with its own set of wavelengths on the spectrum of visible light. Purple is a dichromatic color, made by combining red. Amethyst is a notable violet crystal, its colour arising from iron and other trace elements in quartz. In history and purple have long been associated with royalty and majesty.

The emperors of Rome wore purple togas. During the Middle Ages violet was worn by bishops and university professors and was used in art as the color of the robes of the Virgin Mary. In Chinese painting, the color violet represents the "unity transcending the duality of Yin and yang" and "the ultimate harmony of the universe". In Hinduism and Buddhism violet is associated with the Crown Chakra. According to surveys in Europe and the United States, violet is the color people most associate with extravagance and individualism, the unconventional, the artificial, ambiguity. From the Middle English and old French violette, from the Latin viola, the names of the violet flower; the first recorded use of violet as a color name in English was in 1370. Violet can refer to the first violas which were painted a similar color. In the traditional color wheel used by painters and purple are both placed between red and blue. Purple occupies between crimson and violet. Violet is closer to blue, less intense and bright than purple.

From the point of view of optics, violet is a real color: it occupies its own place at the end of the visible spectrum, was one of the seven spectral colors of the spectrum first described by Isaac Newton in 1672. In the additive color system, used to create colors on a computer screen or on a color television, violet is simulated by purple, by combining blue light at high intensity with a less intense red light on a black screen; the range of purples is created by combining red light of any intensities. Violet is one of the oldest colors used by man. Traces of dark violet, made by grinding the mineral manganese, mixed with water or animal fat and brushed on the cave wall or applied with the fingers, are found in the prehistoric cave art in Pech Merle, in France, dating back about twenty-five thousand years, it has been found in the cave of Altamira and Lascaux. It was sometimes used as an alternative to black charcoal. Sticks of manganese, used for drawing, have been found at sites occupied by Neanderthal man in France and Israel.

From the grinding tools at various sites, it appears it may have been used to color the body and to decorate animal skins. More the earliest dates on cave paintings have been pushed back farther than 35,000 years. Hand paintings on rock walls in Australia may be older, dating back as far as 50,000 years. Berries of the genus rubus, such as blackberries, were a common source of dyes in antiquity; the ancient Egyptians made a kind of violet dye by combining the juice of the mulberry with crushed green grapes. The Roman historian Pliny the Elder reported that the Gauls used a violet dye made from bilberry to color the clothing of slaves; these dyes made a satisfactory purple, but it faded in sunlight and when washed. Violet and purple retained their status as the color of emperors and princes of the church throughout the long rule of the Byzantine Empire. While violet was worn less by Medieval and Renaissance kings and princes, it was worn by the professors of many of Europe's new universities, their robes were modeled after those of the clergy, they wore square violet caps and violet robes, or black robes with violet trim.

Violet played an important part in the religious paintings of the Renaissance. Angels and the Virgin Mary were portrayed wearing violet robes; the 15th-century Florentine painter Cennino Cennini advised artists: "If you want to make a lovely violet colour, take fine lacca, ultramarine blue..." For fresco painters, he advised a less-expensive version, made of a mixture of blue indigo and red hematite. In the 18th century, violet was a color worn by royalty and the wealthy, by both men and women. Good-quality violet fabric was expensive, beyond the reach of ordinary people. Many painters of the 19th century experimented with the uses of the color violet to capture the subtle effects of light. Eugène Delacroix made use of violet in the sky and shadows of many of his works, such as his painting of a tiger; the first cobalt violet, the intensely red-violet cobalt arsenate, was toxic. Although it persisted in some paint lines into the twentieth-century, it was displaced by less toxic cobalt compounds such as cobalt phosphate.

Cobalt violet appeared in the second half of the 19th century. Cobalt violet was used by Paul Signac, Claude Monet, Georges Seurat. Today, cobalt ammonium phosphate, cobalt lithium phosphate

This page lists the regular polytopes and regular polytope compounds in Euclidean and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an n-sphere and hyperbolic spaces. A Schläfli symbol describing an n-polytope equivalently describes a tessellation of an -sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation, called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol, with its octahedral symmetry, or, it is represented by Coxeter diagram; the regular polytopes are grouped by dimension and subgrouped by convex and infinite forms. Nonconvex forms have intersecting facets. Infinite forms tessellate a one-lower-dimensional Euclidean space.

Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale; this allows vertex figures to have negative angle defects, like making a vertex with seven equilateral triangles and allowing it to lie flat. It can be at the right scale of a hyperbolic plane. A more general definition of regular polytopes which do not have simple Schläfli symbols includes regular skew polytopes and regular skew apeirotopes with nonplanar facets or vertex figures; this table shows a summary of regular polytope counts by dimension. There are no Euclidean regular star tessellations in any number of dimensions. A one-dimensional polytope or 1-polytope is a closed line segment, bounded by its two endpoints. A 1-polytope is regular by definition and is represented by Schläfli symbol, or a Coxeter diagram with a single ringed node. Norman Johnson gives it the Schläfli symbol. Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes.

It is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. The two-dimensional polytopes are called polygons. Regular polygons are cyclic. A p-gonal regular polygon is represented by Schläfli symbol. Only convex polygons are considered regular, but star polygons, like the pentagram, can be considered regular, they use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed. Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary; the Schläfli symbol represents a regular p-gon. The regular digon can be considered to be a degenerate regular polygon, it can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a sphere or torus. There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers.

They share the same vertex arrangements of the convex regular polygons. In general, for any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols for all m such that m < n/2 and m and n are coprime. Cases where m and n are not coprime are called compound polygons. Star polygons that can only exist as spherical tilings to the monogon and digon, may exist, however these do not appear to have been studied in detail. There exist failed star polygons, such as the piangle, which do not cover the surface of a circle finitely many times. In 3-dimensional space, a regular skew polygon is called an antiprismatic polygon, with the vertex arrangement of an antiprism, a subset of edges, zig-zagging between top and bottom polygons. In 4-dimensions a regular skew polygon can have vertices on a Clifford torus and related by a Clifford displacement. Unlike antiprismatic skew polygons, skew polygons on double rotations can include an odd-number of sides, they can be seen in the Petrie polygons of the convex regular 4-polytopes, seen as regular plane polygons in the perimeter of Coxeter plane projection: In three dimensions, polytopes are called polyhedra: A regular polyhedron with Schläfli symbol, Coxeter diagrams, has a regular face type, regular vertex figure.

A vertex figure is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular polygon. Existence of a regular polyhedron is constrained by an inequality, related to the vertex figure's angle defect: 1 p + 1 q > 1 2: Polyhedron 1 p + 1 q = 1 2: Euclidean plane tiling 1 p + 1 q < 1 2: Hyperbolic plane tiling

Jeff Rubens is an American bridge player and writer of books including Secrets of Winning Bridge and Expert Bridge Simplified. He is best known for long association with The Bridge World monthly magazine, as co-editor under Edgar Kaplan from 1967 and as editor and publisher since Kaplan's death in 1997. Rubens is from New York. Rubens attended Stuyvesant High School in New York City, where he was captain of the math team in 1957, the year he graduated, he has an undergraduate degree from Cornell University and a graduate degree from Brandeis University. He won seven North American championship events in the 1960s-70s, represented North America in the 1973 world championship, "gave up competitive bridge for family reasons" soon after. Rubens is a retired professor of mathematics and computer science at Pace University in New York. Rubens became an ACBL Life Master at 20 and won two North American championship events at age 23 in 1965, the Men's Pairs and Men's Teams. Seven years he played with B.

Jay Becker on teams that won the 1972 Spingold national championship and the subsequent trial to represent North America in the world championship. Becker was 69 the oldest participant in a Bermuda Bowl tournament, famously conservative. According to Charles Goren's report, Becker is an ultraconservative who has refused to play so accepted a convention as Stayman. Rubens, a math teacher, employs advanced ideas on everything from opening bids to opening leads. Expert selectors would have been hard-pressed to put together a less partnership, yet from their base of operations in the closed room this pair kept sending through perfect results on hand after hand, a performance that the vaunted Blue Team would have found difficult to top. Their opponents in the Trials could not begin to match it. At Guarujá, they finished fourth of five teams in the 1973 Bermuda Bowl. Rubens and Paul Heitner established the short-lived Bridge Journal in the mid-1960s, it is best known for Journalist leads. The Bridge World monthly was established by Ely and Josephine Culbertson in 1929.

Edgar Kaplan acquired it from McCall Corporation in 1966 and served as publisher and editor from the January 1967 issue until his death in September 1997. Some time in 1967 he brought Rubens on board as co-editor, they made the editorial column a prominent feature. ACBL Hall of Fame, Blackwood Award 2004 Precision Award 1977, 1979, 1982 North American Bridge Championships Spingold 1972 Men's Board-a-Match Teams 1965 Men's Pairs 1965 United States Bridge Championships Open Team Trials 1972 Win at Poker. World Bridge Federation. Jeff Rubens at Library of Congress Authorities, with 10 catalog records