Great Grand 120-cell
In geometry, the great grand 120-cell or great grand polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge arrangement as the small stellated 120-cell. See also * List of regular polytopes * Convex regular 4-polytope * Kepler-Poinsot polyhedron – regular star polyhedron * Star polygon – regular star polygons External links Regular polychora Reguläre Polytope References * Edmund Hess, (1883) ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder' * Coxeter, H. S. M. Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. . * John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and c ...
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Ortho Solid 011-uniform Polychoron 53p-t0
Ortho- is a Greek prefix meaning “straight”, “upright”, “right” or “correct”. Ortho may refer to: * Ortho, Belgium, a village in the Belgian province of Luxembourg In science * arene substitution patterns, two substituents that occupy adjacent positions on an aromatic ring * Chlordane, an organochlorine compound that was used as a pesticide In mathematics: * Orthogonal, a synonym for perpendicular * Orthonormal, the property that a collection of vectors are mutually perpendicular and each of unit magnitude * Orthodrome, a synonym for great circle, a geodesic on the sphere * Orthographic projection, a parallel projection onto a perpendicular plane In medicine: * Orthomyxovirus, a family of viruses to which influenza belongs * Orthodontics, a specialty of dentistry concerned with the study and treatment of malocclusions * Orthopedic, the study of the musculoskeletal system * Ortho-DOT, a psychedelic drug * Ortho-cept and Ortho Tri-cyclen, kinds of oral contraceptiv ...
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Orthographic Projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are ''not'' orthogonal to the projection plane. The term ''orthographic'' sometimes means a technique in multiview projection in which principal axes or the planes of the subject are also parallel with the projection plane to create the ''primary views''. If the principal planes or axes of an object in an orthographic projection are ''not'' parallel with the projection plane, the depiction is called ''axonometric'' or an ''auxiliary views''. (''A ...
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Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. Roberts, Siobhan, ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, He felt that mathematics and music were intimately related, outlining his ide ...
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Edmund Hess
Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes. See also * Schläfli–Hess polychoron * Hess polytope References * ''Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...'', (3rd edition, 1973), Dover edition, (p. 28* Hess ''E Uber die regulären Polytope höherer Art'', Sitzungsber Gesells Beförderung gesammten Naturwiss Marburg, 1885, 31-57 19th-century German mathematicians 20th-century German mathematicians 1843 births 1903 deaths People from Marburg University of Marburg alumni {{germany-mathematician-stub ...
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Star Polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations on regular simple and star polygons. Branko Grünbaum identified two primary definitions used by Johannes Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons. The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram. One definition of a ''star polygon'', used in turtle graphics, is a polygon having 2 or more turns (turning number and density), like in spirolaterals.Abelson, Harold, diSessa, Andera, 1980, ''Turtle Geometry'', MIT Press, p.24 Etymology Star polygon names combine a numeral prefix, such as ''penta-'', with the Greek suffix '' -gram'' (in this cas ...
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Star Polyhedron
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains. Mathematical studies of star polyhedra are usually concerned with regular, uniform polyhedra, or the duals of the uniform polyhedra. All these stars are of the self-intersecting kind. Self-intersecting star polyhedra Regular star polyhedra The regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures. There are four regular star polyhedra, known as the Kepler–Poinsot polyhedra. The Schläfli symbol implies faces with ''p'' sides, and vertex figures with ''q'' sides. Two of them have pentag ...
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Convex Regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. History The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: ''F'' − ''E'' + ''V''  2). That excludes cells and vertex figures such as the great dodecahedron and small stellated dodecahedron . Edmund Hess (1843–1903) published th ...
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List Of Regular Polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-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 that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol , and with its octahedral symmetry, [4,3] or , it is represented by Coxeter diagram . The regular polytopes are ...
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Small Stellated 120-cell Ortho-4gon
Small may refer to: Science and technology * SMALL, an ALGOL-like programming language * Small (anatomy), the lumbar region of the back * ''Small'' (journal), a nano-science publication * <small>, an HTML element that defines smaller text Arts and entertainment Fictional characters * Small, in the British children's show Big & Small Other uses * Small, of little size * Small (surname) * "Small", a song from the album ''The Cosmos Rocks'' by Queen + Paul Rodgers See also * Smal (other) * List of people known as the Small * Smalls (other) Smalls may refer to: * Smalls (surname) * Camp Robert Smalls, a United States Naval training facility * Fort Robert Smalls, a Civil War redoubt * Smalls Creek, a northern tributary of the Parramatta River * Smalls Falls, a waterfall in Maine, USA ...

Small Stellated 120-cell Ortho-6gon
Small may refer to: Science and technology * SMALL, an ALGOL-like programming language * Small (anatomy), the lumbar region of the back * ''Small'' (journal), a nano-science publication * <small>, an HTML element that defines smaller text Arts and entertainment Fictional characters * Small, in the British children's show Big & Small Other uses * Small, of little size * Small (surname) * "Small", a song from the album ''The Cosmos Rocks'' by Queen + Paul Rodgers See also * Smal (other) * List of people known as the Small * Smalls (other) Smalls may refer to: * Smalls (surname) * Camp Robert Smalls, a United States Naval training facility * Fort Robert Smalls, a Civil War redoubt * Smalls Creek, a northern tributary of the Parramatta River * Smalls Falls, a waterfall in Maine, USA ...

Small Stellated 120-cell Ortho-10gon
Small may refer to: Science and technology * SMALL, an ALGOL-like programming language * Small (anatomy), the lumbar region of the back * ''Small'' (journal), a nano-science publication * <small>, an HTML element that defines smaller text Arts and entertainment Fictional characters * Small, in the British children's show Big & Small Other uses * Small, of little size * Small (surname) * "Small", a song from the album ''The Cosmos Rocks'' by Queen + Paul Rodgers See also * Smal (other) * List of people known as the Small * Smalls (other) Smalls may refer to: * Smalls (surname) * Camp Robert Smalls, a United States Naval training facility * Fort Robert Smalls, a Civil War redoubt * Smalls Creek, a northern tributary of the Parramatta River * Smalls Falls, a waterfall in Maine, USA ...