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John Flinders Petrie
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides (but no three) belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, , is the Coxeter number of the Coxeter group. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes. Petrie polygons can be defined more generally for any embedded graph. They form the faces of anothe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol . Dimensions If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is :r_u = a\frac \left(1 + \sqrt\right) \approx 1.401\,258\,538 \cdot a and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces) is :r_i = a\frac \sqrt \approx 1.113\,516\,364 \cdot a while the midradius, which touches the middle of each edge, is :r_m = a\frac \left(3 +\sqrt\right) \approx 1.309\,016\,994 \cdot a These quantities may also be expressed as :r_u = a\, \frac \phi :r_i = a\, \frac :r_m = a\, \frac where ''ϕ'' is the golden rat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Embedding
In topological graph theory, an embedding (also spelled imbedding) of a Graph (discrete mathematics), graph G on a surface (mathematics), surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with graph theory, vertices and simple arcs (Homeomorphism, homeomorphic images of [0,1]) are associated with graph theory, edges in such a way that: * the endpoints of the arc associated with an edge e are the points associated with the end vertices of e, * no arcs include points associated with other vertices, * two arcs never intersect at a point which is interior to either of the arcs. Here a surface is a compact space, compact, connected space, connected 2-manifold. Informally, an embedding of a graph into a surface is a drawing of the graph on the surface in such a way that its edges may intersect only at their endpoints. It is well known that any finite graph can be embedded in 3-dimensional Euclidean space \mathbb^3.. A planar graph is one that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Dodecahedron And Icosahedron
In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound. As a compound It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual. It has icosahedral symmetry (I''h'') and the same vertex arrangement as a rhombic triacontahedron. This can be seen as the three-dimensional equivalent of the compound of two pentagons ( " decagram"); this series continues into the fourth dimension as the compound of 120-cell and 600-cell and into higher dimensions as compounds of hyperbolic tilings. As a stellation This polyhedron is the first stellation of the icosidodecahedron, and given as Wenninger model index 47. The stellation facets for construction are: : In popular culture In the film ''Tron'' (1982), the character Bit took this shape when not speaking. In the cartoon series ''Steven Universe'' (2013-2019), Steven's shield bubbl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Cube And Octahedron
The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound. Construction The 14 Cartesian coordinates of the vertices of the compound are. : 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2) : 8: ( ±1, ±1, ±1) As a compound It can be seen as the compound of an octahedron and a cube. It is one of four compounds constructed from a Platonic solid or Kepler-Poinsot polyhedron and its dual. It has octahedral symmetry (O''h'') and shares the same vertices as a rhombic dodecahedron. This can be seen as the three-dimensional equivalent of the compound of two squares ( "octagram"); this series continues on to infinity, with the four-dimensional equivalent being the compound of tesseract and 16-cell. As a stellation It is also the first stellation of the cuboctahedron and given as Wenninger model index 43. It can be seen as a cuboctahedron with square and triangular pyramids added to each face. The stellation facets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Two Tetrahedra
In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra. Stellated octahedron There is only one uniform polyhedral compound, the stellated octahedron, which has octahedral symmetry, order 48. It has a regular octahedron core, and shares the same 8 vertices with the cube. If the edge crossings were treated as their own vertices, the compound would have identical surface topology to the rhombic dodecahedron; were face crossings also considered edges of their own the shape would effectively become a nonconvex triakis octahedron. Lower symmetry constructions There are lower symmetry variations on this compound, based on lower symmetry forms of the tetrahedron. * A facetting of a rectangular cuboid, creating compounds of two tetragonal or two rhombic disphenoids, with a bipyramid or rhombic fusil cores. This is first in a set of uniform compound of two antiprisms. * A facetting of a trigonal trapezohedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiregular Polytope
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polytopes of the Hyperspaces'' which included a wider definition. Gosset's list In three-dimensional space and below, the terms ''semiregular polytope'' and ''uniform polytope'' have identical meanings, because all uniform polygons must be regular. However, since not all uniform polyhedra are regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. The three convex semiregular 4-polytopes are the rectified 5-cell, snub 24-cell and rectified 600-cell. The only semiregular polytopes in higher dimensions are the ''k''21 polytopes, where the rectified 5-cell is the special case of ''k'' = 0. These were all listed by Gosset, but a proof of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension . Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians. Classically, a regular polytope in dimensions may be defined as having regular facets (-faces) and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes. A reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Toronto
The University of Toronto (UToronto or U of T) is a public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution of higher learning in Upper Canada. Originally controlled by the Church of England, the university assumed its present name in 1850 upon becoming a secular institution. As a collegiate university, it comprises eleven colleges each with substantial autonomy on financial and institutional affairs and significant differences in character and history. The university maintains three campuses, the oldest of which, St. George, is located in downtown Toronto. The other two satellite campuses are located in Scarborough and Mississauga. The University of Toronto offers over 700 undergraduate and 200 graduate programs. In all major rankings, the university consistently ranks in the top ten public universities in the world and as the top university ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Fifty-Nine Icosahedra
''The Fifty-Nine Icosahedra'' is a book written and illustrated by Harold Scott MacDonald Coxeter, H. S. M. Coxeter, Patrick du Val, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic regular icosahedron, icosahedron, according to a set of rules put forward by J. C. P. Miller. First published by the University of Toronto in 1938, a Second Edition reprint by Springer-Verlag followed in 1982. Tarquin's 1999 Third Edition included new reference material and photographs by K. and D. Crennell. Authors' contributions Miller's rules Although J. C. P. Miller, Miller did not contribute to the book directly, he was a close colleague of Coxeter and Petrie. His contribution is immortalised in his set of rules for defining which stellation forms should be considered "properly significant and distinct": :''(i) The faces must lie in twenty planes, viz., the bounding planes of the regular icosahedron.'' :''(ii) All parts composing t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patrick Du Val
Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named after him. Early life Du Val was born in Cheadle Hulme, Cheshire. He was the son of a cabinet maker, but his parents' marriage broke up. As a child, he suffered ill-health, in particular asthma, and was educated mostly by his mother. He was awarded a first class honours degree from the University of London External Programme in 1926, which he took by correspondence course. He was a talented linguist, for example teaching himself Norwegian so that he might read Peer Gynt. He also had a strong interest in history but his love of mathematics led him to pursue that as a career. His earliest publications show a leaning towards applied mathematics. His mother moved to a village near Cambridge and he became acquainted with Henry Baker, Lowndean P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sir W
''Sir'' is a formal honorific address in English for men, derived from Sire in the High Middle Ages. Both are derived from the old French "Sieur" (Lord), brought to England by the French-speaking Normans, and which now exist in French only as part of "Monsieur", with the equivalent "My Lord" in English. Traditionally, as governed by law and custom, Sir is used for men titled as knights, often as members of orders of chivalry, as well as later applied to baronets and other offices. As the female equivalent for knighthood is damehood, the female equivalent term is typically Dame. The wife of a knight or baronet tends to be addressed as Lady, although a few exceptions and interchanges of these uses exist. Additionally, since the late modern period, Sir has been used as a respectful way to address a man of superior social status or military rank. Equivalent terms of address for women are Madam (shortened to Ma'am), in addition to social honorifics such as Mrs, Ms or Miss. Etymol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
