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Polyhedra (book)
''Polyhedra'' is a book on polyhedra, by Peter T. Cromwell. It was published by in 1997 by the Cambridge University Press, with an unrevised paperback edition in 1999. Topics The book covers both the mathematics of polyhedra and its historical development, limiting itself only to three-dimensional geometry. The notion of what it means to be a polyhedron has varied over the history of the subject, as have other related definitions, an issue that the book handles largely by keeping definitions informal and flexible, and by pointing out problematic examples for these intuitive definitions. Many digressions help make the material readable, and the book includes many illustrations, including historical reproductions, line diagrams, and photographs of models of polyhedra. ''Polyhedra'' has ten chapters, the first four of which are primarily historical, with the remaining six more technical. The first chapter outlines the history of polyhedra from the ancient world up to Hilbert's third ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flexible Polyhedron
In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex (this is also true in higher dimensions). The first examples of flexible polyhedra, now called Bricard octahedra, were discovered by . They are self-intersecting surfaces isometric to an octahedron. The first example of a flexible non-self-intersecting surface in \mathbb^3, the Connelly sphere, was discovered by . Steffen's polyhedron is another non-self-intersecting flexible polyhedron derived from Bricard's octahedra. Bellows conjecture In the late 1970s Connelly and D. Sullivan formulated the bellows conjecture stating that the volume of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra homeomorphic to a sphere by using elimination theory, and then pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wenzel Jamnitzer
Wenzel Jamnitzer (sometimes Jamitzer, or Wenzel ''Gemniczer'') (1507/1508 – 19 December 1585) was a Northern Mannerist goldsmith, artist, and printmaker in etching, who worked in Nuremberg. He was the best known German goldsmith of his era, and court goldsmith to a succession of Holy Roman Emperors. A native of Vienna, Jamnitzer was a member of a Moravian German family which, for more than 160 years, had produced works under the names ''Jamnitzer, Jemniczer, Gemniczer, and Jamitzer''. Wenzel, with his brother Albrecht, was trained by his father Hans the Elder. Later, Wenzel's son Hans Jamnitzer (1539–1603) and grandson Christof Jamnitzer (1563–1618) continued his business. Jamnitzer worked as a court goldsmith for all the German emperors of his era, including Charles V, Ferdinand I, Maximilian II, and Rudolf II. Also, he probably invented an embossing machine. In 1534, Jamnitzer settled in Nuremberg. He made vases and jewelry boxes with great skill, in a styl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the University of Paris, Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the ''Bulletin of the American Mathematical Society''. Earlier publications (in the original German) appeared in and Nature and influence of the problems Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dehn Invariant
In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissection problem, dissected") into another, and whether a polyhedron or its dissections can Honeycomb (geometry), tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem by proving that not all polyhedra with equal volume could be dissected into each other. Two polyhedra have a dissection into polyhedral pieces that can be reassembled into either one, if and only if their volumes and Dehn invariants are equal. A polyhedron can be cut up and reassembled to tile space if and only if its Dehn invariant is zero, so having Dehn invariant zero is a necessary condition for being a space-filling polyhedron. The Dehn invariant of a self-intersection-free flexible polyhedron is invariant as it flexes. The Dehn invariant is zero for the cube but nonzero for the other Platonic solids, implying that the other solids cannot tile space and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Judith Grabiner
Judith Victor Grabiner (born October 12, 1938) is an American mathematician and historian of mathematics, who is Flora Sanborn Pitzer Professor Emerita of Mathematics at Pitzer College, one of the Claremont Colleges. Her main interest is in mathematics in the eighteenth and nineteenth centuries. Education Grabiner completed a Bachelor of Science degree at the University of Chicago in 1960. She was a graduate student in the history of science at Harvard University, completing a Master of Arts in 1962 and a Ph.D. in 1966, under I. Bernard Cohen. Her PhD dissertation was on Italian mathematician Joseph-Louis Lagrange. Career Grabiner was an instructor at Harvard for several years, before she and her husband Sandy Grabiner moved to California. She was a professor of history at California State University, Dominguez Hills from 1972 to 1985. Grabiner joined the mathematics department at Pitzer College in 1985, and has been the Flora Sanborn Pitzer Professor of Mathematics since 1994. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars. The earliest mathematical texts available are from Mesopotamia and Egypt – '' Plimpton 322'' ( Babylonian c. 2000 – 1900 BC), the ''Rhind Mathematical Papyrus'' ( Egyptian c. 1800 BC) and the '' Moscow Mathematical Papyrus'' (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most anci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter McMullen
Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at the University of Birmingham, where he received his doctorate in 1968. and taught at Western Washington University from 1968 to 1969. In 1978 he earned his Doctor of Science at University College London where he still works as a professor emeritus. In 2006 he was accepted as a corresponding member of the Austrian Academy of Sciences. Contributions McMullen is known for his work in polyhedral combinatorics and discrete geometry, and in particular for proving what was then called the upper bound conjecture and now is the upper bound theorem. This result states that cyclic polytopes have the maximum possible number of faces among all polytopes with a given dimension and number of vertices. McMullen also formulated the g-conjecture, later t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Combinatorics
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex). Additionally, many computer scientists use the phrase “polyhedral combinatorics” to describe research into precise descriptions of the faces of certain specific polytopes (especially 0-1 polytopes, whose vertices are subsets of a hypercube) arising from integer progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bill Casselman (mathematician)
William Allen Casselman (born November 27, 1941) is an American Canadian mathematician who works in representation theory and automorphic forms. He is a Professor Emeritus at the University of British Columbia. He is closely connected to the Langlands program and has been involved in posting all of the work of Robert Langlands on the internet. Career Casselman did his undergraduate work at Harvard College where his advisor was Raoul Bott and received his Ph.D from Princeton University in 1966 where his advisor was Goro Shimura. He was a visiting scholar at the Institute for Advanced Study in 1974, 1983, and 2001. He emigrated to Canada in 1971 and is a Professor Emeritus in mathematics at the University of British Columbia. Research Casselman specializes in representation theory, automorphic forms, geometric combinatorics, and the structure of algebraic groups. He has an interest in mathematical graphics and has been the graphics editor of the ''Notices of the American Mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope Compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected to form a convex polyhedron called its convex hull. A compound is a facetting of its convex hull. Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be used as the core for a set of stellations. Regular compounds A regular polyhedral compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. Unlike the case of polyhedra, this is not equivalent to the symmetry group acting transitively on its flags; the compound of two tetrahedra is the only regular compound with that property. There are five regular compounds of polyhedra: Best known is the regular compound of two tetrahedra, often calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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