
	Where results make sense

Topic: Polychora

Related Topics

Uniform Polychora Project

uniform polychora

nonprismatic polychora

convex uniform polychora

nonconvex uniform polychora

In the News (Mon 22 Mar 10)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Uniform Polychora
		I have been studying polychora, as well as higher dimensional polytopes, since 1990.
		have given all of the uniform polyhedra and uniform polychora (except idcossids and dircospids) short names.
		- this includes the polychora with wedge shaped vertex figures and their facetings (sirgax is one).
	members.aol.com /hedrondude/polychora.html (670 words)

	polychora (Site not responding. Last check: 2007-09-27)
		A polychoron (plural: polychora) (from Greek poly meaning "many" and choros meaning "room" or "space") is a four-dimensional polytope, also known as a 4-polytope, or polyhedroid.
		The use of the term "polychoron" for such figures has been advocated by George Olshevsky, and is also supported by Norman W. Johnson.
		A cell is the three-dimensional analogue of a face, and is therefore a polyhedron.
	www.yourencyclopedia.net /Polychora.html (342 words)

	Polychoron - Wikipedia, the free encyclopedia
		The Uniform Polychora Project has classified the 8,186 currently known uniform polychora into 29 groups.
		The remaining convex uniform polychora may be grouped into two infinite families: the duoprisms and the polyhedral prisms.
		Semiregular 4-polytopes Subset of uniform polychora with regular polyhedron cells.
	en.wikipedia.org /wiki/Polychoron (688 words)

	Talk:Polychoron - InformationBlast
		"Polychora" is a made-up word that, as far as I know, is not used by mathematicians.
		I just did a web search on google for polychora, and found several pages of links, all of which use the word polychora in the meaning used here, some as prestigious as mathworld.com.
		I certainly think that 'polychora' is a much more usable term than '4 dimensional figures', which is the only competition that I know of.
	www.informationblast.com /Talk:Polychoron.html (517 words)

	H4-Zome Polychora
		A Wythoff symbol is an extension of a Coxeter graph, obtained by circling a non-empty subset of its vertices.
		Not many people have seen models of the last 5 polychora in the table, and this is part of the reason why it is difficult to find suitable names for them.
		This phenomenon does not occur for all uniform polychora, and, indeed, it is possible to construct Zome models of other uniform polychora for which each of the vertices is rendered uniquely by a Zome connector ball.
	homepages.wmich.edu /~drichter/h4polychorazome.htm (1557 words)

	Uniform Polychora
		Category 5: Pentagonal Rectates - (Polychora 73 - 132) These are the polychora that belong to the rox army, there are four regiments here, the rox, righi, ragishi, and rigfix regiments, each having 15 members, there are also two coinciding members and five exotic members in each regiment, which are no longer counted as polychora.
		Category 12: Podiumverts - (Polychora 482 - 511) These 30 polychora are grouped into 4 regiments of 7 and the extra two members of the rit regiment (sto and gotto).
		Category 19: Prisms - (Polychora 889 - 962) These 74 polychora are the prisms of 74 of the 75 uniform polyhedra (we excluded the cube since the cube prism is the tesseract).
	www.polytope.net /hedrondude/polychora.htm (4151 words)

	Uniform Polychora Project
		The Uniform Polychora Project in geometry is a collaborative effort to recognize and standardize terms used to describe objects in higher-dimensional spaces.
		John Conway and Michael Guy established by compuer analyssi that there are 64 convex nonprismatic uniform polychora, in the mid-1960s.
		There are only 64 are convex uniform polychora with the other 8126 as nonconvex uniform polychora.
	www.sciencedaily.com /encyclopedia/uniform_polychora_project (302 words)

	Polychoron (Site not responding. Last check: 2007-09-27)
		The Uniform Polychora Project has classified the 186 currently known uniform polychora into 29 There may be more.
		There is a technique called the Coxeter-Dynkin for performing Wythoff's construction for producing uniform This method allows the polychora to be enumerated.
		Another commonly discussed figure that resides in space is the 3-sphere for which the term glome has been proposed.
	www.freeglossary.com /Polychoron (313 words)

	Cross-polytope - InformationBlast
		It is one of six regular, convex polychora, the others being the pentachoron (4-simplex), tesseract (hypercube), 24-cell, 120-cell, and 600-cell.
		These polychora where first described by the Swiss mathematican Ludwig Schläfli in the mid-19th century.
		The hexadecachoron has 16 cells all of which are regular tetrahedra.
	www.informationblast.com /Cross-polytope.html (372 words)

	Four Dimensional Figures Page
		The most recent changes to the roster of uniform polychora begin with two odd figures having the symmetry of a square-octagonal duoprism that Jonathan discovered in January 2002.
		The polychora appear as changing polyhedral sections, as we would see them if they were passing through our real three-space at a uniform rate.
		The numerals on the edges in the pictures denote the number of sides of the corresponding regular-polygonal faces at a vertex of the polychoron; the length of an edge labeled p is thus 2cos(pi/p).
	members.aol.com /Polycell/uniform.html (4264 words)

	The Fourth Dimension (Site not responding. Last check: 2007-09-27)
		This is a way of describing not only regular polyhedra in three dimensions, but also regular polychora in four dimensions, or regular polytopes in any number of dimensions.
		And the next number indicates how many polychora meet at a two-dimensional face to construct a five-dimensional polytope, and so on.
		Since the first number is distinct from the rest, I have adopted the form [(5) 3, 3] for the Schäfli symbol usually written {5, 3, 3} (which, of course, looks like a set, and can therefore cause confusion).
	www.hypermaths.org /quadibloc/math/fdiint.htm (1660 words)

	600-cell
		The reason is that this model only shows the edges and vertices of the 600-cell, not the faces or the cells.
		These 720 edges and 120 vertices coincide with those of 3 other regular polychora.
		This phenomenon may be more familiar in the 3-dimensional case, where, for example, the edges and vertices of the regular icosahedron {3,5} coincide with those of the great dodecahedron {5,5/2}.
	homepages.wmich.edu /~drichter/600cells.htm (1004 words)

	Notes for project (Site not responding. Last check: 2007-09-27)
		A four-dimensional polytope is sometimes referred to as a polychoron (plural is polychora).
		Students study the Platonic solids in geometry, learning that they are the only regular, convex polyhedra (regular means their sides are all the same and are made of regular polygons).
		Remember that, in 3-space, a regular polyhedron is bounded by regular polygons; hence in 4-space a regular polychoron is bounded by regular polyhedra.
	home.comcast.net /~t.downey/dimension_project.htm (4993 words)

	[No title]
		These all arise \ during construction of the star polychora, in the \ \[CloseCurlyDoubleQuote]Solid Shadows of Star Polychora\ \[CloseCurlyDoubleQuote] section.\n\t\n\tOf the nine regular polyhedra, only \ three symmetry groups are involved.
		I have followed the constructions for the vertices of \ the six regular convex polychora, as given in ", StyleBox["Regular Polytopes", FontSlant->"Italic"], ", exactly; and these are cast in such a form that a polychoron and its \ dual are in reciprocal positions.
		There are four such axes, defined by the vectors from the center of the \ polychoron to a vertex, edge center, face center, and cell center.
	www.math.uconn.edu /~rogalski/4d/html/sections/RegularPolytopes.nb (5416 words)

	urticator.net - Notes
		Just as in three dimensions there are polyhedra, so in four dimensions there are polychora.
		I already mentioned the tesseract and the hexadecachoron, but there are others—including one with 600 faces!—that you can read about at the following sites.
		The first has an explanation that is both nice and short, while the second has an unbelievable amount of detail.
	www.urticator.net /essay/4/453.html (471 words)

	About "Uniform Polytopes in Four Dimensions" (Site not responding. Last check: 2007-09-27)
		"Polychoron" (plural: polychora) is Olshevsky's term for a four-dimensional polytope, analogous to polygon in two dimensions and polyhedron in three.
		He includes descriptions and numerical information for all the convex uniform polychora in finite and infinite families: Coxeter-Dynkin graphs, names, symmetry groups, Schlaefli symbol(s), number of elements of each dimension, and vertex figures.
		Nomenclature; uniform polychora based on the regular pentachoron; the hypercube and regular hexakaidecachoron; the regular icosikaitetrachoron; the regular hecatonkaiicosachoron and regular hexacosichoron; anomalous non-Wythoffian polychoron; prismatic polychora; and uniform polychora derived from hyperspherical tetrahedron B4.
	mathforum.org /library/view/5276.html (135 words)

	Polyhedron, Polyhedra, Polytopes, ... - Numericana
		The equivalent of a polyhedron in dimension 4 is called a polychoron (plural polychora).
		Polychora are discussed extensively on beautifully illustrated pages proposed by George Olshevsky and Jonathan Bowers.
		In dimension 4, we have 6 regular polychora.
	home.att.net /~numericana/answer/polyhedra.htm (4643 words)

	Uniform Polychora Project . John Conway . Norman Johnson . Hypersphere
		John Conway and Michael Guy established by computer analysis that there are 64 convex polychora convex nonprismatic polychora nonprismatic uniform polychora, in the mid-1960s.
		John Horton Conway born December 26, 1937, Liverpool, England is a prolific mathematician active in the theory of finite group mathematics groups, knot theory, number theory, combinatorial...
		The village of Le Roy, Michigan Le Roy is located within the township.
	www.uk.fraquisanto.net /Uniform_Polychora_Project (471 words)

	Ditela, polytopes and dyads (Site not responding. Last check: 2007-09-27)
		Modern theories of polygons and polyhedra understand these figures to be two- and three-dimensional examples of more general p-dimensional polytopes, or p-topes.
		In higher dimensions we have four-dimensional polycells or polychora.
		For five dimensions and up, three workers have jointly developed a naming system which is evident from the table and comments below.
	www.queenhill.demon.co.uk /polyhedra/theory/ditela.html (1152 words)

	Mathematical Links
		He has discovered many of the uniform Polychora or 4d polytopes.
		Zvi Har'El is author of the Kaleido program, which gives the uniform polychora and their duals.
		His site has all of the uniform polychora, with the Dynkin Symbol for these.
	www.geocities.com /os2fan2/gloss/pglinks.html (657 words)

	Artifact
		A great book on 3 dimensional polytopes (aka polyhedra) is Alan Holden's Shapes, Space and Symmetry, published by Dover Books.
		George Olshevsky's polychora page- George coined the term "polychora".
		In 2 dimensions you have polygons (many edges), in 3 polyhedra (many faces), and 4 dimensional polytopes are called polychora (many rooms).
	clowder.net /hop/gofix/artifact.html (154 words)

	Schlaefli's Construction (Site not responding. Last check: 2007-09-27)
		A regular polyhderon is made up from regular polygons of one kind, with a constant number at a corner.
		A regular polychora is done in the same manner, with polyhedra of one kind, the same at each edge.
		For the polychora, the notation uses three numbers, eg
	www.geocities.com /os2fan2/schlafli.html (157 words)

	Polyhedra Links
		There are links to polyhedra of the third dimension and polychora of the fourth dimension, which are analogues of polyhedra.
		There are links to regular polychora (called polytopes) and a link to a polytope viewer which can show 3d cross sections of polytopes having four or more dimensions.
		There are also pages on general geometry, which includes links on 3rd and 4th dimension geometry) and just for fun, there are links with info on how polyhedra relate to crystal structure, and the shape of fair dice.
	math.scu.edu /~ffarris/m101/polyhedra.html (702 words)

	Geometry (Site not responding. Last check: 2007-09-27)
		Includes webpages that focus exclusively about two dimension figures, such as lines, circles, and polygons.
		Polytopes include polygons (two-dimensional), polyhedra (three-dimensional), polychora (four-dimensional), and their higher dimensional analogs.
		Thus, polyhedra are built up from polygons, and polychora are built up from polyhedra.
	www.cool-sites-project.com /Science/Math/Geometry (363 words)

	Rebecca Frankel
		The Symmetriad: A Journey of Discovery through the land of the Polychora.
		essay describing these polychora from an artistic point of view.
		I have learned more from Gerry about his vision of the intelligent book, and I implemented for him a
	www.swiss.ai.mit.edu /~rfrankel (695 words)

	[No title] (Site not responding. Last check: 2007-09-27)
		It has the same vertices, edges, and square faces as the Great Prismosaurus, but the pentagons are discarded and replaced by the pentagrams of the regular star-polychoron {5/2,5,3}, the stellated hecatonicosachoron (which has the same vertices and edges as the great grand hecatonicosachoron and both Prismosauri).
		In the first Hybrid, 120 pentagrammatic prisms (ten girdles of twelve) of the Small Prismosaurus are exchanged for 120 pentagonal prisms of the Great Prismosaurus.
		Meanwhile, information on all the convex uniform polychora may be found below.
	www.anomalies.net /ufo/mars/tem-mes/temthreads/a-g/bkparque.etphysics.txt (18969 words)

	Colin Fine (Site not responding. Last check: 2007-09-27)
		I have explored many different areas at different times, but one field I keep returning to is polytopes (polygons, polyhedra etc).
		I discovered for myself the six platonic polychora (4-dimensional polytopes), and the doubly infinite series of archimedean polychora that I call 'bi-prisms', because their cells are polygonal prisms of two kinds.
		I have been worrying for years at the problem of why there are only six polychora, rather than eleven.
	www.kindness.demon.co.uk (1808 words)

	Zometool catalogue
		The 600-cell has the same edge-skeleton as the following polychora:
		The great stellated 120-cell has the same edge-skeleton as the following polychora:
		has the same edge-skeleton as the following polychora:
	math.stanford.edu /~silva/amati/zome.html (52 words)

	etc.html (Site not responding. Last check: 2007-09-27)
		Jonathan Bower's rendering of cross sections of a 4-dimensional star polytope.
		The.1382 cross section of this polychora was used as the artifact in the painting Artifact.
		With Jonathan's help I constructed this model to use for the painting Artifact.
	www.clowder.net /hop/gofix/gofix.html (39 words)

Try your search on: Qwika (all wikis)