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Topic: List of regular polytopes

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Cuboctahedron

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Archimedean solid

	Polytopes :: Geometry : Gourt
		In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions.
		The Platonic solids, or regular polytopes in three dimensions, were a major focus of study of ancient Greek mathematicians (most notably Euclid's Elements), probably because of their intrinsic aesthetic qualities.
		Regular Polytopes - Derivation of volume equations for regular polygons, polyhedra, and polytopes, with images.
	science.gourt.com /Math/Geometry/Polytopes.html (457 words)

	List of regular polytopes - Wikipedia, the free encyclopedia
		The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms.
		The existence of a regular polychoron {p,q,r} is constrained by the existence of the regular polyhedra {p,q},{q,r}.
		In five dimensions, a regular polytope can be named as {p,q,r,s} where {p,q,r} is the hypercell (or teron) type, {p,q} is the cell type, {p} is the face type, and {s} is the face figure, {r,s} is the edge figure, and {q,r,s} is the vertex figure.
	en.wikipedia.org /wiki/List_of_regular_polytopes (1582 words)

	Platonic solid Summary
		Triangular faces: each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
		The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2π.
		The next most regular convex polyhedra after the Platonic solids are the cuboctahedron, which is a rectification of the cube and the octahedron, and the icosidodecahedron, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron).
	www.bookrags.com /Platonic_solid (4116 words)

	Regular polytopes (Platonic solids) in 4D
		Regular polytopes or platonic solids are convex solids (closed) where all the building blocks (vertices, edges, faces, hyperfaces) have the same characteristics.
		Each regular polytope is supplied as data in at least two versions, the first is a simple ascii format listing vertices, edges, and faces.
		These 3 regular polytopes are the equivalent of the tetrahedron, cube, and octahedron in 3 dimensions, they are normally called the n-simplex, n-cube, and n-crosspolytope respectively where n stands for the dimension.
	local.wasp.uwa.edu.au /~pbourke/geometry/platonic4d/index.html (722 words)

	Tesseract (Site not responding. Last check: 2007-10-19)
		The tesseract is to the cube as the cube is to the square; or, more formally, the tesseract can be described as a regular convex 4-polytope whose boundary consists of eight cubical cells.
		Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron.
		The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol {3,3,4}.
	www.tocatch.info /en/Hexachoron.htm (1939 words)

	platonic
		All the edges of a regular polygon are the same length, and all the angles are equal.
		All the faces of a regular polytope must be lower-dimensional regular polytopes of the same size and shape, and all the vertices, edges, etc. have to look identical.
		In general, the 'dual' of a regular polytope is another polytope, also regular, having one vertex in the center of each face of the polytope we started with.
	math.ucr.edu /home/baez/platonic.html (2618 words)

	Geometry of Polyhedrons. (Site not responding. Last check: 2007-10-19)
		Example of a semi-regular polyhedron of 32 faces built of regular hexagons and pentagons.
		Applet: Cross sections of 2 polytopes built of congruent bipyramids (24 and 32 cells).
		Applet: Cross sections of 2 polytopes built of congruent bipyramids (720 and 1200 cells).
	www.mathandcomp.com /list.htm (353 words)

	Building convex polytopes
		By projective duality, we can also reconstruct the polytope from the plane equations of its facets and one interior point, or equivalently, from a list of halfspaces whose intersection is the polytope.
		It is open whether every polytope can be unfolded into a simple net, that is, one that does not overlap itself, by cutting along edges; however, even nonsimple nets can be used to define polyhedral metrics.
		This polytope is unique up to reflections and rotations about the origin, and every combinatorial symmetry of the graph is realized by a symmetry of the polytope.
	compgeom.cs.uiuc.edu /~jeffe/open/makepoly.html (1455 words)

	Alibris: Polytopes
		The simplest and most interesting of these is the moment polytope, a convex polyhedron which sits inside the dual of the Lie algebra of G. One of the main goals of this monograph is to describe what kinds of geometric...
		Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices: Polytopes in Hypercubes and Zn
		Questions arising from linear programming and combinatorial optimization have been a driving force for modern polytope theory, and algorithms now provide the means to computationally study polytopes, to compute their parameters, and to construct examples of large complexity.
	www.alibris.com /search/books/subject/Polytopes (556 words)

	Science - Math - Geometry - Polytopes (Site not responding. Last check: 2007-10-19)
		Polytopes include polygons (two-dimensional), polyhedra (three-dimensional), polychora (four-dimensional), and their higher dimensional analogs.
		A regular polytope is composed of regular (n-1)-dimensional polytopes.
		There are an infinite number of regular convex polygons, five regular convex polyhedra, six regular convex polychora, and three regular convex polytopes for all dimensions five or higher.
	www.inter.co.yu /kategorije/Science/Math/Geometry/Polytopes (539 words)

	Polytopes: Geometry at Canadian Content (Site not responding. Last check: 2007-10-19)
		Additional Information: Polytopes include polygons (two-dimensional), polyhedra (three-dimensional), polychora (four-dimensional), and their higher dimensional analogs.
		Vertex figures, filling, faceting diagrams, stellation, defining polytopes through generators, trimethoric and trisynaptic polyhedra, space-filling polyhedra, lost stellations of the icosahedron, and links.
		Regular, rectified, and truncated polytopes with normal and hidden-detail-removed projections.
	www.canadiancontent.net /dir/Top/Science/Math/Geometry/Polytopes (322 words)

	HyperSpace Polytope Slicer
		A 2-dimensional polytope is a "polygon" -- an area of a 2-dimensional space that is bounded by 1-dimensional line segments.
		The most complex of the 4-dimensional regular convex polytopes is the 600-cell, in which 600 regular tetrahedra enclose a volume of 4-dimensional space.
		The regular convex polytopes in 3 dimensions are the 5 "Platonic Solids" -- the tetrahedron, cube, octahedron, dodecahedron and icosahedron.
	dogfeathers.com /java/hyperslice.html (1396 words)

	DaVinci: Science> Math> Geometry> Polytopes
		- Regular, rectified, and truncated polytopes with normal and hidden-detail-removed projections.
		- Vertex figures, filling, faceting diagrams, stellation, defining polytopes through generators, trimethoric and trisynaptic polyhedra, space-filling polyhedra, lost stellations of the icosahedron, and links.
		Contains a java applet based on a model which allows for generation of multidimensional regular and semi-regular polytopes.
	www.bluegrassdavinci.com /ODP/Science/Math/Geometry/Polytopes (191 words)

	The Geometry Junkyard: Many-dimensional Geometry
		The charged particle model: polytopes and optimal packing of p points in n dimensional spheres.
		Stainless steel 3d model of the 24-cell (one of the six regular polytopes in four dimensions), by Adrian Ocneanu, installed as a sculpture in the Penn State Math Department.
		He lists both general upper and lower bounds as functions of a, b, and c, and specific constructions for specific sizes of box.
	www.ics.uci.edu /~eppstein/junkyard/highdim.html (1236 words)

	Amazon.com: Regular Complex Polytopes: Books: H. S. M. Coxeter (Site not responding. Last check: 2007-10-19)
		The properties of polytopes, the four-dimensional analog of polyhedra, exercise an intellectual fascination that appeals strongly to the mathematically inclined, whether they are professionals, students or amateurs.
		In this classic book, Professor Coxeter explores these properties in easy stages introducing the reader to complex polytopes (a beautiful generalization of regular solids derived from complex numbers) and the unexpected relationships that complex polytopes have with concepts from various branches of mathematics.
		The properties of regular solids exercise a fascination which often appeals strongly to the mathematically inclined, whether they are professionals, students or amateurs.
	www.amazon.com /Regular-Complex-Polytopes-H-Coxeter/dp/0521394902 (820 words)

	Amazon.ca: Abstract Regular Polytopes: Books: Peter McMullen,Egon Schulte (Site not responding. Last check: 2007-10-19)
		Almost everything known about abstract regular polytopes until the date of publication may be found somewhere within its 551 pages.' Zentralblatt MATH
		Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra.
		The rapid development of the subject in the past twenty years has resulted in a rich new theory featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology.
	www.amazon.ca /Abstract-Regular-Polytopes-Peter-McMullen/dp/0521814960 (486 words)

	SiteSearch Directory - Polytopes (Site not responding. Last check: 2007-10-19)
		Regular, rectified, and truncated polytopes with normal and hidden-detail-removed
		Derivation of volume equations for regular polygons, polyhedra, and polytopes, with
		The content of the SiteSearch directory is based on the Open Directory and is enhanced using SiteSearch technology.
	www.sitesearch.ca /?d=science/math/geometry/polytopes (194 words)

	CiteULike: Tag polytopes (Site not responding. Last check: 2007-10-19)
		A Linear Bound On The Diameter Of The Transportation Polytope*
		Lower bound for the maximal number of facets of a 0/1 polytope
		Random walks on the vertices of transportation polytopes with constant number of sources
	www.citeulike.org /tag/polytopes (296 words)

	Amazon.ca: Regular Polytopes: Books: H. S. M. Coxeter (Site not responding. Last check: 2007-10-19)
		Foremost book available on polytopes, incorporating ancient Greek and most modern work done on them.
		Beginning with polygons and polyhedrons, the book moves on to multi-dimensional polytopes in a way that anyone with a basic knowledge of geometry and trigonometry can easily understand.
		The book could be improved with computer graphics of the polytopes, but otherwise it is perfect.
	www.amazon.ca /Regular-Polytopes-H-M-Coxeter/dp/0486614808 (365 words)

	3D Sections of 4D Regular Convex Polytopes
		Then, I define el (short for edgelist) to be a list of all the edges in the particular polytope that I am slicing.
		I got all my edgelists from Russel Towle's RegularPolytopes.nb An edgelist is in the form of a list of pairs of vertices.
		It is not really complete, since it doesnt allow for lists of 2, 3, or more coplanar points.
	www.math.uconn.edu /~rogalski/4d/html/sections/sections2.htm (965 words)

	Science Search > Polytopes
		Collection of discussions regarding volume, vertices, dissection, g-holed tori, and other subjects related to polytopes.
		Geometric model building courses to improve understanding of geometry, using a Matrix kit for sale on the site that can be used in order to create the models yourself.
		List of links to sites on geometric properties of polygons, polyhedra, and higher dimensional polytopes.
	www.science-search.org /index/Math/Geometry/Polytopes (197 words)

	TheMostValuablePage Directory - /Science/Math/Geometry/Polytopes
		» Polygons, Polyhedra, Polytopes - Regular, rectified, and truncated polytopes with normal and hidden-detail-removed projections.
		» Polyhedra, Platonic Solids, Polytopes - Definitions, pictures, templates, and coordinates of the regular 3d and 4d polytopes.
		» Regular Polytopes - Derivation of volume equations for regular polygons, polyhedra, and polytopes, with images.
	www.themostvaluablepage.com /Top/Science/Math/Geometry/Polytopes (303 words)

	SourceForge.net: Polytope Viewer
		This is an attempt to create comprehensive software that will draw the two dimentional projection and rotation of any regular polytope.
		A regular polytope is a convex shape in any dimention with all identical faces.
		View list of RSS feeds available for this project
	sourceforge.net /projects/polytopes (114 words)

	Abstract Regular Polytopes - Adobe Reader PDF eBook - Get eBooks!
		They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations.
		The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology.
		Abstract regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry.
	www.ebookmall.com /ebook/165064-ebook.htm (791 words)

	Polytopes
		This package contains functions that give geometrical characteristics of regular polygons and regular polyhedra.
		Polygons and polyhedra are identified by name (
		These coordinates form the vertices of an octahedron.
	documents.wolfram.com /v4-fr/AddOns/Geometry/Polytopes.html (142 words)

	Amazon.com: Regular Polytopes: Books: H. S. M. Coxeter (Site not responding. Last check: 2007-10-19)
		Be the first person to add product information.
		Books every algebraist should own: A list by "ayrshire1875"
		Polyhedra & Geodesic Dome Resources: A list by "pascalin"
	www.amazon.com /Regular-Polytopes-H-S-Coxeter/dp/0486614808 (908 words)

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