Semiregular polyhedron

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Topic: Semiregular polyhedron

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	Pseudo Rhombicuboctahedra
		He discovered a 14th semiregular polyhedron [figure of pseudo-rhombicuboctahedron] which differs from the one shown in [figure of rhombicuboctahedron] only in that the upper part, consisting of 5 squares and 4 equilateral triangles, is rotated through an angle of pi/4.
		The pseudo-rhombicuboctahedron is not classified as a semi-regular polyhedron, because the essence (and beauty) of the semi-regular polyhedra is not about local properties of each vertex, but the symmetry operations under which which the entire object appears unchanged.
		As far as I can see from studying the uniform polyhedra, these are the only two possible examples of a polyhedron in which all faces are regular and all vertices are locally identical, yet the polyhedron is not uniform; however I know of no simple proof of this.
	euch3i.chem.emory.edu /proposal/www.li.net/~george/virtual-polyhedra/pseudo-rhombicuboctahedra.html (690 words)

	A Fuller Explanation: Chapter Twelve all space filling
		A model of this semiregular polyhedron can be constructed by taping together eight cardboard hexagons and six squares of the same edge length.
		With this generalized principle, we are equipped to determine whether any polyhedron that submits to analysis in terms of octahedral and tetrahedral components is a space filler.
		Lacking one octahedron, the VE must not be a space filler, a conclusion that is consistent with our earlier observation of the IVM.
	www.angelfire.com /mt/marksomers/117.html (1090 words)

	52B: Polytopes and polyhedra
		Schreiber, Peter: "What is the true number of semiregular (Archimedean) solids?", Festschrift on the occasion of the 65th birthday of Otto Krötenheerdt.
		How to compute the volume of a polyhedron?
		"Polyhedron Models", by Magnus J. Wenninger (Cambridge University Press, London-New York, 1971): has models of all 52 uniform polyhedra and some stellations.
	www.math.niu.edu /~rusin/known-math/index/52BXX.html (982 words)

	soccer
		Subject: Re: SOCCER BALLS AND SPHERICAL GEOMETRY Date: Mon, 27 Dec 1999 06:26:00 -0500 Newsgroups: alt.math,alt.math.undergrad,sci.math Keywords: regular, semiregular polyhedra A soccer ball shows the projection of one of the possible "semi-regular" polyhedron onto a sphere.
		Clearly this can be done with any regular polyhedron as the base, thus giving an infinite class of semi-regular polyhedra.
		On the other hand, the usual assumption for semi-regular polyhedra (called Archimedean solids rather than Platonic solids) is that the vertices look the same and the faces are regular.
	www.math.niu.edu /~rusin/known-math/99/soccer (877 words)

	The Many Faces of Polyhedrons
		This is a program that I (Tom York) wrote which allows you to interactively manipulate virtual regular polyhedrons (cube, dodecahedron, etc.) and semiregular polyhedrons (e.g.
		Download the Many Faces of Polyhedrons (165 kB)
		It was originally written for Windows 3.1, but it seems to run fine under later versions of Windows.
	home.twcny.rr.com /fisheryorks/mfop/mfop.html (144 words)

	Pseudo Rhombicuboctahedra
		He discovered a 14th semiregular polyhedron [figure of pseudo-rhombicuboctahedron] which differs from the one shown in [figure of rhombicuboctahedron] only in that the upper part, consisting of 5 squares and 4 equilateral triangles, is rotated through an angle of pi/4.
		The pseudo-rhombicuboctahedron is not classified as a semi-regular polyhedron, because the essence (and beauty) of the semi-regular polyhedra is not about local properties of each vertex, but the symmetry operations under which which the entire object appears unchanged.
		As far as I can see from studying the uniform polyhedra, these are the only two possible examples of a polyhedron in which all faces are regular and all vertices are locally identical, yet the polyhedron is not uniform; however I know of no simple proof of this.
	www.georgehart.com /virtual-polyhedra/pseudo-rhombicuboctahedra.html (691 words)

	Directory - Science: Math: Geometry: Polytopes: Polyhedra
		Polyhedra composed of two different types of regular polygons are called semiregular polyhedra or Archimedean solids, of which there are thirteen.
		Polyhedra can be regular; a regular polyhedron is one composed of regular polygons (polygons where all sides and angles are the same).
		Polyhedra Models by Doug Zongker · cached · Templates for paper models for each of the 5 Platonic solids and the 13 Archimedean semi-regular polyhedra, in pdf format.
	www.incywincy.com /default?p=1211428 (945 words)

	Math Forum - Mathematics Teacher Bibliography: Higher Dimensions
		Semiregular Polyhedra Rick N. Blake and Charles Verhille Activities for use in searching for patterns involved in the structure of polyhedra.
		The Total Angular Deficiency Of Polyhedra William L. Lepowsky Investigates the angles at the vertices of a polyhedron.
		The Algebra and Geometry Of Polyhedra Joseph A. Troccolo Algebraic and geometric approaches to the building of polyhedra.
	mathforum.org /mathed/mtbib/higher.dimensions.html (2050 words)

	Fullerenes: Topology and Structure
		The shape of this highly symmetric polyhedron, having 60 equivalent vertices that can be generated by the 60 operations of the icosahedral point group, was well known to ancient Greek mathematicians, notably to Archimedes who described the 13 possible semiregular polyhedra.
		Semiregular polyhedra, otherwise known as the Archimedean polyhedra, are made of two different kinds of regular polygons and are obtained by vertex truncation of the five regular (Platonic) polyhedra.
		The actual shapes of fullerene clusters are polyhedra whose structural features can be largely understood from pure topological arguments.
	www.dekker.com /sdek/211935813-92967286/abstract~db=enc~content=a713557326 (585 words)

	Middle East Open Encyclopedia: Polyhedron
		Given two polyhedra of equal volume, one may ask whether it is then always possible to cut the first into polyhedral pieces which can be reassembled to yield the second polyhedron.
		It could be defined as the union of a finite number of convex polyhedra, where a convex polyhedron is any set that is the intersection of a finite number of half-spaces.
		Names of polyhedra by number of faces are tetrahedron, pentahedron, hexahedron, etc. Such terms are in particular used with "regular" in front or implied (in the five cases in which this is applicable) because for each there are different types which have not much in common except having the same number of faces.
	www.baghdadmuseum.org /ref/index.php?title=Polyhedron (1752 words)

	Topic: semiregular polyhedron
		Thirteen of the semiregular polyhedra are known as the Archimedean solids.
		A convex polyhedron in which all the faces are regular polygons of more than one type and the faces of different types are arranged in the same way around each vertex.
	www.elko.k12.nv.us /webapps/vmd/full/s/semiregularpolyhedron.htm (42 words)

	Euler Formula
		After class discussion, each student will write and submit a brief statement as to what and why they believe the relationship is between the number of vertices, edges and faces in a polyhedron.
		Each group will be given a set of construction models of polyhedra and a copy of the workheet described previously.
		Students will be directed on the worksheet to summarize what relationship they have discovered based on their results.
	www.trincoll.edu /~czuccote/Eulerformula.htm (223 words)

	Amazon.com: Books: Dual Models
		nonconvex uniform polyhedra, stellation process, stellated forms, deltoidal hexecontahedron, stellation patterns, pentagonal hexecontahedron, pentagonal vertices, final stellation, hidden vertices, stellated dodecahedra, rhombic triacontahedron, semiregular solids, uniform polyhedron, regular dodecahedra, great stellated dodecahedron, facial planes, truncated tetrahedron, rhombic dodecahedron, vertex figures, five regular solids, square prisms, edge model
		In Dual Models, written in the same enthusiastic style as its predecessors Polyhedron Models and Spherical Models, Magnus J. Wenninger presents the complete set of uniform duals of uniform polyhedral, thus rounding out a significant body of knowledge with respect to polyhedral forms.
		The companion to Polyhedron Models, June 13, 2000
	www.amazon.com /exec/obidos/tg/detail/-/0521245249?v=glance (223 words)

	References
		The first illustrates the relation between a polyhedron and its dual by means of a continuous sequence of intermediate rectangle-connected "transpolyhedra." The latter two present graphic arrangements of various polyhedra to illustrate their relationships (akin to the Periodic Table of the Elements.)
		Detailed framework for a general notion of polyhedra in which the faces are basically a path of edges, and so may be nonplanar, or the edges may go around more than once, or may be infinite, e.g., a helix.
		Monograph on infinite polyhedra and space structures, with a crystallographic perspective.
	www.georgehart.com /virtual-polyhedra/references.html (5153 words)

	Tessellation
		Polyhedron which is capable of tessellating space is called a
		In the plane, there are eight such tessellations, illustrated below.
	mathserver.sdu.edu.cn /mathency/math/t/t089.htm (121 words)

	Semiregular Polyhedron
		The usual name for a semiregular polyhedron is an
		Tessellation is called semiregular if its faces are all
	mathserver.sdu.edu.cn /mathency/math/s/s188.htm (60 words)

	What's New on the Mathematics Archives
		Schläfli symbols, Wythoff symbols, Archimedean polyhedra, Platonic polyhedra, Uniform polyhedron, Symmetry groups, Coxeter-Dynkin graphs
		Mathematica, Wythoff Symbol, Regular and Quasi-Regular Polyhedra, Semiregular Polyhedra, Even-Faced Polyhedra, Snub Polyhedra, Great Dirhombicosidodecahedron
		Uniform polyhedra, Duals, Johnson's solids, Stellations of Icosahedron, Stellations of deformed dodecahedron, Spherical Platonic polyhedra, Polyhedral Kaleidoscope, Symmetrical compounds, VRML, Interactive building of polyhedra compounds
	archives.math.utk.edu /whatsnew/dec97.html (1135 words)

	Nuts About Nets!
		regular polyhedron is a polyhedron all of whose faces are regular polygons, the same number of which meet at each corner.
		In regular polyhedra all the dihedral angles are equal (that’s part of what being “regular” is all about), but in semiregular and irregular polyhedra the dihedral angles need not be the same.
		(This, incidentally, has yet to be proved mathematically, but most geometers believe it to be true, since nobody has yet produced a convex polyhedron that cannot be unfolded into a planar net.
	members.aol.com /Polycell/nets.html (1135 words)

	POLYHEDRON - LoveToKnow Article on POLYHEDRON
		Other examples of reciprocal holohedra are: the rhombic dodecahedron and cuboctahedron, with regard to the cube and octahedron; and the semiregular triacontahedron and icosidodecahedron, with regard to the dodecahedron and icosahedron.
		The correspondence of the faces of polyhedra is also of importance, as may be seen from the manner in which one polyhedron may be derived from another.
		The truncated icosahedron is formed similarly to the icosidodecahedron, but the truncation is only carried far enough to leave the original faces hexagons.
	www.1911ency.org /P/PO/POLYHEDRON.htm (3208 words)

	52B: Polytopes and polyhedra
		Schreiber, Peter: "What is the true number of semiregular (Archimedean) solids?", Festschrift on the occasion of the 65th birthday of Otto Krötenheerdt.
		How to compute the volume of a polyhedron?
		"Polyhedron Models", by Magnus J. Wenninger (Cambridge University Press, London-New York, 1971): has models of all 52 uniform polyhedra and some stellations.
	www.math.niu.edu /~rusin/known-math/index/52BXX.html (982 words)

	Citations: Dover Publications - Coxeter, Polytopes (ResearchIndex)
		K s 2 (cross polytope) There are just 3 semiregular 1 polytopes of dimension greater than 3, see [DeSh96] Two of them have equicut skeletons: H(5; 2) and the snub 24 cell s(3; 4; 3) see Figure 1.
		A solution is to create a quasi regular polyhedron with equal distances between each vertex and its neighbours.
		The latter is a 4 dimensional semiregular polytope with 96 vertices (see, for example,
	citeseer.ist.psu.edu /context/144086/0 (570 words)

	DODECAHEDRON - LoveToKnow Article on DODECAHEDRON
		The rhombic dodecahedron, one of the geometrical semiregular solids, is an important crystal form.
		The small stellated dodecahedron, the great dodecahedron and the great stellated dodecahedron are Kepler-Poinsot solids; and the truncated and snub dodecahedra are Archimedean solids (see POLYHEDRON).
		The ordinary dodecahedron is one of the Platonic solids (see POLYHEDRON).
	www.1911encyclopedia.org /D/DO/DODECAHEDRON.htm (172 words)

	The Geometry Junkyard: Polyhedra and Polytopes
		Convex Archimedean polychoremata, 4-dimensional analogues of the semiregular solids, described by Coxeter-Dynkin diagrams representing their symmetry groups.
		V-E+F=2, where V, E, and F are respectively the numbers of vertices, edges, and faces of a convex polyhedron.
		Circumnavigating a cube and a tetrahedron, Henry Bottomley.
	www.ics.uci.edu /~eppstein/junkyard/polytope.html (172 words)

	Polyhedra
		Once we allow multiple kinds of regular polygons to mixed in a single polyhedron (though we still insist that all vertices be congruent) there are many more possibilities.
		In addition to two infinite families of prisms and antiprisms, there are 75 semiregular polyhedra, as proved by Coxeter et al.
		Each face of these four are generalized regular polygons, which means that 5-pointed stars (5/2-gons) are allowed.
	amath.colorado.edu /staff/fast/Polyhedra (172 words)

	Polyhedra
		Once we allow multiple kinds of regular polygons to mixed in a single polyhedron (though we still insist that all vertices be congruent) there are many more possibilities.
		In addition to two infinite families of prisms and antiprisms, there are 75 semiregular polyhedra, as proved by Coxeter et al.
		Each face of these four are generalized regular polygons, which means that 5-pointed stars (5/2-gons) are allowed.
	amath.colorado.edu /staff/fast/Polyhedra (172 words)

	52B: Polytopes and polyhedra
		Just what is a polytope and how does it differ from a polygon or polyhedron?
		Schreiber, Peter: "What is the true number of semiregular (Archimedean) solids?", Festschrift on the occasion of the 65th birthday of Otto Krötenheerdt.
		52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry), See also 06A08, 13F20, 13Hxx
	www.math.niu.edu /~rusin/known-math/index/52BXX.html (172 words)

	Polychoron - Wikipedia, the free encyclopedia
		Semiregular 4-polytopes Subset of uniform polychora with regular polyhedron cells.
		The remaining convex uniform polychora may be grouped into two infinite families: the duoprisms and the polyhedral prisms.
		In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning "many" and choros meaning "room" or "space"), 4-polytope, or polyhedroid.
	en.wikipedia.org /wiki/Polychoron (688 words)

	Polychoron - Wikipedia, the free encyclopedia
		Semiregular 4-polytopes Subset of uniform polychora with regular polyhedron cells.
		The remaining convex uniform polychora may be grouped into two infinite families: the duoprisms and the polyhedral prisms.
		In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning "many" and choros meaning "room" or "space"), 4-polytope, or polyhedroid.
	en.wikipedia.org /wiki/Polychoron (688 words)

	Uniform Polyhedra
		Thus these are semiregular in the same way that the Archimedean solids are, but the faces and vertex figures need not be convex.
		However, the point at which the two triangles join is just an edge-crossing, not a vertex of the polyhedron and so does not correspond to a face of the underlying primal.
		The duals to the uniform polyhedra are facially regular, meaning that they are composed of a single type of face, every face in the same relation to the whole, and their vertex figures are regular polygons.
	www.georgehart.com /virtual-polyhedra/uniform-info.html (688 words)

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