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Topic: Snub hexahedron

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Emperor of Japan

February 2005

Robert Thompson

Bert Sommer

	[No title]
		hexahedron +------------------------------------------------------------ The hexahedron is a polyhedron with 8 vertices and 6 faces.
		snub dodecahedron +------------------------------------------------------------ The snub dodecahedron is a polyhedron with 60 vertices and 92 faces.
		snub square antiprism +------------------------------------------------------------ The snub square antiprism is a polyhedron with 16 vertices and 26 faces.
	www.math.harvard.edu /~knill/sofia/data/polyhedra.txt (2272 words)

	Poliedri
		Hence, the hexahedron is the dual of the octahedron.
		The dual of the tetrakis hexahedron is the truncated octahedron from the tetrakis hexahedron.
		The starred dextro snub hexahedron (in the center of the plate) consists of 36 trigonal and tetragonal pyramids constructed on the faces of the original polyhedron.
	www.gicas.net /poliedri_text.html (4932 words)

	Polyhedron - LoveToKnow Watches
		The names of these five solids are: (r) the tetrahedron, enclosed by four equilateral triangles; (2) the cube or hexahedron, enclosed by 6 squares; (3) the octahedron, enclosed by 8 equilateral triangles; (4) the dodecahedron, enclosed by 12 pentagons; (5) the icosahedron, enclosed by 20 equilateral triangles.
		The snub cube is a 38-faced solid having at each corner 4 triangles and I square; 6 faces belong to a cube, 8 to the coaxial octahedron, and the remaining 24 to no regular solid.
		The snub dodecahedron is a 92-faced solid having 4 triangles and a pentagon at each corner.
	www.1911encyclopedia.org /Polyhedron (2152 words)

	names & notations
		The snub operation introduces "rings" of new faces around the existing faces: two triangles for each edge, and an x-gon for each x-vertex; in this way the snub cube is also the snub octahedron.
		2 3 4 the snub cube, 4/3 3 4
		Used to describe the stellations of a polyhedron, it allows to classify the cells defined by the faces' planes and to specify which ones are used for a given stellation.
	www.ac-noumea.nc /maths/amc/polyhedr/names_.htm (1238 words)

	Wikipedia: Archimedean solid
		snub cube or snub cuboctahedron (2 chiral forms)
		The last two (snub cube and snub dodecahedron) are known as chiral, as they come in a left-handed (latin: levomorph or laevomorph) form and right-handed (latin: dextromorph) form.
		When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs.
	www.factbook.org /wikipedia/en/a/ar/archimedean_solid.html (304 words)

	Regular convex polytopes a short historical overview, Regular Polytopes and n-dimensional packing of points
		The cube or hexahedron, with 6 square faces, 12 edges and 8 vertices, (Schläfli notation {4,3}), which is, proven later, a dual of the octahedron i.e.
		The snub cube, with 38 faces of 32 triangles and 6 squares, 24 vertices and 60 edges.
		The snub dodecahedron, with 92 faces of 80 triangles and 12 pentagons, 60 vertices and 150 edges.
	presh.com /hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.html (2534 words)

	Archimedean Solids
		Seven of the archimedean solids are derived from the platonic solids by the process of ‘truncation’, literally cutting off the corners of each of the platonic solids.
		The first five are the truncated tetrahedron, truncated octahedron, truncated hexahedron, truncated icosahedron and the truncated dodecahedron and these are arrived at by dividing the edges into thirds and cutting off the vertices of these points.
		The term snub refers to a process of surrounding each polygon with a border of triangles as a way of deriving, for example, the snub cube from the cube.
	www.ul.ie /~cahird/polyhedronmode/favorite.htm (482 words)

	The International Bone Rollers' Guild (Site not responding. Last check: 2007-10-24)
		The symmetry of this solid suggests its relationship to the hexahedron and octahedron.
		The Pentagonal Icositetrahedron (left) is dual to the Snub Cube.
		Its twenty-four irregular pentagonal faces are arranged in a matter suggesting its relationship to the Hexahedron and Octahedron.
	members.aol.com /dicetalk/polyh5.htm (282 words)

	Snub Versions
		The snub versions - the snub cube and the snub dodecahedron.
		The term snub can refer to a chiral process (having different left handed and right handed forms).
		The process that is applied to the two solids in this group are identical in theory.
	www.ul.ie /~cahird/polyhedronmode/snub.htm (193 words)

	Krystyna Burczyk's Origami Gallery - regular polyhedra
		There are five regular polyhedra, also called Platonic solids: tetrahedron (triangle faces, 3 in each vertex), hexahedron or cube (square faces, 3 in each vertex), octahedron (triangle faces, 4 in each vertex), dodecahedron (pentagonal faces, 3 in each vertex), icosahedron (triangle faces, 5 in each vertex).
		There are two variations of snub cube: left-handed and right-handed.
		There are two variations of snub dodecahedron: left-handed and right-handed.
	zetosak.zetosa.com.pl /~burczyk/origami/galery1-en.htm (341 words)

	Polyhedron, Polyhedra, Polytopes - Numericana
		In particular, any other hexahedron can be distorted into a shape which is its own mirror image, and the tetragonal antiwedge may thus unambiguously be called the chiral hexahedron.
		It could also be obtained by cutting an elongated square pyramid (the technical name for an obelisk) along a bisecting plane through the apex of the pyramid and the diagonal of the base prism, as pictured at right.
		Snub: Snubbing is an interesting chiral process which, roughly speaking, amounts to loosening all faces of a polyhedron and rotating them all slightly in the same direction (clockwise or counterclockwise), creating 2 triangles for each edge and one m-sided polygon for each vertex of degree m.
	home.att.net /~numericana/answer/polyhedra.htm (5404 words)

	Polyhedra Names (Site not responding. Last check: 2007-10-24)
		The term snub can refer to a chiral process of replacing each edge with a pair of triangles, e.g., as a way of deriving what is usually called the snub cube from the cube.
		To emphasize this equivalence, it is more logical to call the result a snub cuboctahedron but it may take a while for this name to be widely adapted.
		Applying the analogous process to either the dodecahedron or the icosahedron gives the polyhedron usually called the snub dodecahedron, but better called the snub icosidodecahedron.
	www.georgehart.com /virtual-polyhedra/naming.html (1134 words)

	Polyhedron
		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.
		For a tetrahedron this applies to a much lesser extent, it is always a triangular pyramid.
		3 have the symmetry of the snub dodecahedron:
	www.majicape.com /Rol-P/Polyhedron.php (1581 words)

	The Zing-Man Origami Gallery
		I have a new method for folding it which is relatevely easy with a nice strong lock and effeicitent in its utilization of the paper.
		Hexahedron III - This intriguing hexahedron has as faces two equalateral triangles, two rhombi, and two trapezoids, and it has the interesting propery that it can be used as a module to tile space.
		Snub Tetrahedron - This semiregular octohedron has four regular hexagons and four equalateral triangles as faces.
	www.zingman.com /origami/index.html (902 words)

	GEOWOOD (Site not responding. Last check: 2007-10-24)
		When the faces are made up of one set of regular polygons, it is known as a Platonic solid.
		There are five of these types which include tetrahedron (four equilateral triangles), hexahedron (6 cube shapes), octahedron (8 equilateral triangles), icosahedron (20 equilateral triangles), dodecahedron (12 pentagons).
		When the faces are made up of more than one set of polygons, they are known as Archimedian solids.
	geowood.com /PGON.html (360 words)

	Miscellaneous Structures at ASU
		This will eventually be a big page, even though right now there are only a few pictures on it.
		Above is two views of a snub octahedron, a squishy with Sparky, the ASU Mascot, on one of the faces, and an inflatable floating in a fountain at ASU.
		It is also two views of a permutahedron for n=4, since the permutahedron for n=4 is a snub octahedron, one of the Archimedean Solids.
	www.public.asu.edu /~starlite/sierpinskiasu.html (180 words)

	Polyhedra
		Cutting off the vertices of semi-regular polyhedra (and adjusting edge lengths) reveals another set of semi-regular ones.
		The twisted snub forms come from splitting the square rhombus faces into 2 equilateral triangles.
		The rhombus forms can also be arrived at by shaving off the edges of regular polyhedra.
	homepage.ntlworld.com /susan.foord/art/pattern/polyhed.htm (93 words)

	Table of Contents
		Discusses tessellation, or tiling, and how to make spherical models of the semiregular solids and concludes with a discussion of the relationship of polyhedra to geodesic domes and directions for building models of domes.
		Table of Contents for Spherical Models Foreword by Arthur L. Loeb Preface Introduction: Basic properties of the sphere I. The regular spherical models The spherical hexahedron or cube General instructions for making models The spherical octahedron The spherical tetrahedron The spherical icosahedron and dodecahedron The polyhedral kaleidoscope Summary II.
		The semiregular spherical models The spherical cuboctahedron The spherical icosidodecahedron Spherical triangles as characteristic triangles The five truncated regular spherical models The rhombic spherical models The rhombitruncated spherical models The snub forms as a spherical models The spherical duals Summary III.
	www.doverpublications.com /cgi-bin/toc.pl/048640921X (166 words)

	Review and Comparison of the Platonic Solids, the Archimedean Duals, and the Archimedean Solids (Site not responding. Last check: 2007-10-24)
		Image provided by Wikepedia, the free encyclopedia, snub hexahedron clock wise and snub hexahedron counter clockwise.
		Image provided by Wikepedia, the free encyclopedia, snub dodecahedron clockwise and snub dodecahedron counter clockwise.
		Origami paper folding, Snub Dodecahedron, provided by Cecilia Cotton at www.ceciliacotton.ca/origami
	www.thearchimedeandual.com /platonic/Greek/review/review_and_comparison.htm (699 words)

	Computer Graphics - Available Polyhedrons
		Snub Cube (6 squares, 32 triangles; - looks VERY difficult)
		Small Triakis Octahedron (24 triangles; dual of the Truncated Cube; a cumulation of the Octahedron)
		Note that Platonic Solids (Tetrahedron, Hexahedron (Cube), Octahedron, Dodecahedron and Icosahedron), the Cuboctahedron and the Stella Octangula are not available, as they are either too simple or there are too many examples of coding them already available (or both).
	www.soi.city.ac.uk /~dcd/ig2/week1/pol.htm (508 words)

	Polyhedra
		A snub cube (which is # 7 in the list) is written 3
		It can be proven that there are only 13 Archimedean solids, two of which occur in two forms.
		These two are the two 'snubs', and the two forms of each are related to one another like a left-hand and a righthand glove: they are enanttomorphic.
	www.faculty.fairfield.edu /jmac/rs/polyhedra.htm (492 words)

	The Archimedean Solids (Site not responding. Last check: 2007-10-24)
		The Archimedean solids are those geometries that can be inscribed in a sphere and whose faces are made of two or more regular polygons.
		The Archimedean solids can be made by truncating the Platonic solids, by expansive adjustment of certain of the truncations, and by 'snubbing' the faces to the axis of symmetry in certain geometries.
		Images provided by Robert Webb, Cube Minus Snub Cube, Snub Cube, at www.software3d.com/Stella.html
	www.thearchimedeandual.com /platonic/Greek/archimedean_solids/archimedean_solids.htm (1220 words)

	Geometric Solids, Montessori World Educational Institute
		The Platonian regular solids (tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron)
		Hexahedron - 6 faces, each face is a square
		Octahedron - 8 faces, each face is an equilateral triangle
	www.montessoriworld.org /sensfile/sgeosoli.html (398 words)

	Transcending Duality through Tensional Integrity: A lesson in organisation from building design
		(l) Archimedean polyhedra are: cuboctahedron, truncated tetrahedron, truncated octahedron, truncated cube, small rhombicuboctahedron, great rhombicuboctahedron, snub cube, truncated icosahedron, truncated dodecahedron, icosidodecahedron, great rombicosidodecahedron, snub dodecahedron, small rhombicosidodecahedron
		To reduce complication the tendency is to use Platonic polyhedra, only, with faces subdivided as many times as appropriate.
		All but the 2 snub Archimedean figures are in fact generated by 2 to 12 frequency divisions of the Platonic polyhedra
	www.laetusinpraesens.org /docs/systen1.php (5090 words)

	Platonic Solids
		Each one has a single shape and size for the faces, the same number of faces meet at each vertice, and all of the angles are the same across the surface.
		All other polyhedra are irregular (the snub octahedron, for instance, is made up of squares and hexagons) mixing different shapes for the faces and/or different sizes of the same shapes, and at least two different angles are present across the surface.
		In addition to the links above, you can also get around the site using the pulldown menu at the top of the page.
	www.public.asu.edu /~starlite/platonicsolids.html (546 words)

	Archimedean Solids
		The remaining two, however, is not reflexible: the snub cube and the snub dodecahedron.
		for discussions about constructions of these two snub polyhedra.
		In Maple, one can define an Archimedean solid by using the command Archimedean(gon,sch,o,r); where gon is the name of the polyhedron to be defined, sch the Schläfli symbol (Maple's Schläfli), o the center of the polyhedron, and r the radius of the circum-sphere.
	www.cecm.sfu.ca /~hle/polyhedra/archimedean.html (666 words)

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