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Topic: Zonohedron

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	Reference.com/Encyclopedia/Zonohedron
		Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube.
		Each edge in a zonohedron is parallel to at least one of the generators, and has length equal to the sum of the lengths of the generators to which it is parallel.
		Thus, the edges of the zonohedron can be grouped into zones of parallel edges, which correspond to the segments of a common great circle on the Gauss map, and the 1-skeleton of the zonohedron can be viewed as the planar dual graph to an arrangement of great circles on the sphere.
	www.reference.com /browse/wiki/Zonohedron (868 words)

	Polar Zonohedra
		The three-fold zonohedron is a rhombohedron, of which a cube is a special case.
		One special case of the four-fold zonohedron is the rhombic dodecahedron.
		The axial section of the zonohedron is a sine curve.
	www.uwsp.edu /geo/projects/geoweb/participants/Dutch/symmetry/zonohedra.HTM (1443 words)

	Metamorphic tiling patterns based on zonohedra - Patent 5211692
		This application discloses a tiling system for surfaces where the pattern of the tiling changes continuously from one portion of the tiling to another in an Escher-like metamorphoses with the difference the the metamorphoses are based on binary combinations of n transformations on the edges of the tile.
		Accordingly, the tiling is obtained from the n directions of the edges of an underlying zonohedron, a polyhedron derived as a projection of an n-dimensional cube.
		said zonohedron network is based on a 2-dimensional projection of a 4-dimensional cube viewed along its 4-fold axis of symmetry and its edges are in the ratio of 1 to square root of 2 (or 1.414213.
	www.freepatentsonline.com /5211692.html (4625 words)

	Spotlight on Nation (Site not responding. Last check: )
		The Commonwealth of Zonohedron is a massive, socially progressive nation, notable for its complete absence of social welfare.
		Zonohedron's national animal is the rabid gopher, which frolics freely in the nation's many lush forests, and its currency is the spheroid.
		Zonohedron is ranked 11th in the region and 5,780th in the world for Smartest Citizens.
	www.nationstates.net /-1/page=display_nation/nation=zonohedron (220 words)

	The Geometry Junkyard: Zonohedra
		A zonohedron is a polyhedron in which every face is centrally symmetric.
		This zonohedron, computed by a Mathematica notebook I wrote, provides a lower bound for the complexity of the set of centroids of points with approximate weights.
		Zonohedron generated by 30 vectors in a circle, and another generated by 100 random vectors, Paul Heckbert, CMU.
	www.ics.uci.edu /%7eeppstein/junkyard/zono.html (542 words)

	Building systems using saddle polygons and saddle zonohedra based on polyhedral stars - Patent 5155951
		the said plane-faced zonohedron is a parallelopiped and the plurality of said parallelopipeds define a non-periodic lattice with n-directions, wherein the edges of the said parallelopipeds define the edges of a plurality of saddle polygons.
		This particular zonohedron is a deformed rhombic dodecahedron, where the undeformed one is a well-known Archimedean dual polyhedron.
		This zonohedron is an irregular variant of the known Archimedean dual, the rhombic triacontahedron.
	www.freepatentsonline.com /5155951.html?highlight=4686800 (8502 words)

	Extended Polar Zonohedra
		It can be shown that the vertices all lie on equally spaced planes normal to the polar axis, and that the longitudinal cross section of the zonohedron is approximately a sine curve.
		If n is odd, we can cut the zigzag series of edges that bisects the zonohedron, pull the two halves apart parallel to the axis, and add a set of edges joining the two halves.
		For a simple polar zonohedron with n-fold symmetry there are two independent dimensions: the radius R and the polar axis length L. We define angle A = 360/n.
	www.uwsp.edu /geo/projects/geoweb/participants/Dutch/symmetry/ExtZhedra.HTM (595 words)

	Clinton Goveas :: Wikipedia Reference (Site not responding. Last check: )
		The cube is unique among the Platonic solids for being able to tile space regularly.
		It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry).
		The analogue of a cube in four-dimensional Euclidean space has a special name — a tesseract or (rarely) hypercube.
	www.clintongoveas.com /wikipedia/?title=Cube (631 words)

	Scrapbook - Zonohedra
		Polar zonohedron produced from a star of 7 vectors.
		Polar zonohedron produced from a star of 20 vectors and with a proper colouring of the faces
		Polar zonohedron produced from a star of 50 vectors and with a proper colouring of the faces
	www.freewebtown.com /adrian/geom/890_zonohedra/imagelist.html (184 words)

	Quincy Herald-Whig
		The frame of the zonohedron is supported by aluminum bars, shown in this look inside the figure, and the CDs are connected by paper fasteners.
		The zonohedron's frame is an 8 3/4-foot-in-diameter aluminum circle salvaged from the school's scrap heap four years ago and was gathering dust in the hallway.
		What excites Klauser about the final outcome is that the students followed no textbook to make the zonohedron, and that the geometric methods used have practical workplace applications.
	www.whig.com /312593347854447.php (700 words)

	AN ESSENTIALLY-THREE-DIMENSIONAL QUASICRYSTAL
		A zonohedron is a convex polyhedron made of zonogons.
		A golden isozonohedron is a zonohedron whose faces are all golden rhombus.
		Suppose the situation that the symmetry of the outside of a zonohedron is rather high and that of its internal structure low.
	www.mi.sanu.ac.yu /vismath/ogawa/es.htm (2313 words)

	Zonohedra and Zonotopes
		This is the dual to the cuboctahedron, an Archimedean solid formed by combining the six squares of a cube with the eight triangles of an octahedron.
		The cuboctahedron itself is not a zonohedron because of its triangular faces.
		This is the Minkowski sum of a cube, a truncated octahedron, and a rhombic dodecahedron.
	www.ics.uci.edu /~eppstein/junkyard/ukraine/ukraine.html (2108 words)

	Extended Polar Zonohedra
		It can be shown that the vertices all lie on equally spaced planes normal to the polar axis, and that the longitudinal cross section of the zonohedron is approximately a sine curve.
		If n is odd, we can cut the zigzag series of edges that bisects the zonohedron, pull the two halves apart parallel to the axis, and add a set of edges joining the two halves.
		For a simple polar zonohedron with n-fold symmetry there are two independent dimensions: the radius R and the polar axis length L. We define angle A = 360/n.
	www.uwgb.edu /dutchs/symmetry/ExtZhedra.HTM (595 words)

	Quincy High School: The Q-Ball! (Site not responding. Last check: )
		zonohedron designed by George Hart (see the green model in the Chapter 14 section), students in Quincy High School's Creative Problem Solving class decided to make a model of this intriguing figure.
		The angles between the holes as measured from the center of the CDs are indicated by the blue radial lines in the picture.
		If you look at the zonohedron referred to on George Hart's page, you'll see that one CD is needed for each vertex of the zonohedron.
	websites.quincy.edu /~matskvi/qball.html (318 words)

	POLYTOPIA PERFORMANCE
		A zonohedron is a convex polyhedron bounded by parallelograms.
		Johannes Kepler called a zonohedron "a polyhedron wearing a rhombus coat".
		The ratio of frequencies of each solid always corresponds to the number of vertices (or atoms) of a zonohedron.
	members.tripod.com /vismath7/proceedings/schwabe.htm (983 words)

	Count On (Site not responding. Last check: )
		In this model all the faces of the zonohedron are identical.
		The model is made up of identical slices, not only in shape but in the way the slots are cut.
		With more slices the zonohedron looks more solid and it generates more interesting patterns when it is flattened.
	www.mathsyear2000.org /explorer/sliceforms/a-sliceform-zonohedron (794 words)

	The Amazing Q-Ball! (Site not responding. Last check: )
		George Hart designed this zonohedron using the 61 different directions a strut can be placed in a node of the geometrical construction kit, Zometools.
		Eventually a decision was made: use CDs, one for each vertex of the zonohedron.
		Because each CD corresponded to a vertex of the zonohedron, the angles between the edges at each vertex had to be calculated.
	websites.quincy.edu /~matskvi/qball2.html (583 words)

	Zonohedron - Definition, explanation
		A zonohedron is a convex polyhedron where every face is a polygon with point symmetry, or equivalently, symmetry under rotations through 180°.
		The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated:
		Of the Platonic solids, only the cube is a zonohedron
	www.calsky.com /lexikon/en/txt/z/zo/zonohedron.php (185 words)

	Zonohedra
		A zonohedron (by one restrictive definition) is a convex polyhedron all of whose faces are parallelograms.
		This is best understood by examining a model of the same "random" 42-face zonohedron as above, or a rhombic enneacontahedron, or a rhombic tricontahedron, in which one zone is colored.
		Using the diagonals of the octahedron (3 mutually orthogonal vectors) as the star, we construct the 3-zone cube.
	www.georgehart.com /virtual-polyhedra/zonohedra-info.html (965 words)

	Zonohedra
		Every edge lies in one of 24 different directions (the 24 vectors which determine this zonohedron); note that any one edge-direction determines a zone of faces which girdle the zonohedron.
		A zonohedron with octahedral symmetry; we are looking down an axis of 4-fold rotation.
		A zonohedron with icosahedral symmetry, determined by 21 vectors, and an orthogonal shadow of a 21-dimensional cube, cast into three dimensions!
	home.inreach.com /rtowle/Zonohedra/zonohedra.html (187 words)

	Dodecahedra
		The rhombic dodecahedron has all the symmetry of a cube or octahedron.
		It is a zonohedron and the Archimedean dual to the quasi-regularcuboctahedron.
		The rhombic dodecahedron of the second kind is another zonohedron, but with a different rhombus shape, and it only has 2-fold prism symmetry (three mutually orthogonal 2-fold axes and three symmetry planes).
	www.georgehart.com /virtual-polyhedra/dodecahedra.html (1228 words)

	Maths Year 2000
		This means that slicing in two directions gives two sets of slices that are the same.
		In this model all the faces of the zonohedron are identical.
		With more slices the zonohedron looks more solid and it generates more interesting patterns when it is flattened.
	www.mathsyear2000.co.uk /explorer/slice/poly2.shtml (0 words)

	Zome Geometry -- Additional Photos
		Compound of five gb1 tetrahedra, described in Exploration 11H
		Computer model of 61-zone zonohedron, based on all blue, yellow, red, and green directions.
		You could build this if you had enough green struts.
	www.georgehart.com /zomebook/additions.html (92 words)

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