Augmented matrix - Revision history

en>Slawekb at 00:43, 10 February 2014

2014-02-10T00:43:53Z

← Older revision		Revision as of 02:43, 10 February 2014
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	~~{\| class=wikitable align="right"~~		Alyson is the name individuals use to call me and I think it seems fairly great psychics online ([http://breenq.com/index.php?do=/profile-1144/info/ mouse click the next site]) when you say it. Alaska is where he's usually been living. [http://www.prayerarmor.com/uncategorized/dont-know-which-kind-of-hobby-to-take-up-read-the-following-tips/ are psychics real] Since he was 18 he's been operating as an info officer but he ideas on changing it. To climb is something I really appreciate performing.<br><br>Feel free to visit my blog; best psychic readings ([http://www.weddingwall.com.au/groups/easy-advice-for-successful-personal-development-today/ www.weddingwall.com.au])
	~~!bgcolor=#e7dcc3 colspan=2\|Chamfered cube~~
	\|-
	~~\|align=center colspan=2\|~~[~~[Image~~:~~Truncated rhombic dodecahedron~~.~~png\|240px\|Chamfered cube]]~~
	\|-
	~~\|bgcolor=#e7dcc3\|Type\|\|[[Conway polyhedron notation\|Conway polyhedron]]: cC = t4daC~~
	\|-
	~~\|bgcolor=#e7dcc3\|[[Goldberg polyhedron]]\|\|G<sub>IV<~~/~~sub>(2,0)~~
	\|-
	~~\|bgcolor=#e7dcc3\|Faces\|\|6 [[square (geometry)\|square]]s<BR>12 [[hexagon]]s~~
	\|-
	~~\|bgcolor=#e7dcc3\|Edges\|\|48 (2 types)~~
	\|-
	~~\|bgcolor=#e7dcc3\|Vertices\|\|32 (2 types)~~
	\|-
	~~\|bgcolor=#e7dcc3\|[[Vertex configuration]]\|\|(24) 4.6.6<BR>(8) 6~~.~~6.6~~
	\|-
	~~\|bgcolor~~=~~#e7dcc3\|[[List of spherical symmetry groups\|Symmetry]]\|\|[[octahedral symmetry\|''O''<sub>''h''<~~/~~sub>]], [4,3], (*432)~~
	\|-
	~~\|bgcolor=#e7dcc3\|[[Dual polyhedron]]\|\|[[Tetrakis cuboctahedron]]~~
	\|-
	~~\|bgcolor=#e7dcc3\|Properties\|\|[[convex set\|convex]], [[zonohedron]], [[equilateral]]-faced~~
	\|}
	~~[[Image:Truncated rhombic dodecahedron net.png\|250px\|thumb\|[[net (polyhedron)\|A net for chamfered cube]]]]~~
	~~The '''[[chamfer]]ed cube''' (also called '''truncated rhombic dodecahedron''') is a [[convex polygon\|convex]] [[polyhedron]] constructed from~~ the ~~[[rhombic dodecahedron~~]~~] by [[Truncation (geometry~~)~~\|truncating]] the 6 (order 4) vertices.~~

	~~The 6 vertices are truncated such that all edges are equal length. The original 12 [[rhombus\|rhombic]] faces become flattened hexagons, and the truncated vertices become squares.~~

	The hexagonal faces are [[equilateral]] but not [[regular polygon\|regular]]. They are formed by a truncated rhombus, have 2 internal angles of about 109.47 degrees (''arccos(-1/3)'') and 4 internal angles of about 125.26 degrees, while a regular hexagon would have all 120 degree angles.

	~~Because all its faces have an even number of sides with 180 degree rotation symmetry,~~ it ~~is a [[zonohedron]]~~. It is ~~also the [[Goldberg polyhedron]] G<sub>IV</sub>(2,0), containing square and hexagonal faces~~.

	~~== Coordinates ==~~
	~~The ''chamfered cube'' is the~~ [~~[Minkowski sum]] of a rhombic dodecahedron and a cube of side length 1 when~~
	~~eight vertices of the rhombic dodecahedron are at <math>(\pm 1, \pm 1, \pm 1)<~~/~~math> and its~~
	~~six vertices are at the permutations of <math>(\pm 2, 0, 0)<~~/~~math>.~~
	~~== Related polyhedra ==~~

	~~This polyhedron is similar to the uniform [[truncated octahedron]]:~~
	~~{\| class="wikitable"~~
	~~!Truncated rhombic dodecahedron<BR>[[Image:Truncated rhombic dodecahedron2~~.~~png\|120px]]~~
	~~!Truncated octahedron<BR>[[Image:Truncated octahedron~~.~~png\|120px]]~~
	\|}

	~~This polyhedron is also a part of a sequence~~ of ~~truncated rhombic polyhedra and tilings with [n,3] [[Coxeter group]] symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n~~-~~gons at the truncated vertices, and nonregular hexagonal faces. The sequence has two vertex figures (n.6.6) and (6,6,6). The hexagonal tiling can be considered a truncated [[rhombille tiling]].~~

	~~{\| class="wikitable" width=500~~
	~~!colspan=3\|Polyhedra~~
	~~!Euclidean tiling~~
	~~!colspan=2\|Hyperbolic tiling~~
	\|-
	~~![3,3]~~
	~~![4,3]~~
	~~![5,3]~~
	~~![6,3]~~
	~~![7,3]~~
	~~![8,3]~~
	\|- ~~align=center valign=top~~
	~~\|[[File:Hexahedron.svg\|100px]]<BR>[[Cube]]~~
	~~\|[[File:Rhombicdodecahedron.jpg\|100px]]<BR>[[Rhombic dodecahedron]]~~
	~~\|[[File:Rhombictriacontahedron.jpg\|100px]]<BR>[[Rhombic triacontahedron]]~~
	~~\|[[File:Rhombic star tiling.png\|100px]]<BR>[[Rhombille]]~~
	~~\|[[File:Order73 qreg rhombic til.png\|100px]]~~
	~~\|[[File:Uniform dual tiling 433~~-~~t01~~-~~yellow.png\|100px]]~~
	\|- ~~align=center valign=top~~
	~~\|[[File:Alternate truncated cube.png\|100px]]<BR>[[Alternate truncated cube]]~~
	~~\|[[File:Truncated rhombic dodecahedron2.png\|100px]]<BR>Truncated rhombic dodecahedron~~
	~~\|[[File:Truncated rhombic triacontahedron.png\|100px]]<BR>[[Truncated rhombic triacontahedron]]~~
	~~\|[[File:Truncated rhombille tiling.png\|100px]]<BR>[[Hexagonal tiling]]~~
	\|
	\|}

	~~=== Rhombic dodecahedron ===~~
	~~The name ''truncated rhombic dodecahedron'' is ambiguous since only 6 (order~~-~~4) vertices were truncated. A truncation on just the 3~~-~~vertices of the rhombic dodecahedron would cause an icosahedron with 12 equilateral hexagons and 20 triangles, forming 30 vertices in total; this figure~~'s ~~dual can be called the ''triakis cuboctahedron''. Another alternate truncated rhombic dodecahedron can appear by truncating all 14 vertices, yielding 12 irregular octagonal faces~~. ~~The dual of the full truncation~~ is ~~a triangular tetracontaoctahedron labeled the ''tritetrakis cuboctahedron'', which is a complete [[Kleetope]] of the [[cuboctahedron]]~~. ~~The [[tetrahedron]] is to the [[truncated cube]] as the [[cube]] is to the full truncation or a bitruncated cuboctahedron.~~

	<~~gallery~~>
	~~File:rhombic dodecahedron.png\|Rhombic dodecahedron~~
	~~Image:StellaTruncRhombicDodeca.png\|Full truncation of rhombic dodecahedron~~
	~~Image:Truncated rhombic dodecahedron2.png\|Order-4 truncation of rhombic dodecahedron~~<~~BR>(Chamfered cube)~~
	~~File:Chamfered_octahedron.png\|Order-3 truncation of rhombic dodecahedron<BR~~>(~~Chamfered octahedron)~~
	~~</gallery>~~

	~~== See also ==~~
	* [[Chamfered tetrahedron]]
	* [[Chamfered dodecahedron]]
	* [[Near-miss Johnson solid]]

	~~== References==~~
	* Antoine Deza, Michel Deza, Viatcheslav Grishukhin, ''Fullerenes and coordination polyhedra versus half-cube embeddings'', 1998 [[PDF]] [http://www.~~cas~~.~~mcmaster~~.ca/~~~deza~~/~~dm1998.pdf]~~

	~~== External links ==~~
	* [[VRML]] model [http://www.georgehart.com/virtual-~~polyhedra/vrml/zono~~-~~7_from_cube.wrl]~~
	** [http://www.~~georgehart~~.com~~/virtual-polyhedra/conway_notation~~.~~html VTML polyhedral generator] Try "t4daC" ([[Conway polyhedron notation]~~])

	~~{{Near-miss Johnson solids navigator}}~~

	~~[[Category:Zonohedra]]~~

98.229.29.63: /* Existence and number of solutions */

2013-10-18T16:51:08Z

Existence and number of solutions

← Older revision		Revision as of 18:51, 18 October 2013
Line 1:		Line 1:
	They ~~contact me Emilia~~. ~~To collect coins~~ is a ~~thing that I~~'~~m completely addicted~~ to. ~~Puerto Rico~~ is where he and ~~his wife live~~. ~~My day occupation is~~ a ~~librarian~~.<br><br>~~Feel free~~ to ~~visit my website~~ ... [http://www.~~january-yjm~~.~~com~~/xe/~~index~~.~~php?mid~~=~~video&document_srl~~=~~182582~~ http://www.~~january~~-~~yjm~~.com/]		{\| class=wikitable align="right"
			!bgcolor=#e7dcc3 colspan=2\|Chamfered cube
			\|-
			\|align=center colspan=2\|[[Image:Truncated rhombic dodecahedron.png\|240px\|Chamfered cube]]
			\|-
			\|bgcolor=#e7dcc3\|Type\|\|[[Conway polyhedron notation\|Conway polyhedron]]: cC = t4daC
			\|-
			\|bgcolor=#e7dcc3\|[[Goldberg polyhedron]]\|\|G<sub>IV</sub>(2,0)
			\|-
			\|bgcolor=#e7dcc3\|Faces\|\|6 [[square (geometry)\|square]]s<BR>12 [[hexagon]]s
			\|-
			\|bgcolor=#e7dcc3\|Edges\|\|48 (2 types)
			\|-
			\|bgcolor=#e7dcc3\|Vertices\|\|32 (2 types)
			\|-
			\|bgcolor=#e7dcc3\|[[Vertex configuration]]\|\|(24) 4.6.6<BR>(8) 6.6.6
			\|-
			\|bgcolor=#e7dcc3\|[[List of spherical symmetry groups\|Symmetry]]\|\|[[octahedral symmetry\|''O''<sub>''h''</sub>]], [4,3], (*432)
			\|-
			\|bgcolor=#e7dcc3\|[[Dual polyhedron]]\|\|[[Tetrakis cuboctahedron]]
			\|-
			\|bgcolor=#e7dcc3\|Properties\|\|[[convex set\|convex]], [[zonohedron]], [[equilateral]]-faced
			\|}
			[[Image:Truncated rhombic dodecahedron net.png\|250px\|thumb\|[[net (polyhedron)\|A net for chamfered cube]]]]
			The '''[[chamfer]]ed cube''' (also called '''truncated rhombic dodecahedron''') is a [[convex polygon\|convex]] [[polyhedron]] constructed from the [[rhombic dodecahedron]] by [[Truncation (geometry)\|truncating]] the 6 (order 4) vertices.

			The 6 vertices are truncated such that all edges are equal length. The original 12 [[rhombus\|rhombic]] faces become flattened hexagons, and the truncated vertices become squares.

			The hexagonal faces are [[equilateral]] but not [[regular polygon\|regular]]. They are formed by a truncated rhombus, have 2 internal angles of about 109.47 degrees (''arccos(-1/3)'') and 4 internal angles of about 125.26 degrees, while a regular hexagon would have all 120 degree angles.

			Because all its faces have an even number of sides with 180 degree rotation symmetry, it is a [[zonohedron]]. It is also the [[Goldberg polyhedron]] G<sub>IV</sub>(2,0), containing square and hexagonal faces.

			== Coordinates ==
			The ''chamfered cube'' is the [[Minkowski sum]] of a rhombic dodecahedron and a cube of side length 1 when
			eight vertices of the rhombic dodecahedron are at <math>(\pm 1, \pm 1, \pm 1)</math> and its
			six vertices are at the permutations of <math>(\pm 2, 0, 0)</math>.
			== Related polyhedra ==

			This polyhedron is similar to the uniform [[truncated octahedron]]:
			{\| class="wikitable"
			!Truncated rhombic dodecahedron<BR>[[Image:Truncated rhombic dodecahedron2.png\|120px]]
			!Truncated octahedron<BR>[[Image:Truncated octahedron.png\|120px]]
			\|}

			This polyhedron is also a part of a sequence of truncated rhombic polyhedra and tilings with [n,3] [[Coxeter group]] symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n-gons at the truncated vertices, and nonregular hexagonal faces. The sequence has two vertex figures (n.6.6) and (6,6,6). The hexagonal tiling can be considered a truncated [[rhombille tiling]].

			{\| class="wikitable" width=500
			!colspan=3\|Polyhedra
			!Euclidean tiling
			!colspan=2\|Hyperbolic tiling
			\|-
			![3,3]
			![4,3]
			![5,3]
			![6,3]
			![7,3]
			![8,3]
			\|- align=center valign=top
			\|[[File:Hexahedron.svg\|100px]]<BR>[[Cube]]
			\|[[File:Rhombicdodecahedron.jpg\|100px]]<BR>[[Rhombic dodecahedron]]
			\|[[File:Rhombictriacontahedron.jpg\|100px]]<BR>[[Rhombic triacontahedron]]
			\|[[File:Rhombic star tiling.png\|100px]]<BR>[[Rhombille]]
			\|[[File:Order73 qreg rhombic til.png\|100px]]
			\|[[File:Uniform dual tiling 433-t01-yellow.png\|100px]]
			\|- align=center valign=top
			\|[[File:Alternate truncated cube.png\|100px]]<BR>[[Alternate truncated cube]]
			\|[[File:Truncated rhombic dodecahedron2.png\|100px]]<BR>Truncated rhombic dodecahedron
			\|[[File:Truncated rhombic triacontahedron.png\|100px]]<BR>[[Truncated rhombic triacontahedron]]
			\|[[File:Truncated rhombille tiling.png\|100px]]<BR>[[Hexagonal tiling]]
			\|
			\|}

			=== Rhombic dodecahedron ===
			The name ''truncated rhombic dodecahedron'' is ambiguous since only 6 (order-4) vertices were truncated. A truncation on just the 3-vertices of the rhombic dodecahedron would cause an icosahedron with 12 equilateral hexagons and 20 triangles, forming 30 vertices in total; this figure's dual can be called the ''triakis cuboctahedron''. Another alternate truncated rhombic dodecahedron can appear by truncating all 14 vertices, yielding 12 irregular octagonal faces. The dual of the full truncation is a triangular tetracontaoctahedron labeled the ''tritetrakis cuboctahedron'', which is a complete [[Kleetope]] of the [[cuboctahedron]]. The [[tetrahedron]] is to the [[truncated cube]] as the [[cube]] is to the full truncation or a bitruncated cuboctahedron.

			<gallery>
			File:rhombic dodecahedron.png\|Rhombic dodecahedron
			Image:StellaTruncRhombicDodeca.png\|Full truncation of rhombic dodecahedron
			Image:Truncated rhombic dodecahedron2.png\|Order-4 truncation of rhombic dodecahedron<BR>(Chamfered cube)
			File:Chamfered_octahedron.png\|Order-3 truncation of rhombic dodecahedron<BR>(Chamfered octahedron)
			</gallery>

			== See also ==
			* [[Chamfered tetrahedron]]
			* [[Chamfered dodecahedron]]
			* [[Near-miss Johnson solid]]

			== References==
			* Antoine Deza, Michel Deza, Viatcheslav Grishukhin, ''Fullerenes and coordination polyhedra versus half-cube embeddings'', 1998 [[PDF]] [http://www.cas.mcmaster.ca/~deza/dm1998.pdf]

			== External links ==
			* [[VRML]] model [http://www.georgehart.com/virtual-polyhedra/vrml/zono-7_from_cube.wrl]
			** [http://www.georgehart.com/virtual-polyhedra/conway_notation.html VTML polyhedral generator] Try "t4daC" ([[Conway polyhedron notation]])

			{{Near-miss Johnson solids navigator}}

			[[Category:Zonohedra]]

en>Ankit Maity: Reverted edits by 175.141.244.59 (talk) to last revision by ClueBot NG (HG)

2012-03-29T12:37:10Z

Reverted edits by 175.141.244.59 (talk) to last revision by ClueBot NG (HG)

New page

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