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| In [[geometry]], a '''cross-polytope''',<ref>{{citation | last = [[E. L. Elte|Elte]] | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}} Chapter IV, five dimensional semiregular polytope [http://www.amazon.com/Semiregular-Polytopes-Hyperspaces-Emanuel-Lodewijk/dp/141817968X]</ref> '''orthoplex''',<ref>[[John Horton Conway|Conway]] calls it an n-'''orthoplex''' for ''[[orthant]] complex''.
| | Dental Remedy<br><br>Acquiring gracious flannel dentition is non well-fixed for nigh citizenry. Mainly because they do non live what causes teeth to discoloured and are non willing to make up thousands of dollars for a tooth doctor natter. Yet though many celebrities prefer a foresightful lasting business to business marketing tips ([http://www.tropacafe.com/?option=com_k2&view=itemlist&task=user&id=5056 www.tropacafe.com]) in place tooth lightening treatment, about multitude exactly don't get the clock and money to go for it.<br><br>Gingivitis, the mum Orcinus orca of grownup teeth... It is estimated that ended 75% of American adults rich person about fervor of the gums or gingivitis. Unfortunately, the plate marketplace is alive and good crossways the US. Pitiful alveolar hygiene is the origin causa for tooth passing and nates be completely preventable if every day give care is maintained.<br><br>Set uncollectible breath arse non lone be a frustrative metre only it potty be an expensive enterprisingness as fountainhead. With so many varieties of toothpastes, mouthwashes, alveolar consonant flosses and knife scrapers we commode oft close up outlay a deal of money on products that sometimes do not crop.<br><br>You rear end hush up experience the bright, beautiful grin you're wishing for without outlay hundreds of bucks for a laser teeth lightening surgery or for expensive products. At dwelling teeth lightening remedies, for the most part victimization cancel ingredients, or utilizing over-the-buffet products that are very leisurely and commodious to employ are effectual in transforming your dentition into tiptop chopper whites!Don't prove to ameriplan dental read a humble light touch and a nursing bottle of atomic number 1 hydrogen peroxide and hear to white your dentition.<br><br>Mum e'er said, "brush your teeth before going to bed." Haemorrhage gums are usual among Land adults ended 35 due to misfortunate dental consonant hygiene practices. It is estimated that all over 75% of us experience at least a soft shape of gingivitis that bear upon our gums. Bacteria forms course in our mouths owed to convention feeding and dentistry advertising imbibition.<br><br>Tooth lightening is no thirster a decorative luxury, rather it has morphed into a introductory penury of our modernistic life style and give birth begun to fun authoritative roles in our lives. You are literally judged for how whiten your teeth is, which in crook could imply the typecast of modus vivendi you lead, thereby affecting the simulacrum you demo in the eyes of associates and friends.<br><br>At place dentition whitening remedies, generally using cancel ingredients, or utilizing over-the-buffet products that are real tardily and commodious to enforce are in force in transforming your teeth into superintendent chopper whites!Don't hear to accept a lowly encounter and a bottleful of atomic number 1 hydrogen peroxide and effort to white your dentition. |
| </ref> '''hyperoctahedron''', or '''cocube''' is a [[regular polytope|regular]], convex [[polytope]] that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the [[convex hull]] of its vertices. Its facets are [[simplex]]es of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.
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| The ''n''-dimensional cross-polytope can also be defined as the closed [[unit ball]] (or, according to some authors, its boundary) in the [[L1-norm|ℓ<sub>1</sub>-norm]] on '''R'''<sup>''n''</sup>:
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| :<math>\{x\in\mathbb R^n : \|x\|_1 \le 1\}.</math>
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| In 1 dimension the cross-polytope is simply the [[line segment]] [−1, +1], in 2 dimensions it is a [[Square (geometry)|square]] (or diamond) with vertices {(±1, 0), (0, ±1)}. In 3 dimensions it is an [[octahedron]]—one of the five convex regular [[polyhedron|polyhedra]] known as the [[Platonic solid]]s. Higher-dimensional cross-polytopes are generalizations of these.
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| {| style="margin-left: auto; margin-right: auto"
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| |style="text-align: center;"|[[Image:2-orthoplex.svg|120px|A 2-dimensional cross-polytope]]
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| |width="50px"|
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| |style="text-align: center;"|[[Image:Octahedron.png|120px|A 3-dimensional cross-polytope]]
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| |width="50px"|
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| |style="text-align: center;"|[[Image:Schlegel wireframe 16-cell.png|120px|A 4-dimensional cross-polytope]]
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| |-
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| |style="text-align: center;"|2 dimensions<BR>[[square (geometry)|square]]
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| |style="text-align: center;"|3 dimensions<BR>[[octahedron]]
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| |style="text-align: center;"|4 dimensions<BR>[[16-cell]]
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| |}
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| The cross-polytope is the [[dual polytope]] of the [[hypercube]]. The 1-[[Skeleton (topology)|skeleton]] of a ''n''-dimensional cross-polytope is a [[Turán graph]] ''T''(2''n'',''n'').
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| == 4 dimensions ==
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| The 4-dimensional cross-polytope also goes by the name '''hexadecachoron''' or '''[[16-cell]]'''. It is one of six [[convex regular 4-polytope]]s. These [[polychoron|polychora]] were first described by the Swiss mathematician [[Ludwig Schläfli]] in the mid-19th century.
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| == Higher dimensions ==
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| The '''cross polytope''' family is one of three [[regular polytope]] families, labeled by [[Coxeter]] as ''β<sub>n</sub>'', the other two being the [[hypercube]] family, labeled as ''γ<sub>n</sub>'', and the [[simplex|simplices]], labeled as ''α<sub>n</sub>''. A fourth family, the [[hypercubic honeycomb|infinite tessellations of hypercubes]], he labeled as ''δ<sub>n</sub>''.
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| The ''n''-dimensional cross-polytope has 2''n'' vertices, and 2<sup>''n''</sup> facets (''n''−1 dimensional components) all of which are ''n''−1 [[Simplex|simplices]]. The [[vertex figure]]s are all ''n'' − 1 cross-polytopes. The [[Schläfli symbol]] of the cross-polytope is {3,3,…,3,4}. The [[dihedral angle]] of the ''n''-dimensional cross-polytope is
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| :<math>\arccos\left(\frac{2-n}{n}\right)</math>.
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| The number of ''k''-dimensional components (vertices, edges, faces, …, facets) in an ''n''-dimensional cross-polytope is given by (see [[binomial coefficient]]):
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| :<math>2^{k+1}{n \choose {k+1}}</math>
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| The volume of the ''n''-dimensional cross-polytope is
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| :<math>\frac{2^n}{n!}.</math>
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| There are many possible [[orthographic projection]]s that can show the cross-polytopes as 2-dimensional graphs. [[Petrie polygon]] projections map the points into a regular ''2n''-gon or lower order regular polygons. A second projection takes the ''2(n-1)''-gon petrie polygon of the lower dimension, seen as a [[bipyramid]], projected down the axis, with 2 vertices mapped into the center.
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| | |
| {| class="wikitable"
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| |+
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| Cross-polytope elements
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| |-
| |
| ! [[polytope|n]]
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| !β<sub>n</sub><BR>k<sub>11</sub>
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| ! Name(s)<BR>[[Graph (mathematics)|Graph]]
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| !Graph<BR>2n-gon
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| !Graph<BR>2(n-1)-gon
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| ![[Schläfli symbol|Schläfli]]
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| ![[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagrams]]
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| ! [[Vertex (geometry)|Vertices]]
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| ! [[Edge (geometry)|Edges]]
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| ! [[Face (geometry)|Faces]]
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| ! [[Cell (geometry)|Cells]]
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| ! ''4''-faces | |
| ! ''5''-faces
| |
| ! ''6''-faces
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| ! ''7''-faces
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| ! ''8''-faces
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| ! ''9''-faces
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| |-
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| | [[1-polytope|1]]
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| !β<sub>1</sub>
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| |[[Edge (geometry)|Line segment]]<BR>1-orthoplex
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| |[[Image:Cross graph 1.svg|50px]]
| |
| |
| |
| |{}
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| |{{CDD|node_1}}<BR>{{CDD|node_f1}}
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| |align=right| 2
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| |
| |
| |
| |
| |
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| |
| |
| |
| |
| |
| |
| |-
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| | [[2-polytope|2]]
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| !β<sub>2</sub><BR> −1<sub>11</sub>
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| |[[square (geometry)|square]]<BR>2-orthoplex<BR>'''Bicross'''
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| |[[Image:Cross graph 2.png|50px]]
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| |[[Image:2-orthoplex B1.svg|50px]]
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| |{4}<BR>{}+{}
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| |{{CDD|node_1|4|node}}<BR>{{CDD|node_f1|2|node_f1}}
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| |align=right| 4
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| |align=right| 4
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| |
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| |-
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| | [[3-polytope|3]]
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| !β<sub>3</sub><BR>0<sub>11</sub>
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| | [[octahedron]]<BR>3-orthoplex<BR>'''Tricross'''
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| |[[Image:3-orthoplex.svg|50px]]
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| |[[Image:3-orthoplex B2.svg|50px]]
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| |{3,4}<BR>{3<sup>0,1,1</sup>}<BR>{}+{}+{}
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| |{{CDD|node_1|3|node|4|node}}<BR>{{CDD|node_1|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1}}
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| |align=right| 6
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| |align=right| 12
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| |align=right| 8
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| |
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| |
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| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | [[4-polytope|4]]
| |
| !β<sub>4</sub><BR>1<sub>11</sub>
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| |[[16-cell]]<BR>4-orthoplex<BR>'''Tetracross'''
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| |[[Image:4-orthoplex.svg|50px]]
| |
| |[[Image:4-orthoplex B3.svg|50px]]
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| |{3,3,4}<BR>{3<sup>1,1,1</sup>}<BR>4{}
| |
| |{{CDD|node_1|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1}}
| |
| |align=right| 8
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| |align=right| 24
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| |align=right| 32
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| |align=right| 16
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| |
| |
| |
| |
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| |
| |
| |
| |
| |
| |
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| |-
| |
| | [[5-polytope|5]]
| |
| !β<sub>5</sub><BR>2<sub>11</sub>
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| |[[5-orthoplex]]<BR>'''Pentacross'''
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| |[[Image:5-orthoplex.svg|50px]]
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| |[[Image:5-orthoplex B4.svg|50px]]
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| |{3<sup>3</sup>,4}<BR>{3<sup>2,1,1</sup>}<BR>5{}
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| |{{CDD|node_1|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
| |
| |align=right| 10
| |
| |align=right| 40
| |
| |align=right| 80
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| |align=right| 80
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| |align=right| 32
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| |
| |
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| |
| |
| |
| |
| |
| |
| |
| |-
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| | [[6-polytope|6]]
| |
| !β<sub>6</sub><BR>3<sub>11</sub>
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| | [[6-orthoplex]]<BR>'''Hexacross'''
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| |[[Image:6-orthoplex.svg|50px]]
| |
| |[[Image:6-orthoplex B5.svg|50px]]
| |
| |{3<sup>4</sup>,4}<BR>{3<sup>3,1,1</sup>}<BR>6{}
| |
| |{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
| |
| |align=right| 12
| |
| |align=right| 60
| |
| |align=right| 160
| |
| |align=right| 240
| |
| |align=right| 192
| |
| |align=right| 64
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | [[7-polytope|7]]
| |
| !β<sub>7</sub><BR>4<sub>11</sub>
| |
| |[[7-orthoplex]]<BR>'''Heptacross'''
| |
| |[[Image:7-orthoplex.svg|50px]]
| |
| |[[Image:7-orthoplex B6.svg|50px]]
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| |{3<sup>5</sup>,4}<BR>{3<sup>4,1,1</sup>}<BR>7{}
| |
| |{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
| |
| |align=right| 14
| |
| |align=right| 84
| |
| |align=right| 280
| |
| |align=right| 560
| |
| |align=right| 672
| |
| |align=right| 448
| |
| |align=right| 128
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | [[8-polytope|8]]
| |
| !β<sub>8</sub><BR>5<sub>11</sub>
| |
| |[[8-orthoplex]]<BR>'''Octacross'''
| |
| |[[Image:8-orthoplex.svg|50px]]
| |
| |[[Image:8-orthoplex B7.svg|50px]]
| |
| |{3<sup>6</sup>,4}<BR>{3<sup>5,1,1</sup>}<BR>8{}
| |
| |{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
| |
| |align=right| 16
| |
| |align=right| 112
| |
| |align=right| 448
| |
| |align=right| 1120
| |
| |align=right| 1792
| |
| |align=right| 1792
| |
| |align=right| 1024
| |
| |align=right| 256
| |
| |
| |
| |
| |
| |-
| |
| | [[9-polytope|9]]
| |
| !β<sub>9</sub><BR>6<sub>11</sub>
| |
| |[[9-orthoplex]]<BR>'''Enneacross'''
| |
| |[[Image:9-orthoplex.svg|50px]]
| |
| |[[Image:9-orthoplex B8.svg|50px]]
| |
| |{3<sup>7</sup>,4}<BR>{3<sup>6,1,1</sup>}<BR>9{}
| |
| |{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
| |
| |align=right| 18
| |
| |align=right| 144
| |
| |align=right| 672
| |
| |align=right| 2016
| |
| |align=right| 4032
| |
| |align=right| 5376
| |
| |align=right| 4608
| |
| |align=right| 2304
| |
| |align=right| 512
| |
| |
| |
| |-
| |
| ! [[10-polytope|10]]
| |
| ! β<sub>10</sub><BR>7<sub>11</sub>
| |
| |[[10-orthoplex]]<BR>'''Decacross'''
| |
| |[[Image:10-orthoplex.svg|50px]]
| |
| |[[Image:10-orthoplex B9.svg|50px]]
| |
| |{3<sup>8</sup>,4}<BR>{3<sup>7,1,1</sup>}<BR>10{}
| |
| |{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
| |
| |align=right|20
| |
| |align=right|180
| |
| |align=right|960
| |
| |align=right|3360
| |
| |align=right|8064
| |
| |align=right|13440
| |
| |align=right|15360
| |
| |align=right|11520
| |
| |align=right|5120
| |
| |align=right|1024
| |
| |-
| |
| | colspan=17 style="text-align:center" |...
| |
| |-
| |
| ! '''n'''
| |
| ! β<sub>n</sub><BR>''k''<sub>11</sub>
| |
| |''n''-orthoplex<BR>''n''-cross
| |
| |
| |
| |
| |
| |{3<sup>''n'' − 2</sup>,4}<BR>{3<sup>''n'' − 3,1,1</sup>}<BR>n{}
| |
| |{{CDD|node_1|3|node|3|node}}...{{CDD|3|node|3|node|4|node}}<BR>{{CDD|node_1|3|node|3}}...{{CDD|node|3|node|split1|nodes}}<BR>{{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2}}...{{CDD|2|node_f1}}
| |
| | colspan=11 style="text-align:center" | 2''n'' '''0-faces''', ... <math>2^{k+1}{n\choose k+1}</math> '''''k''-faces''' ..., 2<sup>''n''</sup> '''(n-1)-faces'''
| |
| |}
| |
| | |
| The vertices of an axis-aligned cross polytope are all at equal distance from each other in the [[taxicab geometry|Manhattan distance]] ([[Lp space|L<sup>1</sup> norm]]). [[Kusner's conjecture]] states that this set of 2''d'' points is the largest possible equidistant set for this distance.<ref>{{citation
| |
| | last = Guy | first = Richard K.
| |
| | issue = 3
| |
| | journal = American Mathematical Monthly
| |
| | pages = 196–200
| |
| | title = An olla-podrida of open problems, often oddly posed
| |
| | jstor = 2975549
| |
| | volume = 90
| |
| | year = 1983}}.</ref>
| |
| | |
| == See also ==
| |
| | |
| * [[List of regular polytopes]]
| |
| * [[Hyperoctahedral group]], the symmetry group of the cross-polytope
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{cite book | first = H. S. M. | last = Coxeter | authorlink = H. S. M. Coxeter | year = 1973 | title = [[Regular Polytopes (book)|Regular Polytopes]] | edition = 3rd ed. | publisher = Dover Publications | location = New York | pages = 121–122 | isbn = 0-486-61480-8}} p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
| |
| | |
| == External links ==
| |
| {{commons category|Cross-polytope graphs}}
| |
| * {{mathworld | urlname = CrossPolytope | title = Cross polytope}}
| |
| * [http://www.geocities.com/shapirojon34/tesseract/TesseractApplet.html Polytope Viewer]{{dead link|date=November 2010|bot=AnomieBOT}} (Click <polytopes...> to select cross polytope.)
| |
| *{{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
| |
| | |
| {{Dimension topics}}
| |
| {{Polytopes}}
| |
| | |
| {{DEFAULTSORT:Cross-Polytope}}
| |
| [[Category:Polytopes]]
| |
| [[Category:Multi-dimensional geometry]]
| |
| [[Category:Four-dimensional geometry]]
| |
Dental Remedy
Acquiring gracious flannel dentition is non well-fixed for nigh citizenry. Mainly because they do non live what causes teeth to discoloured and are non willing to make up thousands of dollars for a tooth doctor natter. Yet though many celebrities prefer a foresightful lasting business to business marketing tips (www.tropacafe.com) in place tooth lightening treatment, about multitude exactly don't get the clock and money to go for it.
Gingivitis, the mum Orcinus orca of grownup teeth... It is estimated that ended 75% of American adults rich person about fervor of the gums or gingivitis. Unfortunately, the plate marketplace is alive and good crossways the US. Pitiful alveolar hygiene is the origin causa for tooth passing and nates be completely preventable if every day give care is maintained.
Set uncollectible breath arse non lone be a frustrative metre only it potty be an expensive enterprisingness as fountainhead. With so many varieties of toothpastes, mouthwashes, alveolar consonant flosses and knife scrapers we commode oft close up outlay a deal of money on products that sometimes do not crop.
You rear end hush up experience the bright, beautiful grin you're wishing for without outlay hundreds of bucks for a laser teeth lightening surgery or for expensive products. At dwelling teeth lightening remedies, for the most part victimization cancel ingredients, or utilizing over-the-buffet products that are very leisurely and commodious to employ are effectual in transforming your dentition into tiptop chopper whites!Don't prove to ameriplan dental read a humble light touch and a nursing bottle of atomic number 1 hydrogen peroxide and hear to white your dentition.
Mum e'er said, "brush your teeth before going to bed." Haemorrhage gums are usual among Land adults ended 35 due to misfortunate dental consonant hygiene practices. It is estimated that all over 75% of us experience at least a soft shape of gingivitis that bear upon our gums. Bacteria forms course in our mouths owed to convention feeding and dentistry advertising imbibition.
Tooth lightening is no thirster a decorative luxury, rather it has morphed into a introductory penury of our modernistic life style and give birth begun to fun authoritative roles in our lives. You are literally judged for how whiten your teeth is, which in crook could imply the typecast of modus vivendi you lead, thereby affecting the simulacrum you demo in the eyes of associates and friends.
At place dentition whitening remedies, generally using cancel ingredients, or utilizing over-the-buffet products that are real tardily and commodious to enforce are in force in transforming your teeth into superintendent chopper whites!Don't hear to accept a lowly encounter and a bottleful of atomic number 1 hydrogen peroxide and effort to white your dentition.