Editing 16-cell (section)

{{Short description|Four-dimensional analog of the octahedron}}
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{{Infobox polychoron |
  Name=16-cell<br />(4-orthoplex)|
  Image_File=Schlegel wireframe 16-cell.png|
  Image_Caption=[[Schlegel diagram]]<br />(vertices and edges)|
  Type=[[Convex regular 4-polytope]]<br />4-[[orthoplex]]<br />4-[[demihypercube|demicube]]|
  Last=[[Rectified tesseract|11]]|
  Index=12|
  Next=[[Truncated tesseract|13]]|
  Schläfli={3,3,4}|
  CD={{CDD|node_1|3|node|3|node|4|node}} |
  Cell_List=16 [[tetrahedron|{3,3}]] [[File:3-simplex t0.svg|25px]]|
  Face_List=32 [[triangle|{3}]] [[File:2-simplex t0.svg|25px]]|
  Edge_Count= 24|
  Vertex_Count= 8|
  Petrie_Polygon=[[octagon]]|
  Coxeter_Group=B<sub>4</sub>, [3,3,4], order 384<br />D<sub>4</sub>, order 192|
  Vertex_Figure=[[File:16-cell verf.svg|80px]]<br />[[Octahedron]]|
  Dual=[[Tesseract]]|
  Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]], [[regular polytope|regular]], [[Hanner polytope]]
}}
In [[geometry]], the '''16-cell''' is the [[regular convex 4-polytope]] (four-dimensional analogue of a Platonic solid) with [[Schläfli symbol]] {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician [[Ludwig Schläfli]] in the mid-19th century.{{Sfn|Coxeter|1973|p=141|loc=§&nbsp;7-x. Historical remarks}} It is also called '''C<sub>16</sub>''', '''hexadecachoron''',<ref>[[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> or '''hexdecahedroid'''{{sic|?}}.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref>

It is the 4-dimensional member of an infinite family of polytopes called [[cross-polytope]]s, ''orthoplexes'', or ''hyperoctahedrons'' which are analogous to the [[octahedron]] in three dimensions. It is Coxeter's <math>\beta_4</math> polytope.{{Sfn|Coxeter|1973|pp=120–121|loc=§&nbsp;7.2. See illustration Fig 7.2<small>B</small>}} The [[dual polytope]] is the [[tesseract]] (4-[[hypercube|cube]]), which it can be combined with to form a compound figure. The cells of the 16-cell are dual to the 16 vertices of the tesseract.