Editing 16-cell (section)

==== Tetrahedral constructions ====
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|[[File:16-cell net.png|180px|]]
|[[File:16-cell nets.png|180px]]
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The 16-cell has two [[Wythoff construction]]s from regular tetrahedra, a regular form and alternated form, shown here as [[Net (polyhedron)|nets]], the second represented by tetrahedral cells of two alternating colors. The alternated form is a [[#Symmetry constructions|lower symmetry construction]] of the 16-cell called the [[demitesseract]].

Wythoff's construction replicates the 16-cell's [[5-cell#Orthoschemes|characteristic 5-cell]] in a [[kaleidoscope]] of mirrors. Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|An [[orthoscheme]] is a [[chiral]] irregular [[simplex]] with [[right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[facet (geometry)|facets]] (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} There are three regular 4-polytopes with tetrahedral cells: the [[5-cell]], the 16-cell, and the [[600-cell]]. Although all are bounded by ''regular'' tetrahedron cells, their characteristic 5-cells (4-orthoschemes) are different [[5-cell#Isometries|tetrahedral pyramids]], all based on the same characteristic ''irregular'' tetrahedron. They share the same [[Tetrahedron#Orthoschemes|characteristic tetrahedron]] (3-orthoscheme) and characteristic [[right triangle]] (2-orthoscheme) because they have the same kind of cell.{{Efn|A regular polytope of dimension ''k'' has a characteristic ''k''-orthoscheme, and also a characteristic (''k''-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.{{Sfn|Coxeter|1973|p=130|loc=§&nbsp;7.6|ps=; "simplicial subdivision".}} The interior tetrahedra and triangles thus formed will also be orthoschemes.}}

{| class="wikitable floatright"
!colspan=6|Characteristics of the 16-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "16-cell, 𝛽<sub>4</sub>"}}
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!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§&nbsp;7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}}
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!align=right|𝒍
|align=center|<small><math>\sqrt{2} \approx 1.414</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|align=center|<small>120°</small>
|align=center|<small><math>\tfrac{2\pi}{3}</math></small>
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!align=right|𝟀
|align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small>
|align=center|<small>60″</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
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!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small>
|align=center|<small>45″</small>
|align=center|<small><math>\tfrac{\pi}{4}</math></small>
|align=center|<small>45°</small>
|align=center|<small><math>\tfrac{\pi}{4}</math></small>
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!align=right|𝟁
|align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small>
|align=center|<small>30″</small>
|align=center|<small><math>\tfrac{\pi}{6}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
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!align=right|<small><math>_0R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
!align=right|<small><math>_1R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small>
|align=center|<small>45°</small>
|align=center|<small><math>\tfrac{\pi}{4}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
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!align=right|<small><math>_2R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small>
|align=center|<small>30°</small>
|align=center|<small><math>\tfrac{\pi}{6}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
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!align=right|<small><math>_0R^4/l</math></small>
|align=center|<small><math>1</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
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!align=right|<small><math>_1R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
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!align=right|<small><math>_2R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
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!align=right|<small><math>_3R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small>
|align=center|
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The '''characteristic 5-cell of the regular 16-cell''' is represented by the [[Coxeter-Dynkin diagram]] {{CDD|node|3|node|3|node|4|node}}, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular [[Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 16-cell is subdivided by its symmetry hyperplanes into 384 instances of its characteristic 5-cell that all meet at its center.

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 16-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 16-cell has unit radius edge and edge length 𝒍 = <small><math>\sqrt{2}</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small> (edges which are the characteristic radii of the regular 16-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, first from a 16-cell vertex to a 16-cell edge center, then turning 90° to a 16-cell face center, then turning 90° to a 16-cell tetrahedral cell center, then turning 90° to the 16-cell center.