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In four-dimensional geometry, a 16-cell is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid.[1]

Regular hexadecachoron
(16-cell)
(4-orthoplex)
Schlegel wireframe 16-cell.png
Schlegel diagram
(vertices and edges)
Type Convex regular 4-polytope
4-orthoplex
4-demicube
Schläfli symbol {3,3,4}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cells 16 {3,3} 3-simplex t0.svg
Faces 32 {3} 2-simplex t0.svg
Edges 24
Vertices 8
Vertex figure 16-cell verf.png
Octahedron
Petrie polygon octagon
Coxeter group B4, [3,3,4], order 384
D4, order 192
Dual Tesseract
Properties convex, isogonal, isotoxal, isohedral, quasiregular
Uniform index 12

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the tesseract (4-cube). Conway's name for a cross-polytope is orthoplex, for orthant complex. The 16-cell has 16 cells as the tesseract has 16 vertices.

Contents

GeometryEdit

It is bounded by 16 cells, all of which are regular tetrahedra. It has 32 triangular faces, 24 edges, and 8 vertices. The 24 edges bound 6 squares lying in the 6 coordinate planes.

The eight vertices of the 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs.

The Schläfli symbol of the 16-cell is {3,3,4}. Its vertex figure is a regular octahedron. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. Its edge figure is a square. There are 4 tetrahedra and 4 triangles meeting at every edge.

The 16-cell can be decomposed into two similar disjoint circular chains of eight tetrahedrons each, four edges long. Each chain, when stretched out straight, forms a Boerdijk–Coxeter helix. This decomposition can be seen in a 4-4 duoantiprism construction of the 16-cell:         or        , Schläfli symbol {2}⨂{2} or s{2}s{2}, symmetry [[4,2+,4]], order 64.

The 16-cell can be dissected into two octahedral pyramids, which share a new octahedron base through the 16-cell center.

ImagesEdit

 
Stereographic projection
 
A 3D projection of a 16-cell performing a simple rotation.
 
The 16-cell has two Wythoff constructions, a regular form and alternated form, shown here as nets, the second being represented by alternately two colors of tetrahedral cells.

Orthogonal projectionsEdit

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph      
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph    
Dihedral symmetry [12/3] [4]

TessellationsEdit

One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol {3,3,4,3}. Hence, the 16-cell has a dihedral angle of 120°.[2] The dual tessellation, 24-cell honeycomb, {3,4,3,3}, is made of by regular 24-cells. Together with the tesseractic honeycomb {4,3,3,4}, these are the only three regular tessellations of R4. Each 16-cell has 16 neighbors with which it shares a tetrahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point. Twenty-four 16-cells meet at any given vertex in this tessellation.

Boerdijk–Coxeter helixEdit

A 16-cell can be constructed from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each folded into a 4-dimensional ring. The 16 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex. The purple edges represent the Petrie polygon of the 16-cell.

 

ProjectionsEdit

 
Projection envelopes of the 16-cell. (Each cell is drawn with different color faces, inverted cells are undrawn)

The cell-first parallel projection of the 16-cell into 3-space has a cubical envelope. The closest and farthest cells are projected to inscribed tetrahedra within the cube, corresponding with the two possible ways to inscribe a regular tetrahedron in a cube. Surrounding each of these tetrahedra are 4 other (non-regular) tetrahedral volumes that are the images of the 4 surrounding tetrahedral cells, filling up the space between the inscribed tetrahedron and the cube. The remaining 6 cells are projected onto the square faces of the cube. In this projection of the 16-cell, all its edges lie on the faces of the cubical envelope.

The cell-first perspective projection of the 16-cell into 3-space has a triakis tetrahedral envelope. The layout of the cells within this envelope are analogous to that of the cell-first parallel projection.

The vertex-first parallel projection of the 16-cell into 3-space has an octahedral envelope. This octahedron can be divided into 8 tetrahedral volumes, by cutting along the coordinate planes. Each of these volumes is the image of a pair of cells in the 16-cell. The closest vertex of the 16-cell to the viewer projects onto the center of the octahedron.

Finally the edge-first parallel projection has a shortened octahedral envelope, and the face-first parallel projection has a hexagonal bipyramidal envelope.

4 sphere Venn DiagramEdit

The usual projection of the 16-cell   and 4 intersecting spheres (a Venn diagram of 4 sets) form topologically the same object in 3D-space:

   

    

 

      

 

    

 

Symmetry constructionsEdit

There is a lower symmetry form of the 16-cell, called a demitesseract or 4-demicube, a member of the demihypercube family, and represented by h{4,3,3}, and Coxeter diagrams         or      . It can be drawn bicolored with alternating tetrahedral cells.

It can also be seen in lower symmetry form as a tetrahedral antiprism, constructed by 2 parallel tetrahedra in dual configurations, connected by 8 (possibly elongated) tetrahedra. It is represented by s{2,4,3}, and Coxeter diagram:        .

It can also be seen as a snub 4-orthotope, represented by s{21,1,1}, and Coxeter diagram:         or      .

With the tesseract constructed as a 4-4 duoprism, the 16-cell can be seen as its dual, a 4-4 duopyramid.

Name Coxeter diagram Schläfli symbol Coxeter notation Order Vertex figure
Regular 16-cell         {3,3,4} [3,3,4] 384      
Demitesseract
Quasiregular 16-cell
      =        
      =        
h{4,3,3}
{3,31,1}
[31,1,1] = [1+,4,3,3] 192      
Alternated 4-4 duoprism      2s{4,2,4} [[4,2+,4]] 64
Tetrahedral antiprism         s{2,4,3} [2+,4,3] 48
Alternated square prism prism         sr{2,2,4} [(2,2)+,4] 16
Snub 4-orthotope         =       s{21,1,1} [2,2,2]+ = [21,1,1]+ 8      
4-fusil
        {3,3,4} [3,3,4] 384      
        {4}+{4} or 2{4} [[4,2,4]] = [8,2+,8] 128      
        {3,4}+{ } [4,3,2] 96      
     
        {4}+2{ } [4,2,2] 32      
     
        { }+{ }+{ }+{ } or 4{ } [2,2,2] 16      

Related complex polygonsEdit

The Möbius–Kantor polygon is a regular complex polygon 3{3}3,    , in   shares the same vertices as the 16-cell. It has 8 vertices, and 8 3-edges.[3] [4]

The regular complex polygon, 2{4}4,    , in   has a real representation as a 16-cell in 4-dimensional space with 8 vertices, 16 2-edges, only half of the edges of the 16-cell. Its symmetry is 4[4]2, order 32. [5]

Orthographic projections of 2{4}4 polygon
 
In B4 Coxeter plane, 2{4}4 has 8 vertices and 16 2-edges, shown here with 4 sets of colors.
 
The 8 vertices are grouped in 2 sets (shown red and blue), each only connected with edges to vertices in the other set, making this polygon a complete bipartite graph, K4,4.[6]

Related uniform polytopes and honeycombsEdit

The regular 16-cell along with the tesseract exist in a set of 15 uniform 4-polytopes with the same symmetry. It is also a part of the uniform polytopes of D4 symmery.

This 4-polytope is also related to the cubic honeycomb, order-4 dodecahedral honeycomb, and order-4 hexagonal tiling honeycomb which all have octahedral vertex figures.

It is in a sequence to three regular 4-polytopes: the 5-cell {3,3,3}, 600-cell {3,3,5} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have tetrahedral cells.

It is first in a sequence of quasiregular polytopes and honeycombs h{4,p,q}, and a half symmetry sequence, for regular forms {p,3,4}.

See alsoEdit

ReferencesEdit

  1. ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
  2. ^ Coxeter, Regular polygons, p.293
  3. ^ Coxeter and Shephard, 1991, p.30 and p.47
  4. ^ Coxeter and Shephard, 1992
  5. ^ Regular Complex Polytopes, p. 108
  6. ^ Regular Complex Polytopes, p.114
  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • H.S.M. Coxeter:
    • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)

External linksEdit