16-cell


16-cell
Regular hexadecachoron
(16-cell)
(4-orthoplex)
Schlegel wireframe 16-cell.png
Schlegel diagram
(vertices and edges)
Type Convex regular 4-polytope
Schläfli symbol {3,3,4}
{3,31,1}
h{4,3,3}
s{2,2,2}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.png
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.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 C4, [3,3,4]
D4, [31,1,1]
[23] (half)
Symmetry group [3,3,4], order 384
[31,1,1], order 192
[3,4,2+], order 48
[23]+, order 8
Dual Tesseract
Properties convex, isogonal, isotoxal, isohedral
Uniform index 12

In four dimensional geometry, a 16-cell or hexadecachoron 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 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.

Contents

Geometry

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.

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 CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png or CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png. 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 h0,1{2,4,3}, and Coxeter diagram: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.png.

It can also be seen as a snub 4-orthotope, represented by s{2,2,2}, and Coxeter diagram: CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png.

Images

Stereographic polytope 16cell.png
Stereographic projection
16-cell.gif
A 3D projection of a 16-cell performing a simple rotation.
16-cell nets.png
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.
orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t3.svg 4-cube t3 B3.svg 4-cube t3 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph 4-cube t3 F4.svg 4-cube t3 A3.svg
Dihedral symmetry [12/3] [4]
Orthogonal projection graphs
4-demicube graph.png
demitesseract in order-4 Petrie polygon symmetry as an alternated tesseract
Hypercubestar.svg
Tesseract

Tessellations

One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the hexadecachoric honeycomb and has Schläfli symbol {3,3,4,3}. The dual tessellation, icositetrachoric 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 an octahedron, 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.

Projections

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 Diagram

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

Venn 1000 0000 0000 0000.png Venn 0110 1000 1000 0000.png

Venn 0100 0000 0000 0000.pngVenn 0010 0000 0000 0000.pngVenn 0000 1000 0000 0000.pngVenn 0000 0000 1000 0000.png

Venn 0001 0110 0110 1000.png

Venn 0001 0000 0000 0000.pngVenn 0000 0100 0000 0000.pngVenn 0000 0010 0000 0000.pngVenn 0000 0000 0100 0000.pngVenn 0000 0000 0010 0000.pngVenn 0000 0000 0000 1000.png

Venn 0000 0001 0001 0110.png

Venn 0000 0001 0000 0000.pngVenn 0000 0000 0001 0000.pngVenn 0000 0000 0000 0100.pngVenn 0000 0000 0000 0010.png

Venn 0000 0000 0000 0001.png


Related uniform polytopes

Name tesseract rectified
tesseract
truncated
tesseract
cantellated
tesseract
runcinated
tesseract
bitruncated
tesseract
cantitruncated
tesseract
runcitruncated
tesseract
omnitruncated
tesseract
Coxeter-Dynkin
diagram
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schläfli
symbol
{4,3,3} t1{4,3,3} t0,1{4,3,3} t0,2{4,3,3} t0,3{4,3,3} t1,2{4,3,3} t0,1,2{4,3,3} t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
Schlegel wireframe 8-cell.png Schlegel half-solid rectified 8-cell.png Schlegel half-solid truncated tesseract.png Schlegel half-solid cantellated 8-cell.png Schlegel half-solid runcinated 8-cell.png Schlegel half-solid bitruncated 8-cell.png Schlegel half-solid cantitruncated 8-cell.png Schlegel half-solid runcitruncated 8-cell.png Schlegel half-solid omnitruncated 8-cell.png
B4 Coxeter plane graph 4-cube t0.svg 4-cube t1.svg 4-cube t01.svg 4-cube t02.svg 4-cube t03.svg 4-cube t12.svg 4-cube t012.svg 4-cube t013.svg 4-cube t0123.svg
 
Name 16-cell rectified
16-cell
truncated
16-cell
cantellated
16-cell
runcinated
16-cell
bitruncated
16-cell
cantitruncated
16-cell
runcitruncated
16-cell
omnitruncated
16-cell
Coxeter-Dynkin
diagram
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schläfli
symbol
{3,3,4} t1{3,3,4} t0,1{3,3,4} t0,2{3,3,4} t0,3{3,3,4} t1,2{3,3,4} t0,1,2{3,3,4} t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
Schlegel wireframe 16-cell.png Schlegel half-solid rectified 16-cell.png Schlegel half-solid truncated 16-cell.png Schlegel half-solid cantellated 16-cell.png Schlegel half-solid runcinated 16-cell.png Schlegel half-solid bitruncated 16-cell.png Schlegel half-solid cantitruncated 16-cell.png Schlegel half-solid runcitruncated 16-cell.png Schlegel half-solid omnitruncated 16-cell.png
B4 Coxeter plane graph 4-cube t3.svg 4-cube t2.svg 4-cube t23.svg 4-cube t13.svg 4-cube t03.svg 4-cube t12.svg 4-cube t123.svg 4-cube t023.svg 4-cube t0123.svg

See also

References

  • 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, editied 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 links


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