16-CELL

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|16-cell
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|bgcolor=#ffffff align=center colspan=2|
Schlegel diagram
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|bgcolor=#e7dcc3|Type||Regular polychoron
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|bgcolor=#e7dcc3|Families||orthoplex
demihypercube
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|bgcolor=#e7dcc3|Cells||16 (''3.3.3'')
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|bgcolor=#e7dcc3|Faces||32 {3}
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|bgcolor=#e7dcc3|Edges||24
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|bgcolor=#e7dcc3|Vertices||8
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|bgcolor=#e7dcc3|Vertex figure||(3.3.3.3)
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|bgcolor=#e7dcc3|Schläfli symbols||t₀{3,3,4}
t₀{3^1,1,1}
h{4,3,3}
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|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||


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|bgcolor=#e7dcc3|Symmetry group||B₄, [3,3,4]
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|bgcolor=#e7dcc3|Dual||Tesseract
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|bgcolor=#e7dcc3|Properties||convex
|}

In geometry, a '16-cell', or 'orthoplex', is a regular convex polychoron, or polytope existing in four dimensions. It is also known as the 'hexadecachoron'. It is one of the six regular convex polychora first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.

Contents

Geometry

Images

Tessellations

Projections

See also

References

External links

Geometry

The hexadecachoron is a member of the family of polytopes called the cross-polytopes or ''orthoplexes'', which exist in all dimensions. As such, its dual polychoron is the tesseract (the 4-dimensional hypercube).
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 vertices of the hexadecachoron consist of all permutations of (±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 hexadecachoron 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, a member of the demihypercube family, and represented by h{4,3,3}, and can be drawn bicolored with alternating tetrahedra cells.

Images

{| class="prettytable" width=640
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Stereographic projection
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Four orthographic projections
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4-cross-polytope graph
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A 3D projection of a 16-cell performing a double rotation about two orthogonal planes.
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Tessellations

One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the hexadecachoronic tetracomb and has Schläfli symbol {3,3,4,3}. The dual tessellation, icositetrachoronic tetracomb, {3,4,3,3}, is made of by regular 24-cells. Together with the tesseractic tetracomb {4,3,3,4}, these are the only three regular tessellations of 'R'⁴. 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

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 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.

See also

★ 24-cell

★ Polychoron

References

★ H. S. M. Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.

External links

★

★

★ Der 16-Zeller (16-cell) Marco Möller's Regular polytopes in R⁴ (German)

★ Description and diagrams of 16-cell projections