16-cell

Regular hexadecachoron (16-cell) (4-orthoplex)
Schlegel diagram (vertices and edges)
Type	Convex regular 4-polytope
Vertices	8
Edges	24
Faces	32 {3}
Cell	16 {3,3}
Vertex figure	Octahedron
Schläfli symbol	{3,3,4} {31,1,1} h{4,3,3} s{2,2,2}
Coxeter-Dynkin diagram
Petrie polygon	octagon
Coxeter group	C4, [3,3,4] D4, [31,1,1]
Dual	Tesseract
Properties	convex, isogonal, isotoxal, isohedral
Uniform index	11 12 13

In four dimensional geometry, a 16-cell, is a regular convex 4-polytope. It is also known as the hexadecachoron. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.

Conway calls it an orthoplex for orthant complex, as well as the entire class of cross-polytopes.

Geometry

The hexadecachoron is a member of the family of polytopes called the cross-polytopes, 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 eight vertices of the hexadecachoron 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 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 or 4-demicube, a member of the demihypercube family, and represented by h{4,3,3}, and can be drawn bicolored with alternating tetrahedral cells.

Images

Stereographic projection	Four orthographic projections	A skew orthogonal projection inside its regular octagonal Petrie polygon, connecting all vertices except opposite ones.
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.		A 3D projection of a 16-cell performing a double rotation about two orthogonal planes.

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.

See also

• 24-cell
• Polychoron

References

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

External links

• Weisstein, Eric W., "16-Cell" from MathWorld.
• Olshevsky, George, Hexadecachoron at Glossary for Hyperspace.
  • 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 12, George Olshevsky.
• Der 16-Zeller (16-cell) Marco Möller's Regular polytopes in R⁴ (German)
• Description and diagrams of 16-cell projections