16-cell

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16-cell
Schlegel diagram
Type	Regular polychoron
Cells	16 (3.3.3)
Faces	32 {3}
Edges	24
Vertices	8
Vertex figure	(3.3.3.3)
Schläfli symbols	t₀{3,3,4} t₀{3^1,1,1} h{4,3,3}
Coxeter-Dynkin diagrams
Symmetry group	B₄, [3,3,4]
Dual	Tesseract
Properties	convex

Vertex figure: octahedron

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. Two abstract polytopes have also been discovered.

There is a lower symmetry form of the 16-cell, called a demitesseract, represented by h{4,3,3}, and can be drawn bicolored with alternating tetrahedra cells.

1 Geometry
2 Images
3 Tessellations
4 Projections
5 See also
6 References
7 External links

[edit] 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 vertices of the hexadecachoron consist of all permutations of (±1, 0, 0, 0).

The Schläfli symbol of the hexadecachoron is {3,3,4}. Its vertex figures are all regular octahedra. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. There are 4 tetrahedra and 4 triangles meeting at every edge.

[edit] Images

Stereographic projection

Four orthographic projections

4-cross-polytope graph

A 3D projection of a 16-cell performing a double rotation about two orthogonal planes.

[edit] 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 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.

[edit] Projections

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.

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.