16cell facts for kids
In four dimensional geometry, a 16cell, 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 mid19th century.
Conway calls it an orthoplex for orthant complex, as well as the entire class of crosspolytopes.
Geometry
The hexadecachoron is a member of the family of polytopes called the crosspolytopes, which exist in all dimensions. As such, its dual polychoron is the tesseract (the 4dimensional 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 16cell, called a demitesseract or 4demicube, a member of the demihypercube family, and represented by h{4,3,3}, and can be drawn bicolored with alternating tetrahedral cells.
Images for kids

A skew orthogonal projection inside its regular octagonal Petrie polygon, connecting all vertices except opposite ones.
Tessellations
One can tessellate 4dimensional Euclidean space by regular 16cells. 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 24cells. Together with the tesseractic honeycomb {4,3,3,4}, these are the only three regular tessellations of R^{4}. Each 16cell 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. Twentyfour 16cells meet at any given vertex in this tessellation.
Projections
The cellfirst parallel projection of the 16cell into 3space 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 (nonregular) 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 16cell, all its edges lie on the faces of the cubical envelope.
The cellfirst perspective projection of the 16cell into 3space has a triakis tetrahedral envelope. The layout of the cells within this envelope are analogous to that of the cellfirst parallel projection.
The vertexfirst parallel projection of the 16cell into 3space 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 16cell. The closest vertex of the 16cell to the viewer projects onto the center of the octahedron.
Finally the edgefirst parallel projection has a shortened octahedral envelope, and the facefirst parallel projection has a hexagonal bipyramidal envelope.
Related pages
 24cell
 Polychoron
 H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN: 0486614808.
 Olshevsky, George, Hexadecachoron at Glossary for Hyperspace.