Hexadecachoron



A hexadecachoron (also called tetracross) is the four-dimensional cross polytope and the dual of a tesseract. It is also called a 16-cell, due to its 16 tetrahedral cells. It has the schläfli symbol $$\{3, 3, 4\}$$, meaning that 4 tetrahedra meet at each edge. Its Bowers acronym is "hex".

Properties
The hexadecachoron is the 4-D demihypercube, meaning it can be constructed by taking half the vertices of a tesseract. It is also a bipyramid of an octahedron, or a square duopyramid. Eight tetrahedra join at each vertex.

Structure
When seen from one of its cells, the hexadecachoron first has one cell, then four attached to its faces. The next six share an edge with the top cell and are perpendicular to it. The remaining four side cells share only a vertex with the top cell, and are themselves joined to the final cell, in dual orientation to the top cell.

In vertex-first orientation, it is just two octahedral pyramids joined together.

Hypervolumes

 * vertex count = $$8$$
 * edge length = $$24l$$
 * surface area = $$8{ l }^{ 2 }\sqrt { 3 }$$
 * surcell volume = $$\frac { 4\sqrt { 2 } }{ 3 } { l }^{ 3 }$$
 * surteron bulk = $$\frac{1}{6}l^4$$

Subfacets

 * 8 points (0D)
 * 24 line segments (1D)
 * 32 triangles (2D)
 * 16 tetrahedra (3D)