Tesseract

The tesseract, or tes, also sometimes called the 8-cell or octachoron, is one of the 6 convex regular polychora. It has 8 cubes as cells, joining 3 to an edge and 4 to a vertex. It is the 4-dimensional hypercube and also the square duoprism, digonal duoantitegum, digonal diswirltegum and the 8-3 gyrochoron. Together with its dual, the tesseract is the first in a series of tetrahedral and digonal antiprismatic swirlchora and the first in a series of square dihedral swirlchora.

It is one of the three regular polychora that can tile 4D space, similar to hypercubes of any other dimension.

The tesseract has the same circumradius as its edge length.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Cube (8): Hexadecachoron
 * Square (24): Icositetrachoron
 * Edge (32): Rectified tesseract

Vertex coordinates
The vertices of a tesseract of edge length 1, centered at the origin, are given by:
 * (±1/2, ±1/2, ±1/2, ±1/2).