A tesseract, also known as a hypercube, is a four dimensional object created by attaching an edge at each vertex of a cube such that these edges are oriented orthogonally to the other edges of the cube. Tesseracts inhabit a space with four dimensions, which is not easy for many people without mathematical training to visualize. |

A tesseract, or hypercube, is a four-dimensional analogue of a cube.In a square, each vertex has two perpendicular edges incident to it, while a cube has three. A hypercube has four. So, canonical coordinates for the vertices of a tesseract centered at the origin are (±1, ±1, ±1, ±1), while the interior of the same consists of all points (x _{0}, x_{1}, x_{2}, x_{3}) with -1 < x_{i} < 1. |

In a cube, every edge is shared by 2 squares. In a tesseract, 3 squares meet at every edge. A tesseract has 16 vertices, 32 edges, 24 squares, and 8 cubes. |

A tesseract is bound by eight hyperplanes, each of which intersects it to form a cube. Two cubes, and so three squares, intersect at each edge. There are three cubes meeting at every vertex, the vertex polyhedron of which is a regular tetrahedron. Thus the tesseract is given Schläfi notation {4,3,3}. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices. The square, cube, and tesseracts are all examples of measure polytopes in their respective dimensions. |

A tesseract is defined by the set of points:{(x, y, z, h): 0 < x < 1, 0 < y < 1, 0 < z < 1, 0 < h < 1} |

External link:*http://pweb.netcom.com/~hjsmith/WireFrame4/tesseract.html has an illustration (requires Java). |

See also [1] for an illustration. /talk |

/Talk |

A **tesseract**, or **hypercube**, is a four-dimensional analogue of a cube.
In a square, each vertex has two perpendicular edges incident to it, while a cube has three.
A hypercube has four.
So, canonical coordinates for the vertices of a tesseract centered at the origin are (±1, ±1, ±1, ±1), while the interior of the same consists of all points (x_{0}, x_{1}, x_{2}, x_{3}) with -1 < x_{i} < 1.

/Talk

A tesseract is bound by eight hyperplanes, each of which intersects it to form a cube.
Two cubes, and so three squares, intersect at each edge.
There are three cubes meeting at every vertex, the vertex polyhedron of which is a regular tetrahedron.
Thus the tesseract is given Schläfi notation {4,3,3}.
All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.
The square, cube, and tesseracts are all examples of *measure polytopes* in their respective dimensions.

**External link:**

- http://pweb.netcom.com/~hjsmith/WireFrame4/tesseract.html has an illustration (requires Java).

/Talk