# Extensive Definition

In geometry, the tesseract, also
called an 8-cell or regular octachoron, is the four-dimensional
analog of the cube, which
is in turn the three dimensional analog of the square.
The tesseract is to the cube as the cube is to the square;
or, more formally, the tesseract can be described as a regular
convex 4-polytope whose boundary consists of eight cubical cells.

A generalization of the cube to dimensions
greater than three is called a “hypercube”, “n-cube” or
“measure polytope”. The tesseract is the four-dimensional hypercube
or 4-cube.

According to the Oxford English
Dictionary, the word tesseract was coined and first used in
1888 by Charles
Howard Hinton in his book A
New Era of Thought, from the Greek “”
(“four rays”), referring to the four lines from each vertex to
other vertices. Some people have called the same figure a
“tetracube”, and also simply a "hypercube" (although a hypercube
can be of any dimension).

## Geometry

The tesseract can be constructed in a number of
different ways. As a regular
polytope constructed by three cubes folded together around every
edge, it has Schläfli
symbol . Constructed as a 4D hyperprism made of two
parallel cubes, it can be named as a composite Schläfli symbol x.
As a duoprism, a
Cartesian
product of two squares,
it can be named by a composite Schläfli symbol x.

Since each vertex of a tesseract is adjacent to
four edges, the vertex
figure of the tesseract is a regular tetrahedron. The dual
polytope of the tesseract is called the hexadecachoron, or
16-cell, with Schläfli symbol .

The standard tesseract in Euclidean
4-space is given as the convex hull
of the points (±1, ±1, ±1, ±1). That is, it consists of the points:

- \.

A tesseract is bounded by eight hyperplanes (xi = ±1). Each
pair of non-parallel hyperplanes intersects to form 24 square faces
in a tesseract. Three cubes and three squares intersect at each
edge. There are four cubes, six squares, and four edges meeting at
every vertex. All in all, it consists of 8 cubes, 24 squares, 32
edges, and 16 vertices.

### Projections to 2 dimensions

The construction of a hypercube can be imagined the following way:- 1-dimensional: Two points A and B can be connected to a line, giving a new line AB.
- 2-dimensional: Two parallel lines AB and CD can be connected to become a square, with the corners marked as ABCD.
- 3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
- 4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.

This structure is not easily imagined but it is
possible to project tesseracts into three- or two-dimensional
spaces. Furthermore, projections on the 2D-plane become more
instructive by rearranging the positions of the projected vertices.
In this fashion, one can obtain pictures that no longer reflect the
spatial relationships within the tesseract, but which illustrate
the connection structure of the vertices, such as in the following
examples:

A tesseract is in principle obtained by combining
two cubes. The scheme is similar to the construction of a cube from
two squares: juxtapose two copies of the lower dimensional cube and
connect the corresponding vertices. Each edge of a tesseract is of
the same length. A multitude of cubes that are nicely
interconnected. The vertices of the tesseract with respect to the
distance along the edges, with respect to the bottom point. This
view is of interest when using tesseracts as the basis for a
network
topology to link multiple processors in parallel
computing: the distance between two nodes is at most 4 and
there are many different paths to allow weight balancing.

Tesseracts are also bipartite
graphs, just as a path, square, cube and tree are.

### Projections to 3 dimensions

The cell-first parallel projection of the tesseract into 3-dimensional space has a cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining 6 cells are projected onto the 6 square faces of the cube.The face-first parallel projection of the
tesseract into 3-dimensional space has a cuboidal envelope. Two
pairs of cells project to the upper and lower halves of this
envelope, and the 4 remaining cells project to the side
faces.

The edge-first parallel projection of the
tesseract into 3-dimensional space has an envelope in the shape of
a hexagonal prism. Six cells project onto rhombic prisms, which are
laid out in the hexagonal prism in a way analogous to how the faces
of the 3D cube project onto 6 rhombs in a hexagonal envelope under
vertex-first projection. The two remaining cells project onto the
prism bases.

The vertex-first parallel projection of the
tesseract into 3-dimensional space has a rhombic
dodecahedral envelope. There are exactly two ways of
decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving a
total of 8 possible parallelepipeds. The images of the tesseract's
cells under this projection are precisely these 8 parallelepipeds.
This projection is also the one with maximal volume.

### Unfolding the tesseract

The tesseract can be unfolded into eight cubes,
just as the cube can be unfolded into six squares. An unfolding of
a polytope is called a net.
There are 261 distinct nets of the tesseract. The unfoldings of the
tesseract can be counted by mapping the nets to paired trees (a
tree
together with a perfect matching
in its complement).