Tesseract

A cell-on 3-D projection of a tesseract.

A **tesseract** or **octachoron** is a fourth-dimensional hypercube. Since the number of dimensions is a square number, the diagonal length of a tesseract is an integer - in this case, 2. Its Bowers acronym is "**tes**". It is one of the three regular polychora that can tile 4-dimensional space, forming the tesseractic tetracomb. Under the elemental naming scheme it is called a **geochoron**.

Tesseract Rubik's cubes can found online, but cannot be built in the 3D world.

## Hypercube Products

The tesseract can be expressed as a hypercube product, potentially with less symmetry than the uniform and regular ideal tesseract, in five different ways:

###

{
4
,
3
,
3
}

{\displaystyle \{4, 3, 3\}}

 - tesseract

As a tesseract, the hypervolumes can be expressed in terms of a single variable, the edge length l. This is the most symmetrical variant of the tesseract.

###

{
4
,
3
}
×
{
}

{\displaystyle \{4,3\} \times \{\}}

 - cube prism

As a cube prism, the hypervolumes require two lengths to express: the edge length a of the cube, and the height b of the prism.

* edge length =

8

(

3
a
b

)

{\displaystyle 8\left(3a + b\right)}
* surface area =

12
a

(

a
b

)

{\displaystyle 12a \left( a + b \right)}
* surcell volume =

2

a
2

(

a
3
b

)

{\displaystyle 2a^2 \left( a + 3b \right)}
* surteron bulk =

a
3

b

{\displaystyle a^3 b}

When a=b, this becomes the symmetrical tesseract.

###

{
4
}
×
{

}
2

{\displaystyle \{4\} \times \{\}^2}

 - square prism prism

As a square prism prism, the hypervolumes require three lengths to express: the edge length a of the square, and the seperate heights b and c of the two prisms.

* edge length =

8

(

2
a
b
c

)

{\displaystyle 8 \left( 2a + b + c \right)}
* surface area =

4

(

a
2

2
a
b
2
a
c
b
c

)

{\displaystyle 4 \left( a^2 + 2 ab + 2 ac + bc \right)}
* surcell volume =

2
a

(

a
b
a
c
2
b
c

)

{\displaystyle 2a \left( ab + ac + 2bc \right)}
* surteron bulk =

a
2

b
c

{\displaystyle a^2 b c}

When a=b xor a=c, this becomes the cubic prism. When b=c, this becomes the square duoprism. When a=b=c, this becomes the symmetrical tesseract.

###

{

}
4

{\displaystyle \{\}^4}

 - line prism prism prism

As a line prism prism prism, the hypervolumes require four lengths to express. This is the least symmetrical variant of the tesseract.

* edge length =

8

(

a
b
c
d

)

{\displaystyle 8\left( a + b + c + d \right)}
* surface area =

4

(

a
b
a
c
a
d
b
c
b
d
c
d

)

{\displaystyle 4 \left( ab + ac + ad + bc + bd + cd \right) }
* surcell volume =

2

(

a
b
c
a
b
d
a
c
d
b
c
d

)

{\displaystyle 2 \left( abc + abd + acd + bcd \right)}
* surteron bulk =

a
b
c
d

{\displaystyle abcd}

When a=b and c=d, a=c and b=d, xor a=d and b=c, this becomes the square duoprism. When a=b=c, b=c=d, a=c=d xor a=b=d, this becomes the cubic prism. When a=b, a=c, a=d, b=c, b=d xor c=d, this becomes the square prism prism. When a=b=c=d, this becomes the symmetrical tesseract.

###

{
4

}
2

{\displaystyle \{4\}^2}

 - square duoprism

As a square duoprism, the hypervolumes require two lengths to express: the seperate edge lengths a and b of the two squares.

* edge length =

16

(

a
b

)

{\displaystyle 16 \left( a + b \right)}
* surface area =

4

(

a
2

4
a
b

b
2

)

{\displaystyle 4 \left( a^2 + 4 ab + b^2 \right)}
* surcell volume =

4
a
b

(

a
b

)

{\displaystyle 4ab \left( a + b \right)}
* surteron bulk =

a
2

b
2

{\displaystyle a^2 b^2}

When a=b, this becomes the symmetrical tesseract.

## Properties

The tesseract can be exactly decomposed into eight cubic pyramids with unit side length. This is because the distance between a vertex and the center is the same as the edge length. If these pyramids are joined to the cubes of the tesseract the result is the icositetrachoron - the square pyramidal cells merge into octahedra.

## Symbols

Dynkin symbols of the tesseract include:

* x4o3o3o (regular)
* x x4o3o (cubic prism)
* x4o x4o (square duoprism)
* x x x4o (square diprism)
* x x x x (tesseractic block)
* xx4oo3oo&#xt, xx xx4oo&#xt, xx xx xx&#xt (as cube atop cube)
* oqooo3ooqoo3oooqo&#xt (vertex first)
* xxx4ooo oqo&#xt, xxx xxx oqo&#xt (square first)
* xxxx oqoo3ooqo&#xt (edge first)
* qo3oo3oq \*b3oo&#zx (sum of two demitesseracts)
* xx xx qo oq&#zx (rhombic diprism)
* xx qo3oo3oq&#zx \*prism of sum of two tetrahedra)

## Structure and Sections

### Structure

The tesseract is composed of 8 cubic cells. Two of these cubes line in parallel 3-D spaces, while the remaining six connect the faces of the cubes. Four cubes meet at each vertex.

In cube-first position, it is a sequence of identical cubes. In square-centered orientation, it is a square which expands to a square prism and back. When seen line-first it is a line that expands to a triangular prism, then turns to a hexagonal prism, and then back. Finally in corner first orientation, it goes through the entire tetrahedral truncation series, from point to tetrahedron to octahedron in the middle and then back.

### Hypervolumes

* vertex count =

16

{\displaystyle 16}
* edge length =

32
l

{\displaystyle 32l}
* surface area =

24

l
2

{\displaystyle 24l^2}
* surcell volume =

8

l
3

{\displaystyle 8l^3}
* surteron bulk =

l
4

{\displaystyle l^4}

### Subfacets

* 16 points (0D)
* 32 line segments (1D)
* 24 squares (2D)
* 8 cubes (3D)
* 1 tesseract (4D)

### Radii

* Vertex radius:

l

{\displaystyle l}
* Edge radius:

3

2

l

{\displaystyle \frac{\sqrt{3}}{2}l}
* Face radius:

2

2

l

{\displaystyle \frac{\sqrt{2}}{2}l}
* Cell radius:

1

/

2
l

{\displaystyle 1/2l}

### Vertex coordinates

The vertices of a tesseract with side 2 can be denoted on a 4D Cartesian plane by (±1,±1,±1,±1).

### Equations

The surface of a tesseract can be graphed by the equation

m
a
x
(

x
2

,

y
2

,

z
2

,

w
2

# )
1

{\displaystyle max(x^2,y^2,z^2,w^2) = 1}

### Notations

1111

{\displaystyle 1111}

### Related shapes

* Dual: Hexadecachoron
* Vertex figure: Tetrahedron, edge length

2

{\displaystyle \sqrt{2}}

## See Also

**Self-intersecting regular polychora:** fix · gohi · gahi · sishi · gaghi · gishi · gashi · gofix · gax · gogishi
**Tesseractihemioctachoron:** tho

{
2
,
3
,
3
}

{\displaystyle \{2,3,3\}}

{
3
,
3
,
3
}

{\displaystyle \{3,3,3\}}

{
4
,
3
,
3
}

{\displaystyle \{4, 3, 3\}}

{
5
,
3
,
3
}

{\displaystyle \{5,3,3\}}

{
6
,
3
,
3
}

{\displaystyle \{6,3,3\}}

{
4
,
3
,
2
}

{\displaystyle \{4,3,2\}}

{
4
,
3
,
3
}

{\displaystyle \{4, 3, 3\}}

{
4
,
3
,
4
}

{\displaystyle \{4,3,4\}}

{
4
,
3
,
5
}

{\displaystyle \{4,3,5\}}

{
4
,
3
,
6
}

{\displaystyle \{4,3,6\}}

t
0

{
4
,
3
,
3
}

{\displaystyle t\_0 \{4,3,3\}}

t
1

{
4
,
3
,
3
}

{\displaystyle t\_1 \{4,3,3\}}

t
2

{
4
,
3
,
3
}

{\displaystyle t\_2 \{4,3,3\}}

t
3

{
4
,
3
,
3
}

{\displaystyle t\_3 \{4,3,3\}}

t

0
,
1

{
4
,
3
,
3
}

{\displaystyle t\_{0,1} \{4,3,3\}}

t

1
,
2

{
4
,
3
,
3
}

{\displaystyle t\_{1,2} \{4,3,3\}}

t

2
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{2,3} \{4,3,3\}}

t

0
,
2

{
4
,
3
,
3
}

{\displaystyle t\_{0,2} \{4,3,3\}}

t

1
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{1,3} \{4,3,3\}}

t

0
,
1
,
2

{
4
,
3
,
3
}

{\displaystyle t\_{0,1,2} \{4,3,3\}}

t

1
,
2
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{1,2,3} \{4,3,3\}}

t

0
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{0,3} \{4,3,3\}}

t

0
,
2
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{0,2,3} \{4,3,3\}}

t

0
,
1
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{0,1,3} \{4,3,3\}}

t

0
,
1
,
2
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{0,1,2,3} \{4,3,3\}}

t
0

{
4
,
3
,
2
}

{\displaystyle t\_0 \{4,3,2\}}

t
1

{
4
,
3
,
2
}

{\displaystyle t\_1 \{4,3,2\}}

t
2

{
4
,
3
,
2
}

{\displaystyle t\_2 \{4,3,2\}}

t
3

{
4
,
3
,
2
}

{\displaystyle t\_3 \{4,3,2\}}

t

0
,
1

{
4
,
3
,
2
}

{\displaystyle t\_{0,1} \{4,3,2\}}

t

1
,
2

{
4
,
3
,
2
}

{\displaystyle t\_{1,2} \{4,3,2\}}

t

2
,
3

{
4
,
3
,
2
}

{\displaystyle t\_{2,3} \{4,3,2\}}

t

0
,
2

{
4
,
3
,
2
}

{\displaystyle t\_{0,2} \{4,3,2\}}

t

1
,
3

{
4
,
3
,
2
}

{\displaystyle t\_{1,3} \{4,3,2\}}

t

0
,
1
,
2

{
4
,
3
,
2
}

{\displaystyle t\_{0,1,2} \{4,3,2\}}

t

1
,
2
,
3

{
4
,
3
,
2
}

{\displaystyle t\_{1,2,3} \{4,3,2\}}

t

0
,
3

{
4
,
3
,
2
}

{\displaystyle t\_{0,3} \{4,3,2\}}

t

0
,
2
,
3

{
4
,
3
,
2
}

{\displaystyle t\_{0,2,3} \{4,3,2\}}

t

0
,
1
,
3

{
4
,
3
,
2
}

{\displaystyle t\_{0,1,3} \{4,3,2\}}

t

0
,
1
,
2
,
3

{
4
,
3
,
2
}

{\displaystyle t\_{0,1,2,3} \{4,3,2\}}

t
0

{
4
,
2
,
4
}

{\displaystyle t\_0 \{4,2,4\}}

t
1

{
4
,
2
,
4
}

{\displaystyle t\_1 \{4,2,4\}}

t
2

{
4
,
2
,
4
}

{\displaystyle t\_2 \{4,2,4\}}

t
3

{
4
,
2
,
4
}

{\displaystyle t\_3 \{4,2,4\}}

t

0
,
1

{
4
,
2
,
4
}

{\displaystyle t\_{0,1} \{4,2,4\}}

t

1
,
2

{
4
,
2
,
4
}

{\displaystyle t\_{1,2} \{4,2,4\}}

t

2
,
3

{
4
,
2
,
4
}

{\displaystyle t\_{2,3} \{4,2,4\}}

t

0
,
2

{
4
,
2
,
4
}

{\displaystyle t\_{0,2} \{4,2,4\}}

t

1
,
3

{
4
,
2
,
4
}

{\displaystyle t\_{1,3} \{4,2,4\}}

t

0
,
1
,
2

{
4
,
2
,
4
}

{\displaystyle t\_{0,1,2} \{4,2,4\}}

t

1
,
2
,
3

{
4
,
2
,
4
}

{\displaystyle t\_{1,2,3} \{4,2,4\}}

t

0
,
3

{
4
,
2
,
4
}

{\displaystyle t\_{0,3} \{4,2,4\}}

t

0
,
2
,
3

{
4
,
2
,
4
}

{\displaystyle t\_{0,2,3} \{4,2,4\}}

t

0
,
1
,
3

{
4
,
2
,
4
}

{\displaystyle t\_{0,1,3} \{4,2,4\}}

t

0
,
1
,
2
,
3

{
4
,
2
,
4
}

{\displaystyle t\_{0,1,2,3} \{4,2,4\}}

t
0

{
4
,
2
,
2
}

{\displaystyle t\_0 \{4,2,2\}}

t
1

{
4
,
2
,
2
}

{\displaystyle t\_1 \{4,2,2\}}

t
2

{
4
,
2
,
2
}

{\displaystyle t\_2 \{4,2,2\}}

t
3

{
4
,
2
,
2
}

{\displaystyle t\_3 \{4,2,2\}}

t

0
,
1

{
4
,
2
,
2
}

{\displaystyle t\_{0,1} \{4,2,2\}}

t

1
,
2

{
4
,
2
,
2
}

{\displaystyle t\_{1,2} \{4,2,2\}}

t

2
,
3

{
4
,
2
,
2
}

{\displaystyle t\_{2,3} \{4,2,2\}}

t

0
,
2

{
4
,
2
,
2
}

{\displaystyle t\_{0,2} \{4,2,2\}}

t

1
,
3

{
4
,
2
,
2
}

{\displaystyle t\_{1,3} \{4,2,2\}}

t

0
,
1
,
2

{
4
,
2
,
2
}

{\displaystyle t\_{0,1,2} \{4,2,2\}}

t

1
,
2
,
3

{
4
,
2
,
2
}

{\displaystyle t\_{1,2,3} \{4,2,2\}}

t

0
,
3

{
4
,
2
,
2
}

{\displaystyle t\_{0,3} \{4,2,2\}}

t

0
,
2
,
3

{
4
,
2
,
2
}

{\displaystyle t\_{0,2,3} \{4,2,2\}}

t

0
,
1
,
3

{
4
,
2
,
2
}

{\displaystyle t\_{0,1,3} \{4,2,2\}}

t

0
,
1
,
2
,
3

{
4
,
2
,
2
}

{\displaystyle t\_{0,1,2,3} \{4,2,2\}}

t
0

{
2
,
2
,
2
}

{\displaystyle t\_0 \{2,2,2\}}

t
1

{
2
,
2
,
2
}

{\displaystyle t\_1 \{2,2,2\}}

t
2

{
2
,
2
,
2
}

{\displaystyle t\_2 \{2,2,2\}}

t
3

{
2
,
2
,
2
}

{\displaystyle t\_3 \{2,2,2\}}

t

0
,
1

{
2
,
2
,
2
}

{\displaystyle t\_{0,1} \{2,2,2\}}

t

1
,
2

{
2
,
2
,
2
}

{\displaystyle t\_{1,2} \{2,2,2\}}

t

2
,
3

{
2
,
2
,
2
}

{\displaystyle t\_{2,3} \{2,2,2\}}

t

0
,
2

{
2
,
2
,
2
}

{\displaystyle t\_{0,2} \{2,2,2\}}

t

1
,
3

{
2
,
2
,
2
}

{\displaystyle t\_{1,3} \{2,2,2\}}

t

0
,
1
,
2

{
2
,
2
,
2
}

{\displaystyle t\_{0,1,2} \{2,2,2\}}

t

1
,
2
,
3

{
2
,
2
,
2
}

{\displaystyle t\_{1,2,3} \{2,2,2\}}

t

0
,
3

{
2
,
2
,
2
}

{\displaystyle t\_{0,3} \{2,2,2\}}

t

0
,
2
,
3

{
2
,
2
,
2
}

{\displaystyle t\_{0,2,3} \{2,2,2\}}

t

0
,
1
,
3

{
2
,
2
,
2
}

{\displaystyle t\_{0,1,3} \{2,2,2\}}

t

0
,
1
,
2
,
3

{
2
,
2
,
2
}

{\displaystyle t\_{0,1,2,3} \{2,2,2\}}