Acronym cube, 4-p
TOCID symbol C, (4)P
Name cube,
hexahedron,
3D measure-polytope3),
geohedron,
square prism,
triangular antitegum,
Voronoi cell of primitive cubical lattice,
Delone cell of primitive cubical lattice

` © ©`
Circumradius sqrt(3)/2 = 0.866025
Inradius 1/2
Vertex figure [43] = q3o
Vertex layers
 Layer Symmetry Subsymmetries o3o4o o3o . o . o . o4o 1 o3o4x o3o .vertex first o . xedge first . o4x{4} first 2 o3q . q . x . o4xopposite {4} 3 q3o . o . xopposite edge 4 o3o .opposite vertex o o4o o o . o . o . o4o 1 x x4o x x .{4} first x . oedge first . x4o{4} first 2 x x .opposite {4} x . q . x4oopposite {4} 3 x . oopposite edge o o o o o . o . o . o o 1 x x x x x .{4} first x . x{4} first . x x{4} first 2 x x .opposite {4} x . xopposite {4} . x xopposite {4}
Lace city
in approx. ASCII-art
```x x
x x
```
```o q o
o q o
```
Coordinates (1/2, 1/2, 1/2)   & all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex – no other uniform polyhedral members)
Dual oct
Confer
general prisms:
n-p   n/d-p
variations:
q x4o   f x4o   u x4o   w x4o
Grünbaumian relatives:
cadditradid   gicdatrid   sicdatrid
compounds:
rhom   rah   risdoh
general polytopal classes:
regular   hypercube
External
links

Incidence matrix according to Dynkin symbol

```o3o4x

. . . | 8 |  3 | 3
------+---+----+--
. . x | 2 | 12 | 2
------+---+----+--
. o4x | 4 |  4 | 6
```

```o3/2o4x

.   . . | 8 |  3 | 3
--------+---+----+--
.   . x | 2 | 12 | 2
--------+---+----+--
.   o4x | 4 |  4 | 6
```

```x4/3o3o

.   . . | 8 |  3 | 3
--------+---+----+--
x   . . | 2 | 12 | 2
--------+---+----+--
x4/3o . | 4 |  4 | 6
```

```x4/3o3/2o

.   .   . | 8 |  3 | 3
----------+---+----+--
x   .   . | 2 | 12 | 2
----------+---+----+--
x4/3o   . | 4 |  4 | 6
```

```x x4o

. . . | 8 | 1 2 | 2 1
------+---+-----+----
x . . | 2 | 4 * | 2 0
. x . | 2 | * 8 | 1 1
------+---+-----+----
x x . | 4 | 2 2 | 4 *
. x4o | 4 | 0 4 | * 2
```

```x x x

. . . | 8 | 1 1 1 | 1 1 1
------+---+-------+------
x . . | 2 | 4 * * | 1 1 0
. x . | 2 | * 4 * | 1 0 1
. . x | 2 | * * 4 | 0 1 1
------+---+-------+------
x x . | 4 | 2 2 0 | 2 * *
x . x | 4 | 2 0 2 | * 2 *
. x x | 4 | 0 2 2 | * * 2
```

```xx4oo&#x   → height = 1
({4} || {4})

o.4o.    | 4 * | 2 1 0 | 1 2 0
.o4.o    | * 4 | 0 1 2 | 0 2 1
---------+-----+-------+------
x. ..    | 2 0 | 4 * * | 1 1 0
oo4oo&#x | 1 1 | * 4 * | 0 2 0
.x ..    | 0 2 | * * 4 | 0 1 1
---------+-----+-------+------
x.4o.    | 4 0 | 4 0 0 | 1 * *
xx ..&#x | 2 2 | 1 2 1 | * 4 *
.x4.o    | 0 4 | 0 0 4 | * * 1
```

```xx xx&#x   → height = 1
({4} || {4})

o. o.    | 4 * | 1 1 1 0 0 | 1 1 1 0
.o .o    | * 4 | 0 0 1 1 1 | 0 1 1 1
---------+-----+-----------+--------
x. ..    | 2 0 | 2 * * * * | 1 1 0 0
.. x.    | 2 0 | * 2 * * * | 1 0 1 0
oo oo&#x | 1 1 | * * 4 * * | 0 1 1 0
.x ..    | 0 2 | * * * 2 * | 0 1 0 1
.. .x    | 0 2 | * * * * 2 | 0 0 1 1
---------+-----+-----------+--------
x. x.    | 4 0 | 2 2 0 0 0 | 1 * * *
xx ..&#x | 2 2 | 1 0 2 1 0 | * 2 * *
.. xx&#x | 2 2 | 0 1 2 0 1 | * * 2 *
.x .x    | 0 4 | 0 0 0 2 2 | * * * 1
```

```oqoo3ooqo&#xt   → all heights = 1/sqrt(3) = 0.577350
(pt || pseudo q-{3} || pseudo dual q-{3} || pt)

o...3o...     | 1 * * * | 3 0 0 | 3 0
.o..3.o..     | * 3 * * | 1 2 0 | 2 1
..o.3..o.     | * * 3 * | 0 2 1 | 1 2
...o3...o     | * * * 1 | 0 0 3 | 0 3
--------------+---------+-------+----
oo..3oo..&#x  | 1 1 0 0 | 3 * * | 2 0
.oo.3.oo.&#x  | 0 1 1 0 | * 6 * | 1 1
..oo3..oo&#x  | 0 0 1 1 | * * 3 | 0 2
--------------+---------+-------+----
oqo. ....&#xt | 1 2 1 0 | 2 2 0 | 3 *
.... .oqo&#xt | 0 1 2 1 | 0 2 2 | * 3

or
o...3o...      & | 2 * | 3 0 | 3
.o..3.o..      & | * 6 | 1 2 | 3
-----------------+-----+-----+--
oo..3oo..&#x   & | 1 1 | 6 * | 2
.oo.3.oo.&#x     | 0 2 | * 6 | 2
-----------------+-----+-----+--
oqo. ....&#xt  & | 1 3 | 2 2 | 6
```

```xxx oqo&#xt   → both heights = 1/sqrt(2) = 0.707107
(line || pseudo (x,q)-{4} || line)

o.. o..     | 2 * * | 1 2 0 0 0 | 2 1 0
.o. .o.     | * 4 * | 0 1 1 1 0 | 1 1 1
..o ..o     | * * 2 | 0 0 0 2 1 | 0 1 2
------------+-------+-----------+------
x.. ...     | 2 0 0 | 1 * * * * | 2 0 0
oo. oo.&#x  | 1 1 0 | * 4 * * * | 1 1 0
.x. ...     | 0 2 0 | * * 2 * * | 1 0 1
.oo .oo&#x  | 0 1 1 | * * * 4 * | 0 1 1
..x ...     | 0 0 2 | * * * * 1 | 0 0 2
------------+-------+-----------+------
xx. ...&#x  | 2 2 0 | 1 2 1 0 0 | 2 * *
... oqo&#xt | 1 2 1 | 0 2 0 2 0 | * 2 *
.xx ...&#x  | 0 2 2 | 0 0 1 2 1 | * * 2

or
o.. o..      & | 4 * | 1 2 0 | 2 1
.o. .o.        | * 4 | 0 2 1 | 2 1
---------------+-----+-------+----
x.. ...      & | 2 0 | 2 * * | 2 0
oo. oo.&#x   & | 1 1 | * 8 * | 1 1
.x. ...        | 0 2 | * * 2 | 2 0
---------------+-----+-------+----
xx. ...&#x   & | 2 2 | 1 2 1 | 4 *
... oqo&#xt    | 2 2 | 0 4 0 | * 2
```

```oqooqoo&#xt   → height(1,2) = height(5,6) = 1/2
height(2,3) = height(4,5) = (sqrt(2)-1)/2 = 0.207107
height(3,4) = (2-sqrt(2))/2 = 0.292893
(pt || pseudo q-line || pt || pt || pseudo q-line || pt)

o.....     | 1 * * * * * | 2 1 0 0 0 0 0 | 1 2 0 0
.o....     | * 2 * * * * | 1 0 1 1 0 0 0 | 1 1 1 0
..o...     | * * 1 * * * | 0 1 0 0 2 0 0 | 0 2 0 1
...o..     | * * * 1 * * | 0 0 2 0 0 1 0 | 1 0 2 0
....o.     | * * * * 2 * | 0 0 0 1 1 0 1 | 0 1 1 1
.....o     | * * * * * 1 | 0 0 0 0 0 1 2 | 0 0 2 1
-----------+-------------+---------------+--------
oo....&#x  | 1 1 0 0 0 0 | 2 * * * * * * | 1 1 0 0
o.o...&#x  | 1 0 1 0 0 0 | * 1 * * * * * | 0 2 0 0
.o.o..&#x  | 0 1 0 1 0 0 | * * 2 * * * * | 1 0 1 0
.o..o.&#x  | 0 1 0 0 1 0 | * * * 2 * * * | 0 1 1 0
..o.o.&#x  | 0 0 1 0 1 0 | * * * * 2 * * | 0 1 0 1
...o.o&#x  | 0 0 0 1 0 1 | * * * * * 1 * | 0 0 2 0
....oo&#x  | 0 0 0 0 1 1 | * * * * * * 2 | 0 0 1 1
-----------+-------------+---------------+--------
oq.o..&#xt | 1 2 0 1 0 0 | 2 0 2 0 0 0 0 | 1 * * *
ooo.o.&#xt | 1 1 1 0 1 0 | 1 1 0 1 1 0 0 | * 2 * *
.o.ooo&#xt | 0 1 0 1 1 1 | 0 0 1 1 0 1 1 | * * 2 *
..o.qo&#xt | 0 0 1 0 2 1 | 0 0 0 0 2 0 2 | * * * 1

or
o.....      & | 2 * * | 2 1 0 0 | 1 2
.o....      & | * 4 * | 1 0 1 1 | 1 2
..o...      & | * * 2 | 0 1 2 0 | 1 2
--------------+-------+---------+----
oo....&#x   & | 1 1 0 | 4 * * * | 1 1
o.o...&#x   & | 1 0 1 | * 2 * * | 0 2
.o.o..&#x   & | 0 1 1 | * * 4 * | 1 1
.o..o.&#x     | 0 2 0 | * * * 2 | 0 2
--------------+-------+---------+----
oq.o..&#xt  & | 1 2 1 | 2 0 2 0 | 2 *
ooo.o.&#xt  & | 1 2 1 | 1 1 1 1 | * 4
```

 © 2004-2013 top of page