cube

Acronym cube, 4-p

TOCID symbol C, (4)P

Name cube,
hexahedron,
3D measure-polytope (γ₃),
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 . x` edge first	`. o4x` {4} first
2		`o3q .`	`q . x`	`. o4x` opposite {4}
3		`q3o .`	`o . x` opposite edge
4		`o3o .` opposite vertex
	`o o4o`	`o o .`	`o . o`	`. o4o`
1	`x x4o`	`x x .` {4} first	`x . o` edge first	`. x4o` {4} first
2		`x x .` opposite {4}	`x . q`	`. x4o` opposite {4}
3			`x . o` opposite 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 x`	`x x .` opposite {4}	`x . x` opposite {4}	`. x x` opposite {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

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