cube

Acronym cube (alt: squip)

TOCID symbol C, (4)P

Name cube,
hexahedron,
3D measure-polytope (γ₃),
geochor(id),
square prism,
cantellated square dihedron,
triangular antitegum,
Voronoi cell of primitive cubical lattice,
Delone cell of primitive cubical lattice,
terminally chamfered tetrahedron,
surtegmated tetrahedron

|,>,O device line prism prism = |||

 © ©

Circumradius sqrt(3)/2 = 0.866025

Edge radius 1/sqrt(2) = 0.707107

Inradius 1/2

Vertex figure [4³] = q3o

Snub derivation

  
      x2s4x                         x2s4s                         s2s4x

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

Volume 1

Surface 6

Rel. Roundness π/6 = 52.359878 %

General of army (is itself convex)

Colonel of regiment (is itself locally convex – no other uniform polyhedral members)

Dual oct

Dihedral angles

between {4} and {4}: 90°

Face vector 8, 12, 6

Confer

more general:: x4oPo x4o2Po n-p n/d-p 4,n-dip
variations:: q x4o f x4o u x4o w x4o x q4o recta trazip
Grünbaumian relatives:: cadditradid gicdatrid sicdatrid
compounds:: rhom rah risdoh
unit-edged relatives:: patex cube
axial segments:: qo3oo&#x qo3oq&#x oqo3ooq&#xt
ambification:: co
complex polytopes:: Shephard's generalized cube
general polytopal classes:: Wythoffian polyhedra Catalan polyhedra regular noble polytopes hypercube partial Stott expansions segmentohedra bistratic lace towers lace simplices Hanner polytopes
analogs:: regular hypercube C_n birectified orthoplex brO_n

External
links

This polyhedron can be obtained as the convex hull of the 2 tet compound (so).

The number of ways to color the cube with different colors per face is 6!/24 = 30. – This is because the color group is the permutation group of 6 elements and has size 6!, while the order of the pure rotational octahedral group is 24. (The reflectional octahedral group would have twice as many, i.e. 48 elements.)

Being the dual of oct and considering that one's coordinates, it is apparent that this solid is nothing but a ball wrt. the norm max(|x|, |y|, |z|).

Incidence matrix according to Dynkin symbol

o3o4x

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

snubbed forms: o3o4s, o3o4β

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

snubbed forms: s2s4o, β2β4o

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

snubbed forms: s2s2s, β2β2β

x2s8o

demi( . . . ) | 8 | 1 2 | 1 2
--------------+---+-----+----
demi( x . . ) | 2 | 4 * | 0 2
sefa( . s8o ) | 2 | * 8 | 1 1
--------------+---+-----+----
      . s8o   | 4 | 0 4 | 2 *
sefa( x2s8o ) | 4 | 2 2 | * 4

starting figure: x x8o

x2s4x

demi( . . . ) | 8 | 1 1 1 | 1 1 1
--------------+---+-------+------
demi( x . . ) | 2 | 4 * * | 1 1 0
demi( . . x ) | 2 | * 4 * | 1 0 1
sefa( . s4x ) | 2 | * * 4 | 0 1 1
--------------+---+-------+------
demi( x . x ) | 4 | 2 2 0 | 2 * *
      . s4x   | 4 | 2 0 2 | * 2 *
sefa( x2s4x ) | 4 | 0 2 2 | * * 2

starting figure: x x4x

x2s4s

demi( . . . ) | 8 | 1 2 | 1 2
--------------+---+-----+----
demi( x . . ) | 2 | 4 * | 0 2
sefa( . s4s ) | 2 | * 8 | 1 1
--------------+---+-----+----
      . s4s   | 4 | 0 4 | 2 *
sefa( x2s4s ) | 4 | 2 2 | * 4

starting figure: x x4x

s2s4x

demi( . . . ) | 8 | 1 1 1 | 1 2
--------------+---+-------+----
demi( . . x ) | 2 | 4 * * | 1 1
      s2s .   | 2 | * 4 * | 0 2
sefa( . s4x ) | 2 | * * 4 | 1 1
--------------+---+-------+----
      . s4x   | 4 | 2 0 2 | 2 *
sefa( s2s4x ) | 4 | 1 2 1 | * 4

starting figure: x x4x

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

oqooqo&#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

qo3oo3oq&#zx   → height = 0
(tegum sum of 2 dual q-tets)

o.3o.3o.     | 4 * |  3 | 3
.o3.o3.o     | * 4 |  3 | 3
-------------+-----+----+--
oo3oo3oo&#x  | 1 1 | 12 | 2
-------------+-----+----+--
qo .. oq&#zx | 2 2 |  4 | 6

or
o.3o.3o.   & | 8 |  3 | 3
-------------+---+----+--
oo3oo3oo&#x  | 2 | 12 | 2
-------------+---+----+--
qo .. oq&#zx | 4 |  4 | 6

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