Acronym cube (alt: squip)
TOCID symbol C, (4)P
Name cube,
hexahedron,
3D measure-polytope3),
geochor(id),
square prism,
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
LayerSymmetrySubsymmetries
 o3o4oo3o .o . o. o4o
1o3o4xo3o .
vertex first
o . x
edge first
. o4x
{4} first
2o3q .q . x. o4x
opposite {4}
3q3o .o . x
opposite edge
 
4o3o .
opposite vertex
 
 o o4oo o .o . o. o4o
1x x4ox x .
{4} first
x . o
edge first
. x4o
{4} first
2x x .
opposite {4}
x . q. x4o
opposite {4}
3 x . o
opposite edge
 
 o o oo o .o . o. o o
1x x xx x .
{4} first
x . x
{4} first
. x x
{4} first
2x 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°
Confer
more general:
x4oPo   x4o2Po   n-p   n/d-p   4,n-dip  
variations:
q x4o   f x4o   u x4o   w x4o  
Grünbaumian relatives:
cadditradid   gicdatrid   sicdatrid  
compounds:
rhom   rah   risdoh  
unit-edged relatives:
patex cube  
axial segments:
qo3oo&#x   qo3oq&#x   oqo3ooq&#xt  
general polytopal classes:
regular   noble polytopes   hypercube   partial Stott expansions   segmentohedra   bistratic lace towers   lace simplices  
External
links
hedrondude   wikipedia   WikiChoron   mathworld   quickfur

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.)


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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