Acronym oct (alt.: trap)
TOCID symbol O, TT, (3)Q
Name octahedron,
rectified tetrahedron,
tricross3),
tetratetrahedron,
aerochor(id),
trigonal antiprism,
larger Delone cell of face-centered cubic (fcc) lattice,
equatorial cross-section of (vertex first) 1/q-tes,
vertex figure of hex,
Gosset polytope 01,1,
lattice C3 contact polytope (span of its small roots)
 
 © ©
Circumradius 1/sqrt(2) = 0.707107
Edge radius 1/2
Inradius 1/sqrt(6) = 0.408248
Vertex figure [34] = x4o
Snub derivation
Vertex layers
LayerSymmetrySubsymmetries
 o3o4oo3o .o . o. o4o
1x3o4ox3o .
{3} first
x . o
edge first
. o4o
vertex first
2o3x .
opposite {3}
o . q. x4o
vertex figure
3 x . o
opposite edge
. o4o
opposite vertex
 o3o3oo3o .o . o. o3o
1o3x3oo3x .
{3} first
o . o
vertex first
. x3o
{3} first
2x3o .
opposite {3}
x . x
vertex figure
. o3x
opposite {3}
3 o . o
opposite vertex
 
Lace city
in approx. ASCII-art
 x o
o x 
  o  
o q o
  o  
Coordinates (1/sqrt(2), 0, 0)   & all permutations, all changes of sign
Volume sqrt(2)/3 = 0.471405
Surface 2 sqrt(3) = 3.464102
Rel. Roundness π sqrt(3)/9 = 60.459979 %
General of army (is itself convex)
Colonel of regiment (is itself locally convex – other uniform polyhedral member: thah – other edge facetings)
Dual cube
Dihedral angles
  • between {3} and {3}:   arccos(-1/3) = 109.471221°
Confer
general antiprisms:
n-ap   n/2-ap   n/d-ap  
special bipyramids:
m mNo  
variations:
xo3ox&#q   qo3oq&#x  
Grünbaumian relatives:
oct+6{4}   2oct   2oct+6{4}   2oct+8{3}   2oct+12{4}   4oct  
related Johnson solids:
squippy  
compounds:
se   sno   doso   dissit   si   gissi   addasi   dasi  
general polytopal classes:
deltahedra   regular   noble polytopes   orthoplex   partial Stott expansions   segmentohedra   fundamental lace prisms   bistratic lace towers   lace simplices   Coxeter-Elte-Gosset polytopes  
External
links
hedrondude   wikipedia   WikiChoron   mathworld   Polyedergarten   quickfur

The number of ways to color the octahedron with different colors per face is 8!/24 = 1 680. – This is because the color group is the permutation group of 8 elements and has size 8!, while the order of the pure rotational octahedral group is 24. (The reflectional octahedral group would have twice as many, i.e. 48 elements.)

When considered as a trigonal antiprism s s3s (and scaled by h = sqrt(3)), then the prismatic compound with its mirrored copy will have for hull metrically exact the hip-variant q x3x. Stated the other way round, the mere alternating faceting of that variant results in this (scaled) regular shape.


Incidence matrix according to Dynkin symbol

x3o4o

. . . | 6 |  4 | 4
------+---+----+--
x . . | 2 | 12 | 2
------+---+----+--
x3o . | 3 |  3 | 8

snubbed forms: β3o4o

x3/2o4o

.   . . | 6 |  4 | 4
--------+---+----+--
x   . . | 2 | 12 | 2
--------+---+----+--
x3/2o . | 3 |  3 | 8

snubbed forms: β3/2o4o

o4/3o3x

.   . . | 6 |  4 | 4
--------+---+----+--
.   . x | 2 | 12 | 2
--------+---+----+--
.   o3x | 3 |  3 | 8

snubbed forms: o4/3o3β

o4/3o3/2x

.   .   . | 6 |  4 | 4
----------+---+----+--
.   .   x | 2 | 12 | 2
----------+---+----+--
.   o3/2x | 3 |  3 | 8

snubbed forms: o4/3o3/2β

o3x3o

. . . | 6 |  4 | 2 2
------+---+----+----
. x . | 2 | 12 | 1 1
------+---+----+----
o3x . | 3 |  3 | 4 *
. x3o | 3 |  3 | * 4

snubbed forms: o3β3o

o3/2x3o

.   . . | 6 |  4 | 2 2
--------+---+----+----
.   x . | 2 | 12 | 1 1
--------+---+----+----
o3/2x . | 3 |  3 | 4 *
.   x3o | 3 |  3 | * 4

snubbed forms: o3/2β3o

o3/2x3/2o

.   .   . | 6 |  4 | 2 2
----------+---+----+----
.   x   . | 2 | 12 | 1 1
----------+---+----+----
o3/2x   . | 3 |  3 | 4 *
.   x3/2o | 3 |  3 | * 4

snubbed forms: o3/2β3/2o

s2s3s

demi( . . .  ) | 6 | 1 1 2 | 1 3
---------------+---+-------+----
      s2s .    | 2 | 3 * * | 0 2
      s . s2*a | 2 | * 3 * | 0 2
sefa( . s3s  ) | 2 | * * 6 | 1 1
---------------+---+-------+----
      . s3s     3 | 0 0 3 | 2 *
sefa( s2s3s  ) | 3 | 1 1 1 | * 6
or
demi( . . . )            | 6 | 2 2 | 1 3
-------------------------+---+-----+----
      s2s .  &  s . s2*a | 2 | 6 * | 0 2
sefa( . s3s )            | 2 | * 6 | 1 1
-------------------------+---+-----+----
      . s3s               3 | 0 3 | 2 *
sefa( s2s3s )            | 3 | 2 1 | * 6

starting figure: x x3x

s2s6o

demi( . . . ) | 6 | 2 2 | 1 3
--------------+---+-----+----
      s2s .   | 2 | 6 * | 0 2
sefa( . s6o ) | 2 | * 6 | 1 1
--------------+---+-----+----
      . s6o )  3 | 0 3 | 2 *
sefa( s2s6o ) | 3 | 2 1 | * 6

starting figure: x x6o

xo3ox&#x   → height = sqrt(2/3) = 0.816497
({3} || dual {3})

o.3o.    | 3 * | 2 2 0 | 1 2 1 0
.o3.o    | * 3 | 0 2 2 | 0 1 2 1
---------+-----+-------+--------
x. ..    | 2 0 | 3 * * | 1 1 0 0
oo3oo&#x | 1 1 | * 6 * | 0 1 1 0
.. .x    | 0 2 | * * 3 | 0 0 1 1
---------+-----+-------+--------
x.3o.    | 3 0 | 3 0 0 | 1 * * *
xo ..&#x | 2 1 | 1 2 0 | * 3 * *
.. ox&#x | 1 2 | 0 2 1 | * * 3 *
.o3.x    | 0 3 | 0 0 3 | * * * 1

oxo4ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo {4} || pt)

o..4o..    | 1 * * | 4 0 0 | 4 0
.o.4.o.    | * 4 * | 1 2 1 | 2 2
..o4..o    | * * 1 | 0 0 4 | 0 4
-----------+-------+-------+----
oo.4oo.&#x | 1 1 0 | 4 * * | 2 0
.x. ...    | 0 2 0 | * 4 * | 1 1
.oo4.oo&#x | 0 1 1 | * * 4 | 0 2
-----------+-------+-------+----
ox. ...&#x | 1 2 0 | 2 1 0 | 4 *
.xo ...&#x | 0 2 1 | 0 1 2 | * 4
or
o..4o..    & | 2 * | 4 0 | 4
.o.4.o.      | * 4 | 2 2 | 4
-------------+-----+-----+--
oo.4oo.&#x & | 1 1 | 8 * | 2
.x. ...      | 0 2 | * 4 | 2
-------------+-----+-----+--
ox. ...&#x & | 1 2 | 2 1 | 8

oxo oxo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo {4} || pt)

o.. o..    | 1 * * | 4 0 0 0 | 2 2 0 0
.o. .o.    | * 4 * | 1 1 1 1 | 1 1 1 1
..o ..o    | * * 1 | 0 0 0 4 | 0 0 2 2
-----------+-------+---------+--------
oo. oo.&#x | 1 1 0 | 4 * * * | 1 1 0 0
.x. ...    | 0 2 0 | * 2 * * | 1 0 1 0
... .x.    | 0 2 0 | * * 2 * | 0 1 0 1
.oo .oo&#x | 0 1 1 | * * * 4 | 0 0 1 1
-----------+-------+---------+--------
ox. ...&#x | 1 2 0 | 2 1 0 0 | 2 * * *
... ox.&#x | 1 2 0 | 2 0 1 0 | * 2 * *
.xo ...&#x | 0 2 1 | 0 1 0 2 | * * 2 *
... .xo&#x | 0 2 1 | 0 0 1 2 | * * * 2
or
o.. o..    & | 2 * | 4 0 0 | 2 2
.o. .o.      | * 4 | 2 1 1 | 2 2
-------------+-----+-------+----
oo. oo.&#x & | 1 1 | 8 * * | 1 1
.x. ...      | 0 2 | * 2 * | 2 0
... .x.      | 0 2 | * * 2 | 0 2
-------------+-----+-------+----
ox. ...&#x & | 1 2 | 2 1 0 | 4 *
... ox.&#x & | 1 2 | 2 0 1 | * 4

xox oqo&#xt   → both heights = 1/2
(line || perp pseudo q-line || line)

o.. o..     | 2 * * | 1 2 1 0 0 | 2 2 0
.o. .o.     | * 2 * | 0 2 0 2 0 | 1 2 1
..o ..o     | * * 2 | 0 0 1 2 1 | 0 2 2
------------+-------+-----------+------
x.. ...     | 2 0 0 | 1 * * * * | 2 0 0
oo. oo.&#x  | 1 1 0 | * 4 * * * | 1 1 0
o.o o.o&#x  | 1 0 1 | * * 2 * * | 0 2 0
.oo .oo&#x  | 0 1 1 | * * * 4 * | 0 1 1
..x ...     | 0 0 2 | * * * * 1 | 0 0 2
------------+-------+-----------+------
xo. ...&#x  | 2 1 0 | 1 2 0 0 0 | 2 * *
ooo ooo&#xt | 1 1 1 | 0 1 1 1 0 | * 4 *
.ox ...&#x  | 0 1 2 | 0 0 0 2 1 | * * 2
or
o.. o..     & | 4 * | 1 2 1 | 2 2
.o. .o.       | * 2 | 0 4 0 | 2 2
--------------+-----+-------+----
x.. ...     & | 2 0 | 2 * * | 2 0
oo. oo.&#x  & | 1 1 | * 8 * | 1 1
o.o o.o&#x    | 2 0 | * * 2 | 0 2
--------------+-----+-------+----
xo. ...&#x  & | 2 1 | 1 2 0 | 4 *
ooo ooo&#xt   | 2 1 | 0 2 1 | * 4

oxox&#xr   → all cyclical heights = sqrt(3)/2 = 0.866025
             in fact this lace simplex degenerates into a rhomb with diagonals:
             height(1,3) = sqrt(2) = 1.414214
             height(2,4) = 1

o...    | 1 * * * | 2 2 0 0 0 0 0 | 1 2 1 0 0 0
.o..    | * 2 * * | 1 0 1 1 1 0 0 | 1 1 0 1 1 0
..o.    | * * 1 * | 0 0 0 2 0 2 0 | 0 0 0 1 2 1
...o    | * * * 2 | 0 1 0 0 1 1 1 | 0 1 1 0 1 1
--------+---------+---------------+------------
oo..&#x | 1 1 0 0 | 2 * * * * * * | 1 1 0 0 0 0
o..o&#x | 1 0 0 1 | * 2 * * * * * | 0 1 1 0 0 0
.x..    | 0 2 0 0 | * * 1 * * * * | 1 0 0 1 0 0
.oo.&#x | 0 1 1 0 | * * * 2 * * * | 0 0 0 1 1 0
.o.o&#x | 0 1 0 1 | * * * * 2 * * | 0 1 0 0 1 0
..oo&#x | 0 0 1 1 | * * * * * 2 * | 0 0 0 0 1 1
...x    | 0 0 0 2 | * * * * * * 1 | 0 0 1 0 0 1
--------+---------+---------------+------------
ox..&#x | 1 2 0 0 | 2 0 1 0 0 0 0 | 1 * * * * *
oo.o&#x | 1 1 0 1 | 1 1 0 0 1 0 0 | * 2 * * * *
o..x&#x | 1 0 0 2 | 0 2 0 0 0 0 1 | * * 1 * * *
.xo.&#x | 0 2 1 0 | 0 0 1 2 0 0 0 | * * * 1 * *
.ooo&#x | 0 1 1 1 | 0 0 0 1 1 1 0 | * * * * 2 *
..ox&#x | 0 0 1 2 | 0 0 0 0 0 2 1 | * * * * * 1

qo ox4oo&#zx   → height = 0
(tegum sum of q-line and perp {4})
(tegum product of q-line with {4})

o. o.4o.    | 2 * | 4 0 | 4
.o .o4.o    | * 4 | 2 2 | 4
------------+-----+-----+--
oo oo4oo&#x | 1 1 | 8 * | 2
.. .x ..    | 0 2 | * 4 | 2
------------+-----+-----+--
.. ox ..&#x | 1 2 | 2 1 | 8

qo ox ox&#zx   → height = 0
(tegum sum of q-line and perp {4})
(tegum product of q-line with {4})

... 

qoo oqo ooq&#zx   → all heights = 0
(tegum sum of 3 perp q-lines)
(tegum product of 3 q-lines)

o.. o.. o..    | 2 * * | 2 2 0 | 4
.o. .o. .o.    | * 2 * | 2 0 2 | 4
..o ..o ..o    | * * 2 | 0 2 2 | 4
---------------+-------+-------+--
oo. oo. oo.&#x | 1 1 0 | 4 * * | 2
o.o o.o o.o&#x | 1 0 1 | * 4 * | 2
.oo .oo .oo&#x | 0 1 1 | * * 4 | 2
---------------+-------+-------+--
ooo ooo ooo&#x | 1 1 1 | 1 1 1 | 8

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