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Acronym	oct (alt.: trap, tatet)
TOCID symbol	O,TT, (3)Q
Name	octahedron, rectifiedtetrahedron, tricross (β3), tetratetrahedron, aerochor(id), trigonalantiprism, snubbed triangular dihedron, larger Delone cell of face-centered cubic (fcc)lattice, equatorial cross-section of (vertex first) 1/q-tes, vertex figure ofhex, Gosset polytope 01,1, lattice C3 contact polytope (span of its small roots), equatorial cross-section of vertex-firsthex
	©
VRML	⭳©
Circumradius	1/sqrt(2) = 0.707107
Edge radius	1/2
Inradius	1/sqrt(6) = 0.408248
Vertex figure	[34] = x4o
Snub derivation / VRML	⭳
Vertex layers	Layer Symmetry Subsymmetries   o3o4o o3o . o . o . o4o 1 x3o4o x3o . {3} first x . o edge first . o4o vertex first 2 o3x . opposite {3} o . q . x4o vertex figure 3   x . o opposite edge . o4o opposite vertex   o3o3o o3o . o . o . o3o 1 o3x3o o3x . {3} first o . o vertex first . x3o {3} first 2 x3o . opposite {3} x . x vertex figure . o3x opposite {3} 3   o . o opposite vertex
Lace city in approx. ASCII-art	© x oo x
	© o o q o o
Lace hyper city in approx. ASCII-art	© o o o o 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– otheredge facetings)
Dual	cube
Dihedral angles	between {3} and {3}:   arccos(-1/3) = 109.471221°
Face vector	6, 12, 8
Confer	more general: xPo3o...o3o4o   general antiprisms: n-ap  n/2-ap  n/d-ap   special bipyramids: m mNo   variations: qo3oq&#x  xo3ox&#q  xo3ox&#h  ho3oh&#q  squit   Grünbaumian relatives: oct+6{4}  2oct  2oct+6{4}  2oct+8{3}  2oct+12{4}  4oct   uniform relative: thah   relatedJohnson solids: squippy   relatedscaliform: bobipyr   compounds: se  sno  daso  dissit  si  gissi  addasi  dasi  oct+4gyro3py   ambification: co   ambification pre-image: tet   complex polytopes: Shephard's generalized oct   general polytopal classes: Wythoffian polyhedra  Catalan polyhedra  deltahedra  regular  noble polytopes  orthoplex  partial Stott expansions  segmentohedra  fundamental lace prisms  bistratic lace towers  lace simplices  Coxeter-Elte-Gosset polytopes  Hanner polytopes   analogs: rectified simplex rSn  mid-rectified simplex mrSn  regular orthoplex On  birectified hypercube brCn
External links

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 antiprisms s3s (and scaled byh = sqrt(3)), then the prismaticcompound with its mirrored copy will have for hull metrically exact thehip-variantq x3x. Stated the other way round, the merealternating faceting of thatvariant results in this (scaled) regular shape.

The acronymoct is being used for the full symmetrical (octahedral) variant,trap refers to the trigonal antiprismatic subsymmetry,andtatet refers to the tetratetrahedron, i.e. its tetrahedral subsymmetry.

Being the dual ofcube and considering that one's coordinates, it is apparent that this solid is nothing but a ball wrt.the norm |x|+|y|+|z|.

Somehow off-topic there are some neet number relations between the oct and thetet:

• face count ratio: F_oct / F_tet = 8/4 = 2
• face type: n_oct = n_tet = 3 (triangles each)
• radius ratio: (circumradius_oct / inradius_oct)² = circumradius_tet/inradius_tet = 3
• volume ratio: V_oct / V_tet = 2

Incidence matrix according toDynkin symbol

x3o4o. . . | 6 |  4 | 4------+---+----+--x . . | 2 | 12 | 2------+---+----+--x3o . | 3 |  3 | 8snubbed forms:β3o4o

x3/2o4o.   . . | 6 |  4 | 4--------+---+----+--x   . . | 2 | 12 | 2--------+---+----+--x3/2o . | 3 |  3 | 8snubbed forms:β3/2o4o

o4/3o3x.   . . | 6 |  4 | 4--------+---+----+--.   . x | 2 | 12 | 2--------+---+----+--.   o3x | 3 |  3 | 8snubbed forms:o4/3o3β

o4/3o3/2x.   .   . | 6 |  4 | 4----------+---+----+--.   .   x | 2 | 12 | 2----------+---+----+--.   o3/2x | 3 |  3 | 8snubbed forms:o4/3o3/2β

o3x3o. . . | 6 |  4 | 2 2------+---+----+----. x . | 2 | 12 | 1 1------+---+----+----o3x . | 3 |  3 | 4 *. x3o | 3 |  3 | * 4snubbed forms:o3β3o

o3/2x3o.   . . | 6 |  4 | 2 2--------+---+----+----.   x . | 2 | 12 | 1 1--------+---+----+----o3/2x . | 3 |  3 | 4 *.   x3o | 3 |  3 | * 4snubbed forms:o3/2β3o

o3/2x3/2o.   .   . | 6 |  4 | 2 2----------+---+----+----.   x   . | 2 | 12 | 1 1----------+---+----+----o3/2x   . | 3 |  3 | 4 *.   x3/2o | 3 |  3 | * 4snubbed forms:o3/2β3/2o

s2s3sdemi( . . .  ) |6 | 1 1 2 | 1 3---------------+---+-------+----      s2s .    | 2 | 3 * * | 0 2      s . s2*a | 2 | * 3 * | 0 2sefa( . s3s  ) | 2 | * * 6 | 1 1---------------+---+-------+----      . s3s♦ 3 | 0 0 3 | 2 *sefa( s2s3s  ) | 3 | 1 1 1 | * 6

ordemi( . . . )            |6 | 2 2 | 1 3-------------------------+---+-----+----      s2s .  &  s . s2*a | 2 | 6 * | 0 2sefa( . s3s )            | 2 | * 6 | 1 1-------------------------+---+-----+----      . s3s♦ 3 | 0 3 | 2 *sefa( s2s3s )            | 3 | 2 1 | * 6starting figure:x x3x

s2s6odemi( . . . ) |6 | 2 2 | 1 3--------------+---+-----+----      s2s .   | 2 | 6 * | 0 2sefa( . s6o ) | 2 | * 6 | 1 1--------------+---+-----+----      . s6o♦ 3 | 0 3 | 2 *sefa( s2s6o ) | 3 | 2 1 | * 6starting 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 0oo3oo&#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

oro..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

oro.. 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 0oo. oo.&#x  | 1 1 0 | * 4 * * * | 1 1 0o.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

oro.. o..     & | 4 * | 1 2 1 | 2 2.o. .o.       | * 2 | 0 4 0 | 2 2--------------+-----+-------+----x.. ...     & | 2 0 |2 * * | 2 0oo. oo.&#x  & | 1 1 | * 8 * | 1 1o.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) = 1o...    | 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 0o..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})o. o. o.    | 2 * | 4 0 0 | 2 2.o .o .o    | * 4 | 2 1 1 | 2 2------------+-----+-------+----oo 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

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 * * | 2o.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

oooooo&#xr   → all consecutive pairwise heights = all alternating pairwise heights = 1               Note: these lengths show that this cycle isnot flat, rather it is wobbling up and down!o.....     & | 6 | 2 2 | 3 1-------------+---+-----+----oo....&#x  & | 2 | 6 * | 2 0o.o...&#x  & | 2 | * 6 | 1 1-------------+---+-----+----ooo...&#x  & | 3 | 2 1 | 6 *o.o.o.&#x  & | 3 | 0 3 | * 2

oxxo&#xt   → height(1,2) = height(3,4) = 1/sqrt(12) = 0.288675             height(2,3) = 1/sqrt(3) = 0.577350             Note: these lengths show that this tower isnot flat, rather it has additional leporello folds!o...     & |2 * | 2 2 0 0 | 1 1 2.o..     & | * 4 | 1 1 1 1 | 1 1 2-----------+-----+---------+------oo..&#x  & | 1 1 | 4 * * * | 1 0 1o.o.&#x  & | 1 1 | * 4 * * | 0 1 1.x..     & | 0 2 | * * 2 * | 1 1 0.oo.&#x    | 0 2 | * * * 2 | 0 0 2-----------+-----+---------+------ox..&#x  & | 1 2 | 2 0 1 0 | 2 * *o.x.&#x  & | 1 2 | 0 2 1 0 | * 2 *ooo.&#x  & | 1 2 | 1 1 0 1 | * * 4