Acronym | oct (alt.: trap, tatet) | |||||||||||||||||||||||||||||||||||||||||
TOCID symbol | O, TT, (3)Q | |||||||||||||||||||||||||||||||||||||||||
Name |
octahedron, rectified tetrahedron, tricross (β3), 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) | |||||||||||||||||||||||||||||||||||||||||
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Circumradius | 1/sqrt(2) = 0.707107 | |||||||||||||||||||||||||||||||||||||||||
Edge radius | 1/2 | |||||||||||||||||||||||||||||||||||||||||
Inradius | 1/sqrt(6) = 0.408248 | |||||||||||||||||||||||||||||||||||||||||
Vertex figure | [34] = x4o | |||||||||||||||||||||||||||||||||||||||||
Snub derivation |
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Vertex layers |
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Lace city in approx. ASCII-art |
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Lace hyper city in approx. ASCII-art |
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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 |
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Confer |
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External links |
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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
oooooo&#xr → all consecutive pairwise heights = all alternating pairwise heights = 1 Note: these lengths show that this cycle is not flat, rather it is wobbling up and down! o..... & | 6 | 2 2 | 3 1 -------------+---+-----+---- oo....&#x & | 2 | 6 * | 2 0 o.o...&#x & | 2 | * 6 | 1 1 -------------+---+-----+---- ooo...&#x & | 3 | 2 1 | 6 * o.o.o.&#x & | 3 | 0 3 | * 2
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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