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Acronym	toe (alt.: gratet)
TOCID symbol	tO, tTT
Name	truncatedoctahedron, omnitruncatedtetrahedron, great rhombitetratetrahedron, Voronoi cell of body-centered cubic (bcc)lattice, Kelvin's tetrakaidecahedron, Waterman polyhedron number 10 wrt. face-centered cubiclattice A3 centered at a lattice point, permutohedron of 4 elements, equatorial cross-section ofoct-firstthex
	©
VRML	⭳©
Circumradius	sqrt(5/2) = 1.581139
Inradius wrt. {4}	sqrt(2) = 1.414214
Inradius wrt. {6}	sqrt(3/2) = 1.224745
Vertex figure	[4,62] = qo&#h
Snub derivation / VRML	⭳
Vertex layers	Layer Symmetry Subsymmetries   o3o4o o3o . o . o . o4o 1 x3x4o x3x . {6} first x . o edge first . x4o {4} first 2 u3x . u . q . u4o 3a x3u . w . o . x4q 3b x . Q 4a x3x . opposite {6} w . o . u4o 4b x . Q 5   u . q . x4o opposite {4} 6 x . o     o3o3o o3o . o . o . o3o 1 x3x3x x3x . {6} first x . x {4} first . x3x {6} first 2 x3u . u . u . u3x 3a u3x . x . w . x3u 3b w . x 4 x3x . opposite {6} u . u . x3x opposite {6} 5   x . x
Lace city in approx. ASCII-art	o q o o Q o (Q=2q)q Q Q qo Q o o q o
	x u u x w x x w x u u x
Coordinates	(2, 1, 0)/sqrt(2)   & all permutations, all changes of sign
Volume	8 sqrt(2) = 11.313708
Surface	6+12 sqrt(3) = 26.784610
General of army	(is itself convex)
Colonel of regiment	(is itself locally convex – no other uniform polyhedral members)
Dual	tekah
Dihedral angles	between {4} and {6}:   arccos[-1/sqrt(3)] = 125.264390° between {6} and {6}:   arccos(-1/3) = 109.471221°
Face vector	24, 36, 14
Confer	variations: a3b3c  x3x3u  a3b4c  x3f4o  x3u4o  x3w4o  u3x4o  (-x)3x4o   facetings: tithah  pabditoe   decompositions: octatoe   ambification: retoe   general polytopal classes: Wythoffian polyhedra  lace simplices  partial Stott expansions   analogs: omnitruncated simplex otSn  truncated orthoplex tOn  bitruncated hypercube btCn
External links

⭳©

Note that toe can be thought of as the externalblend of 1oct + 8tricues + 6squippies,cf. the rightSteward toroid K3 \ 4Q3(S3).Thisdecomposition is also described as the degeneratesegmentochoronxx3ox4oo&#xt.

The second picture shows how the volume of toe is intimely related to half the volume of thecube: each eighth is its vertex-first half.Moreover it displays the interrelation between the primitve cubic and the body-centered cubiclattice, resp. theirVoronoi honeycombs each: i.e.chon andbatch.

In 1887 LordKelvin conjectured that this tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. But in 1994 he finally got kind of disproven, cf. Weaire, D. and Phelan, R. "A Counter-Example to Kelvin's Conjecture on Minimal Surfaces." Philos. Mag. Let. 69, 107-110, 1994.They presented acounter-example of a space-filling geometry with even smaller surface to volume ratio, thereby however using non-flat bounding manifolds.

Incidence matrix according toDynkin symbol

x3x4o. . . | 24 |  1  2 | 2 1------+----+-------+----x . . |  2 | 12  * | 2 0. x . |  2 |  * 24 | 1 1------+----+-------+----x3x . |  6 |  3  3 | 8 *. x4o |  4 |  0  4 | * 6snubbed forms:β3x4o,x3β4o,s3s4o (or as mere facetingqQo oqQ Qoq&#zh),β3β4o

x3x4/3o. .   . | 24 |  1  2 | 2 1--------+----+-------+----x .   . |  2 | 12  * | 2 0. x   . |  2 |  * 24 | 1 1--------+----+-------+----x3x   . |  6 |  3  3 | 8 *. x4/3o |  4 |  0  4 | * 6snubbed forms:s3s4/3o

x3x3x. . . | 24 |  1  1  1 | 1 1 1------+----+----------+------x . . |  2 | 12  *  * | 1 1 0. x . |  2 |  * 12  * | 1 0 1. . x |  2 |  *  * 12 | 0 1 1------+----+----------+------x3x . |  6 |  3  3  0 | 4 * *x . x |  4 |  2  0  2 | * 6 *. x3x |  6 |  0  3  3 | * * 4snubbed forms:β3x3x,x3β3x,β3β3x,β3x3β,s3s3s (or as mere facetingqQo oqQ Qoq&#zh),β3β3β

s4x3xdemi( . . . ) | 24 |  1  1  1 | 1 1 1--------------+----+----------+------demi( . x . ) |  2 | 12  *  * | 1 1 0demi( . . x ) |  2 |  * 12  * | 0 1 1sefa( s4x . ) |  2 |  *  * 12 | 1 0 1--------------+----+----------+------      s4x .♦  4 |  2  0  2 | 6 * *demi( . x3x ) |  6 |  3  3  0 | *4 *sefa( s4x3x ) |  6 |  0  3  3 | * * 4starting figure:x4x3x

xuxux4ooqoo&#xt   → all heights = 1/sqrt(2) = 0.707107({4}|| pseudo u-{4}|| pseudo (x,q)-{8}|| pseudo u-{4}|| {4})o....4o....     | 4 * * * * | 2 1 0 0 0 0 0 | 1 2 0 0 0.o...4.o...     | * 4 * * * | 0 1 2 0 0 0 0 | 0 2 1 0 0..o..4..o..     | * * 8 * * | 0 0 1 1 1 0 0 | 0 1 1 1 0...o.4...o.     | * * * 4 * | 0 0 0 0 2 1 0 | 0 0 1 2 0....o4....o     | * * * * 4 | 0 0 0 0 0 1 2 | 0 0 0 2 1----------------+-----------+---------------+----------x.... .....     | 2 0 0 0 0 | 4 * * * * * * | 1 1 0 0 0oo...4oo...&#x  | 1 1 0 0 0 | * 4 * * * * * | 0 2 0 0 0.oo..4.oo..&#x  | 0 1 1 0 0 | * * 8 * * * * | 0 1 1 0 0..x.. .....     | 0 0 2 0 0 | * * * 4 * * * | 0 1 0 1 0..oo.4..oo.&#x  | 0 0 1 1 0 | * * * * 8 * * | 0 0 1 1 0...oo4...oo&#x  | 0 0 0 1 1 | * * * * * 4 * | 0 0 0 2 0....x .....     | 0 0 0 0 2 | * * * * * * 4 | 0 0 0 1 1----------------+-----------+---------------+----------x....4o....     | 4 0 0 0 0 | 4 0 0 0 0 0 0 |1 * * * *xux.. .....&#xt | 2 2 2 0 0 | 1 2 2 1 0 0 0 | * 4 * * *..... .oqo.&#xt | 0 1 2 1 0 | 0 0 2 0 2 0 0 | * * 4 * *..xux .....&#xt | 0 0 2 2 2 | 0 0 0 1 2 2 1 | * * * 4 *....x4....o     | 0 0 0 0 4 | 0 0 0 0 0 0 4 | * * * *1

oro....4o....      & | 8 * * | 2 1  0 0 | 1 2 0.o...4.o...      & | * 8 * | 0 1  2 0 | 0 2 1..o..4..o..        | * * 8 | 0 0  2 1 | 0 2 1-------------------+-------+----------+------x.... .....      & | 2 0 0 | 8 *  * * | 1 1 0oo...4oo...&#x   & | 1 1 0 | * 8  * * | 0 2 0.oo..4.oo..&#x   & | 0 1 1 | * * 16 * | 0 1 1..x.. .....        | 0 0 2 | * *  * 4 | 0 2 0-------------------+-------+----------+------x....4o....      & | 4 0 0 | 4 0  0 0 |2 * *xux.. .....&#xt  & | 2 2 2 | 1 2  2 1 | * 8 *..... .oqo.&#xt    | 0 2 2 | 0 0  4 0 | * * 4

xxux3xuxx&#xt   → all heights = sqrt(2/3) = 0.816497({6}|| pseudo (x,u)-{6}|| pseudo (u,x)-{6}|| {6})o...3o...     | 6 * * * | 1 1 1 0 0 0 0 0 0 | 1 1 1 0 0 0.o..3.o..     | * 6 * * | 0 0 1 1 1 0 0 0 0 | 0 1 1 1 0 0..o.3..o.     | * * 6 * | 0 0 0 0 1 1 1 0 0 | 0 0 1 1 1 0...o3...o     | * * * 6 | 0 0 0 0 0 0 1 1 1 | 0 0 0 1 1 1--------------+---------+-------------------+------------x... ....     | 2 0 0 0 | 3 * * * * * * * * | 1 1 0 0 0 0.... x...     | 2 0 0 0 | * 3 * * * * * * * | 1 0 1 0 0 0oo..3oo..&#x  | 1 1 0 0 | * * 6 * * * * * * | 0 1 1 0 0 0.x.. ....     | 0 2 0 0 | * * * 3 * * * * * | 0 1 0 1 0 0.oo.3.oo.&#x  | 0 1 1 0 | * * * * 6 * * * * | 0 0 1 1 0 0.... ..x.     | 0 0 2 0 | * * * * * 3 * * * | 0 0 1 0 1 0..oo3..oo&#x  | 0 0 1 1 | * * * * * * 6 * * | 0 0 0 1 1 0...x ....     | 0 0 0 2 | * * * * * * * 3 * | 0 0 0 1 0 1.... ...x     | 0 0 0 2 | * * * * * * * * 3 | 0 0 0 0 1 1--------------+---------+-------------------+------------x...3x...     | 6 0 0 0 | 3 3 0 0 0 0 0 0 0 |1 * * * * *xx.. ....&#x  | 2 2 0 0 | 1 0 2 1 0 0 0 0 0 | * 3 * * * *.... xux.&#xt | 2 2 2 0 | 0 1 2 0 2 1 0 0 0 | * * 3 * * *.xux ....&#xt | 0 2 2 2 | 0 0 0 1 2 0 2 1 0 | * * * 3 * *.... ..xx&#x  | 0 0 2 2 | 0 0 0 0 0 1 2 0 1 | * * * * 3 *...x3...x     | 0 0 0 6 | 0 0 0 0 0 0 0 3 3 | * * * * *1

oro...3o...      & | 12  * | 1 1  1 0 0 | 1 1 1.o..3.o..      & |  * 12 | 0 0  1 1 1 | 0 1 2-----------------+-------+------------+------x... ....      & |  2  0 | 6 *  * * * | 1 1 0.... x...      & |  2  0 | * 6  * * * | 1 0 1oo..3oo..&#x   & |  1  1 | * * 12 * * | 0 1 1.x.. ....      & |  0  2 | * *  * 6 * | 0 1 1.oo.3.oo.&#x     |  0  2 | * *  * * 6 | 0 0 2-----------------+-------+------------+------x...3x...      & |  6  0 | 3 3  0 0 0 |2 * *xx.. ....&#x   & |  2  2 | 1 0  2 1 0 | * 6 *.... xux.&#xt  & |  2  4 | 0 1  2 1 2 | * * 6

oqQ qoo4xux&#zxt   → all existing heights = 0, Q = 2q = 2.828427o.. o..4o..     | 8 * * | 1  2 0 0 | 1 2 0.o. .o.4.o.     | * 8 * | 0  2 1 0 | 1 2 0..o ..o4..o     | * * 8 | 0  0 1 2 | 0 2 1----------------+-------+----------+------... ... x..     | 2 0 0 | 4  * * * | 0 2 0oo. oo.4oo.&#x  | 1 1 0 | * 16 * * | 1 1 0.oo .oo4.oo&#x  | 0 1 1 | *  * 8 * | 0 2 0... ... ..x     | 0 0 2 | *  * * 8 | 0 1 1----------------+-------+----------+------oq. qo. ...&#zx | 2 2 0 | 0  4 0 0 | 4 * *... ... xux&#xt | 2 2 2 | 1  2 2 1 | * 8 *... ..o4..x     | 0 0 4 | 0  0 0 4 | * * 2