Acronym	toe (alt.: gratet)
TOCID symbol	tO, tTT
Name	truncated octahedron, omnitruncated tetrahedron, great rhombitetratetrahedron, Voronoi cell of body-centered cubic (bcc) lattice, Kelvin's tetrakaidecahedron, Waterman polyhedron number 10 wrt. face-centered cubic lattice A3 centered at a lattice point, permutohedron of 4 elements, equatorial cross-section of oct-first thex
	© © ©
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
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 q o Q o o q o
	x u u x w x x w x u u x
Coordinates	(sqrt(2), 1/sqrt(2), 0)   & 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 external blend of 1 oct + 8 tricues + 6 squippies, cf. the Steward toroid K3 \ 4Q3(S3). This decomposition is also described as the degenerate segmentochoron xx3ox4oo&#xt.

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

In 1887 Lord Kelvin 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 a counter-example of a space-filling geometry with even smaller surface to volume ratio, thereby however using non-flat bounding manifolds.

Incidence matrix according to Dynkin 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 | * 6

snubbed forms: β3x4o, x3β4o, s3s4o (or as mere faceting qQo 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 | * 6

snubbed 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 | * * 4

snubbed forms: β3x3x, x3β3x, β3β3x, β3x3β, s3s3s (or as mere faceting qQo oqQ Qoq&#zh), β3β3β

s4x3x

demi( . . . ) | 24 |  1  1  1 | 1 1 1
--------------+----+----------+------
demi( . x . ) |  2 | 12  *  * | 1 1 0
demi( . . x ) |  2 |  * 12  * | 0 1 1
sefa( 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 | * * 4

starting 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 0
oo...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

or
o....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 0
oo...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 0
oo..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

or
o...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 1
oo..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 = 2.828427

o.. 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 0
oo. 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