Acronym tut
TOCID symbol tT
Name truncated tetrahedron,
cantic cube

` © ©`
wrt. {3}
5/sqrt(24) = 1.020621
wrt. {6}
sqrt(3/8) = 0.612372
Vertex figure [3,6,6] = xo&#h
Snub derivation
Vertex layers
 Layer Symmetry Subsymmetries o3o3o o3o . o . o . o3o 1 x3x3o x3x .{6} first x . oedge first . x3o{3} first 2 u3o . u . x . u3o 3 x3o .opposite {3} x . u . x3xopposite{6} 4 o . xopposite edge
Lace city
in approx. ASCII-art
```   o     o

x           x

u     u

x
```
Coordinates (3/sqrt(8), 1/sqrt(8), 1/sqrt(8))   all permutations, even changes of sign
Volume 23 sqrt(2)/12 = 2.710576
Surface 7 sqrt(3) = 12.124356
General of army (is itself convex)
Colonel of regiment (is itself locally convex – no other uniform polyhedral members)
Dihedral angles
• between {3} and {6}:   arccos(-1/3) = 109.471221°
• between {6} and {6}:   arccos(1/3) = 70.528779°
Dual tikit
Confer
Grünbaumian relatives:
2tut
variations:
a3b3c   x3q3o   x3v3o   x3u3o   x3w3o   v3x3o   q3x3o   u3x3o   w3x3o   (-x)3x3o
compounds:
tisso   taki   te
blends:
tutut
general polytopal classes:
partial Stott expansions   bistratic lace towers   lace simplices
analogs:
truncated simplex tSn   truncated demihypercube tDn   maximal expanded demihypercube eDn
External

Note that tut can be thought of as the external blend of 1 tet + 4 octs + 4 hippies. This decomposition is described as the degenerate segmentochoron ox3ox3xo&#xt.

Incidence matrix according to Dynkin symbol

```x3x3o

. . . | 12 | 1  2 | 2 1
------+----+------+----
x . . |  2 | 6  * | 2 0
. x . |  2 | * 12 | 1 1
------+----+------+----
x3x . |  6 | 3  3 | 4 *
. x3o |  3 | 0  3 | * 4

snubbed forms: β3x3o, x3β3o, β3β3o
```

```o3/2x3x

.   . . | 12 |  2 1 | 1 2
--------+----+------+----
.   x . |  2 | 12 * | 1 1
.   . x |  2 |  * 6 | 0 2
--------+----+------+----
o3/2x . |  3 | 3  0 | 4 *
.   x3x |  6 | 3  3 | * 4

snubbed forms: o3/2x3β, o3/2β3x, o3/2β3β
```

```s4o3x

demi( . . . ) | 12 | 1  2 | 1 2
--------------+----+------+----
s4o . ) ♦  2 | 6  * | 0 2
demi( . . x ) |  2 | * 12 | 1 1
--------------+----+------+----
demi( . o3x ) |  3 | 0  3 | 4 *
sefa( s4o3x ) |  6 | 3  3 | * 4

starting figure: x4o3x
```

```xux3oox&#xt   → both heights = sqrt(2/3) = 0.816497
({3} || pseudo u-{3} || {6})

o..3o..     | 3 * * | 2 1 0 0 0 | 1 2 0 0
.o.3.o.     | * 3 * | 0 1 2 0 0 | 0 2 1 0
..o3..o     | * * 6 | 0 0 1 1 1 | 0 1 1 1
------------+-------+-----------+--------
x.. ...     | 2 0 0 | 3 * * * * | 1 1 0 0
oo.3oo.&#x  | 1 1 0 | * 3 * * * | 0 2 0 0
.oo3.oo&#x  | 0 1 1 | * * 6 * * | 0 1 1 0
..x ...     | 0 0 2 | * * * 3 * | 0 1 0 1
... ..x     | 0 0 2 | * * * * 3 | 0 0 1 1
------------+-------+-----------+--------
x..3o..     | 3 0 0 | 3 0 0 0 0 | 1 * * *
xux ...&#xt | 2 2 2 | 1 2 2 1 0 | * 3 * *
... .ox&#x  | 0 1 2 | 0 0 2 0 1 | * * 3 *
..x3..x     | 0 0 6 | 0 0 0 3 3 | * * * 1
```

```xuxo oxux&#xt   → all heights = 1/sqrt(2) = 0.707107
(line || pseudo (u,x)-{4} || pseudo (x,u)-{4} || ortho line)

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