Acronym	cytatoh (old: batatoh)
Name	cyclotruncated tetrahedral-octahedral honeycomb (but not the hyperbolic cyclotruncated octahedral-tetrahedral honeycomb cytoth), bitruncated tetrahedral-octahedral honeycomb, bitruncated alternated-cubic honeycomb, quarter cubic honeycomb
	©
VRML	⭳ ©
Vertex figure	©
Confer	ambification: recytatoh   more general: xPo3o...o3oPxQ*a
External links

 ©

This honeycomb also can be described as s4o3o4s', i.e. as sequential applications of alternated facetings, according to x4o3o4x → s4o3o4x (= x3o3o *b4x) → x3o3o *b4s' (cf. below).

By virtue of an outer symmetry this is a non-quasiregular monotoxal honeycomb, that is all edges belong to the same equivalence class.

Dissecting the tets into their facial centri-pyramids and re-adjoining those bits to the neighbouring tuts each, produces the right shown monotopal triakis truncated tetrahedron honeycomb, which in turn is nothing but the Voronoi complex of the Diamond "lattice".

Incidence matrix according to Dynkin symbol

x3x3o3o3*a   (N → ∞)

. . . .    | 4N ♦  3  3 |  6  3  3 | 3 3 1 1
-----------+----+-------+----------+--------
x . . .    |  2 | 6N  * |  2  2  0 | 1 2 1 0
. x . .    |  2 |  * 6N |  2  0  2 | 2 1 0 1
-----------+----+-------+----------+--------
x3x . .    |  6 |  3  3 | 4N  *  * | 1 1 0 0
x . . o3*a |  3 |  3  0 |  * 4N  * | 0 1 1 0
. x3o .    |  3 |  0  3 |  *  * 4N | 1 0 0 1
-----------+----+-------+----------+--------
x3x3o .    ♦ 12 |  6 12 |  4  0  4 | N * * *
x3x . o3*a ♦ 12 | 12  6 |  4  4  0 | * N * *
x . o3o3*a ♦  4 |  6  0 |  0  4  0 | * * N *
. x3o3o    ♦  4 |  0  6 |  0  0  4 | * * * N

or
. . . .       | 2N ♦  6 |  6  6 | 6 2
--------------+----+----+-------+----
x . . .     & |  2 | 6N |  2  2 | 3 1
--------------+----+----+-------+----
x3x . .       |  6 |  6 | 2N  * | 2 0
x . . o3*a  & |  3 |  3 |  * 4N | 1 1
--------------+----+----+-------+----
x3x3o .     & ♦ 12 | 18 |  4  4 | N *
x . o3o3*a  & ♦  4 |  6 |  0  4 | * N

x3x3o3/2o3/2*a   (N → ∞)

. . .   .      | 4N ♦  3  3 |  6  3  3 | 3 3 1 1
---------------+----+-------+----------+--------
x . .   .      |  2 | 6N  * |  2  2  0 | 1 2 1 0
. x .   .      |  2 |  * 6N |  2  0  2 | 2 1 0 1
---------------+----+-------+----------+--------
x3x .   .      |  6 |  3  3 | 4N  *  * | 1 1 0 0
x . .   o3/2*a |  3 |  3  0 |  * 4N  * | 0 1 1 0
. x3o   .      |  3 |  0  3 |  *  * 4N | 1 0 0 1
---------------+----+-------+----------+--------
x3x3o   .      ♦ 12 |  6 12 |  4  0  4 | N * * *
x3x .   o3/2*a ♦ 12 | 12  6 |  4  4  0 | * N * *
x . o3/2o3/2*a ♦  4 |  6  0 |  0  4  0 | * * N *
. x3o3/2o      ♦  4 |  0  6 |  0  0  4 | * * * N

x3o3o *b4s   (N → ∞)

demi( . . .    . ) | 4N ♦  3  3 |  3  6  3 | 1 3 1 3
-------------------+----+-------+----------+--------
demi( x . .    . ) |  2 | 6N  * |  2  2  0 | 1 2 0 1
      . o . *b4s   |  2 |  * 6N |  0  2  2 | 0 1 1 2
-------------------+----+-------+----------+--------
demi( x3o .    . ) |  3 |  3  0 | 4N  *  * | 1 1 0 0
sefa( x3o . *b4s ) |  6 |  3  3 |  * 4N  * | 0 1 0 1
sefa( . o3o *b4s ) |  3 |  0  3 |  *  * 4N | 0 0 1 1
-------------------+----+-------+----------+--------
demi( x3o3o    . ) ♦  4 |  6  0 |  4  0  0 | N * * *
      x3o . *b4s   ♦ 12 | 12  6 |  4  4  0 | * N * *
      . o3o *b4s   ♦  4 |  0  6 |  0  0  4 | * * N *
sefa( x3o3o *b4s ) ♦ 12 |  6 12 |  0  4  4 | * * * N

starting figure: x3o3o *b4x