Acronym octet
Name alternated cubic honeycomb,
tetrahedral-octahedral honeycomb,
Delone complex of face-centered cubic (fcc) lattice
 
 © ©    ©    ©
Vertex figure
 ©
Coordinates (i/sqrt(2), j/sqrt(2), k/sqrt(2))           for integers i,j,k with i+j+k even
Dual radh
Confer
more general:
s4oPo4s  
related tesselations:
Delone complex of primitive cubic lattice   Voronoi complex of primitive cubic lattice   Voronoi complex of bcc lattice   gytoh   etoh  
related CRF honeycombs:
10Y4-8T-0  
general polytopal classes:
partial Stott expansions  
External
links
wikipedia
  ©

When each oct would be considered an exterior blend of 2 squippies (using parallel intersection planes), this very honeycomb becomes 10Y4-8T-0.

In order to understand the relation between the octet and fcc, take a first oct completely inscribed into the cube (its face centers). Next attach 8 tets onto its faces. Those then would connect to the corners of the cube. Their outer edges then describe an x-cross onto every face of the cube. The remainder of the cubical domain is filled along either edge by exactly one quarter of a further oct. As the primitive cubical honycomb (chon) has 4 cubes around each edge, this completes then those octs again. Thus indeed, the vertex set of octet is the fcc lattice.


Incidence matrix according to Dynkin symbol

x3o3o *b4o   (N → ∞)

. . .    . | N  12 | 24 |  8 6
-----------+---+----+----+-----
x . .    . | 2 | 6N |  4 |  2 2
-----------+---+----+----+-----
x3o .    . | 3 |  3 | 8N |  1 1
-----------+---+----+----+-----
x3o3o    .  4 |  6 |  4 | 2N *
x3o . *b4o  6 | 12 |  8 |  * N

x3o3o *b4/3o   (N → ∞)

. . .      . | N  12 | 24 |  8 6
-------------+---+----+----+-----
x . .      . | 2 | 6N |  4 |  2 2
-------------+---+----+----+-----
x3o .      . | 3 |  3 | 8N |  1 1
-------------+---+----+----+-----
x3o3o      .  4 |  6 |  4 | 2N *
x3o . *b4/3o  6 | 12 |  8 |  * N

x3o3o3o3*a   (N → ∞)

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

x3o3o3/2o3/2*a   (N → ∞)

. . .   .      | N  12 | 12 12 | 4 6 4
---------------+---+----+-------+------
x . .   .      | 2 | 6N |  2  2 | 1 2 1
---------------+---+----+-------+------
x3o .   .      | 3 |  3 | 4N  * | 1 1 0
x . .   o3/2*a | 3 |  3 |  * 4N | 0 1 1
---------------+---+----+-------+------
x3o3o   .       4 |  6 |  4  0 | N * *
x3o .   o3/2*a  6 | 12 |  4  4 | * N *
x . o3/2o3/2*a  4 |  6 |  0  4 | * * N

x3o3/2o3/2o3*a   (N → ∞)

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

x3/2o3o3o3/2*a   (N → ∞)

.   . . .      | N  12 | 12 12 | 4 6 4
---------------+---+----+-------+------
x   . . .      | 2 | 6N |  2  2 | 1 2 1
---------------+---+----+-------+------
x3/2o . .      | 3 |  3 | 4N  * | 1 1 0
x   . . o3/2*a | 3 |  3 |  * 4N | 0 1 1
---------------+---+----+-------+------
x3/2o3o .       4 |  6 |  4  0 | N * *
x3/2o . o3/2*a  6 | 12 |  4  4 | * N *
x   . o3o3/2*a  4 |  6 |  0  4 | * * N

s4o3o4o   (N → ∞)

demi( . . . . ) | N  12 | 24 |  8 6
----------------+---+----+----+-----
      s4o . .   | 2 | 6N |  4 |  2 2
----------------+---+----+----+-----
sefa( s4o3o . ) | 3 |  3 | 8N |  1 1
----------------+---+----+----+-----
      s4o3o .    4 |  6 |  4 | 2N *
sefa( s4o3o4o )  6 | 12 |  8 |  * N

starting figure: x4o3o4o

s4o3o4s   (N → ∞)

demi( . . . . )    | 2N   6  6 |  6  18 | 2  6  6
-------------------+----+-------+--------+--------
      s4o . .    & |  2 | 6N  * |  2   2 | 1  1  2
      s . . s    & |  2 |  * 6N |  0   4 | 0  2  2
-------------------+----+-------+--------+--------
sefa( s4o3o . )  & |  3 |  3  0 | 4N   * | 1  0  1
sefa( s4o . s )  & |  3 |  1  2 |  * 12N | 0  1  1
-------------------+----+-------+--------+--------
      s4o3o .    &   4 |  6  0 |  4   0 | N  *  *
      s4o . s    &   4 |  2  4 |  0   4 | * 3N  *
sefa( s4o3o4s )      6 |  6  6 |  2   6 | *  * 2N

starting figure: x4o3o4x

o3o3o *b4s   (N → ∞)

demi( . . .    . ) | N  12 | 12 12 | 4 4 6
-------------------+---+----+-------+------
      . o . *b4s   | 2 | 6N |  2  2 | 1 1 2
-------------------+---+----+-------+------
sefa( o3o . *b4s ) | 3 |  3 | 4N  * | 1 0 1
sefa( . o3o *b4s ) | 3 |  3 |  * 4N | 0 1 1
-------------------+---+----+-------+------
      o3o . *b4s    4 |  6 |  4  0 | N * *
      . o3o *b4s    4 |  6 |  0  4 | * N *
sefa( o3o3o *b4s )  6 | 12 |  4  4 | * * N

starting figure: o3o3o *b4x

s∞o2s4o4o   (N → ∞)

demi( . . . . . ) | N   8  4 | 24 |  8 6
------------------+---+-------+----+-----
      s 2 s . .   | 2 | 4N  * |  4 |  2 2
      . . s4o .   | 2 |  * 2N |  4 |  2 2
------------------+---+-------+----+-----
sefa( s 2 s4o . ) | 3 |  2  1 | 8N |  1 1
------------------+---+-------+----+-----
      s 2 s4o .    4 |  4  2 |  4 | 2N *
sefa( s∞o2s4o4o )  6 |  8  4 |  8 |  * N

starting figure: x∞o x4o4o

s∞o2o4s4o   (N → ∞)

demi( . . . . . ) | N   8 2 2 | 12 12 | 4 4 6
------------------+---+--------+-------+------
      s 2 . s .   | 2 | 4N * * |  2  2 | 1 1 2
      . . o4s .   | 2 |  * N * |  4  0 | 2 0 2
      . . . s4o   | 2 |  * * N |  0  4 | 0 2 2
------------------+---+--------+-------+------
sefa( s 2 o4s . ) | 3 |  2 1 0 | 4N  * | 1 0 1
sefa( s 2 . s4o ) | 3 |  2 0 1 |  * 4N | 0 1 1
------------------+---+--------+-------+------
      s 2 o4s .    4 |  4 2 0 |  4  0 | N * *
      s 2 . s4o    4 |  4 0 2 |  0  4 | * N *
sefa( s∞o2o4s4o )  6 |  8 2 2 |  4  4 | * * N

starting figure: x∞o o4x4o

:xoo:3:oxo:3:oox:3*a&##x   (N → ∞)   → all heights = sqrt(2/3) = 0.816497

 o.. 3 o.. 3 o.. 3*a    | N * *   6  3  3  0  0  0 | 3 3  6  3  6  3 0 0  0  0 0 0 | 3 3 1 3 3 1 0 0 0
 .o. 3 .o. 3 .o. 3*a    | * N *   0  3  0  6  3  0 | 0 0  3  6  0  0 3 3  6  3 0 0 | 3 1 3 0 0 0 3 1 3
 ..o 3 ..o 3 ..o 3*a    | * * N   0  0  3  0  3  6 | 0 0  0  0  3  6 0 0  3  6 3 3 | 0 0 0 1 3 3 1 3 3
------------------------+-------+-------------------+-------------------------------+------------------
 x..   ...   ...        | 2 0 0 | 3N  *  *  *  *  * | 1 1  1  0  1  0 0 0  0  0 0 0 | 1 1 0 1 1 0 0 0 0
 oo. 3 oo. 3 oo. 3*a&#x | 1 1 0 |  * 3N  *  *  *  * | 0 0  2  2  0  0 0 0  0  0 0 0 | 2 1 1 0 0 0 0 0 0
:o.o:3:o.o:3:o.o:3*a&#x | 1 0 1 |  *  * 3N  *  *  * | 0 0  0  0  2  2 0 0  0  0 0 0 | 0 0 0 1 2 1 0 0 0
 ...   .x.   ...        | 0 2 0 |  *  *  * 3N  *  * | 0 0  0  1  0  0 1 1  1  0 0 0 | 1 0 1 0 0 0 1 0 1
 .oo 3 .oo 3 .oo 3*a&#x | 0 1 1 |  *  *  *  * 3N  * | 0 0  0  0  0  0 0 0  2  2 0 0 | 0 0 0 0 0 0 1 1 2
 ...   ...   ..x        | 0 0 2 |  *  *  *  *  * 3N | 0 0  0  0  0  1 0 0  0  1 1 1 | 0 0 0 0 1 1 0 1 1
------------------------+-------+-------------------+-------------------------------+------------------
 x.. 3 o..   ...        | 3 0 0 |  3  0  0  0  0  0 | N *  *  *  *  * * *  *  * * * | 1 0 0 1 0 0 0 0 0
 x..   ...   o.. 3*a    | 3 0 0 |  3  0  0  0  0  0 | * N  *  *  *  * * *  *  * * * | 0 1 0 0 1 0 0 0 0
 xo.   ...   ...    &#x | 2 1 0 |  1  2  0  0  0  0 | * * 3N  *  *  * * *  *  * * * | 1 1 0 0 0 0 0 0 0
 ...   ox.   ...    &#x | 1 2 0 |  0  2  0  1  0  0 | * *  * 3N  *  * * *  *  * * * | 1 0 1 0 0 0 0 0 0
:x.o: :...: :...:   &#x | 2 0 1 |  1  0  2  0  0  0 | * *  *  * 3N  * * *  *  * * * | 0 0 0 1 1 0 0 0 0
:...: :...: :o.x:   &#x | 1 0 2 |  0  0  2  0  0  1 | * *  *  *  * 3N * *  *  * * * | 0 0 0 0 1 1 0 0 0
 .o. 3 .x.   ...        | 0 3 0 |  0  0  0  3  0  0 | * *  *  *  *  * N *  *  * * * | 1 0 0 0 0 0 1 0 0
 ...   .x. 3 .o.        | 0 3 0 |  0  0  0  3  0  0 | * *  *  *  *  * * N  *  * * * | 0 0 1 0 0 0 0 0 1
 ...   .xo   ...    &#x | 0 2 1 |  0  0  0  1  2  0 | * *  *  *  *  * * * 3N  * * * | 0 0 0 0 0 0 1 0 1
 ...   ...   .ox    &#x | 0 1 2 |  0  0  0  0  2  1 | * *  *  *  *  * * *  * 3N * * | 0 0 0 0 0 0 0 1 1
 ..o   ...   ..x 3*a    | 0 0 3 |  0  0  0  0  0  3 | * *  *  *  *  * * *  *  * N * | 0 0 0 0 1 0 0 1 0
 ...   ..o 3 ..x 3*a    | 0 0 3 |  0  0  0  0  0  3 | * *  *  *  *  * * *  *  * * N | 0 0 0 0 0 1 0 0 1
------------------------+-------+-------------------+-------------------------------+------------------
 xo. 3 ox.   ...    &#x  3 3 0 |  3  6  0  3  0  0 | 1 0  3  3  0  0 1 0  0  0 0 0 | N * * * * * * * *
 xo.   ...   oo. 3*a&#x  3 1 0 |  3  3  0  0  0  0 | 0 1  3  0  0  0 0 0  0  0 0 0 | * N * * * * * * *
 ...   ox.   oo.    &#x  1 3 0 |  0  3  0  3  0  0 | 0 0  0  3  0  0 0 1  0  0 0 0 | * * N * * * * * *
:x.o:3:o.o: :...:   &#x  3 0 1 |  3  0  3  0  0  0 | 1 0  0  0  3  0 0 0  0  0 0 0 | * * * N * * * * *
:x.o: :...: :o.x:3*a&#x  3 0 3 |  3  0  6  0  0  3 | 0 1  0  0  3  3 0 0  0  0 1 0 | * * * * N * * * *
:...: :o.o: :o.x:   &#x  1 0 3 |  0  0  3  0  0  3 | 0 0  0  0  0  3 0 0  0  0 0 1 | * * * * * N * * *
 .oo 3 .xo   ...    &#x  0 3 1 |  0  0  0  3  3  0 | 0 0  0  0  0  0 1 0  3  0 0 0 | * * * * * * N * *
 .oo   ...   .ox 3*a&#x  0 1 3 |  0  0  0  0  3  3 | 0 0  0  0  0  0 0 0  0  3 1 0 | * * * * * * * N *
 ...   .xo 3 .ox    &#x  0 3 3 |  0  0  0  3  6  3 | 0 0  0  0  0  0 0 1  3  3 0 1 | * * * * * * * * N

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