| Acronym | octet |
| Name |
alternated cubic honeycomb, tetrahedral-octahedral honeycomb, Delone complex of face-centered cubic (fcc) lattice |
| |
| VRML |
|
| 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 |
|
|
External links |
|
When each oct would be considered an exterior blend of 2 squippies (using parallel intersection planes), this very honeycomb becomes 10Y4-8T-0.
This honeycomb can be considered as the inifinite blend (or stack) of a single monostratic slab thereof, which is squatap. Likewise it can be considered as the inifinite blend (or stack) of an other single monostratic slab thereof, which is tratap.
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.
As finite modwrap of this honeycomb occurs the tomotope, when restricting the below N to the value 4 instead; B. Manson and E. Schulte further described a different modwrap for N equals 8 too. However, both replace therein the octs by their hemiversion, i.e. by ellocts.
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
xo3xo3ox3ox3*a&#zx (N → ∞) → height = 0 o.3o.3o.3o.3*a & | 4N ♦ 3 3 6 | 3 3 9 9 | 1 1 6 3 3 ---------------------+----+-----------+---------------+------------- x. .. .. .. & | 2 | 6N * * | 2 0 2 0 | 1 0 1 2 0 .. x. .. .. & | 2 | * 6N * | 0 2 0 2 | 0 1 1 0 2 oo3oo3oo3oo3*a&#x | 2 | * * 12N | 0 0 2 2 | 0 0 2 1 1 ---------------------+----+-----------+---------------+------------- x. .. .. o.3*a & | 3 | 3 0 0 | 4N * * * | 1 0 0 1 0 .. x.3o. .. & | 3 | 0 3 0 | * 4N * * | 0 1 0 0 1 xo .. .. .. &#x & | 3 | 1 0 2 | * * 12N * | 0 0 1 1 0 .. xo .. .. &#x & | 3 | 0 1 2 | * * * 12N | 0 0 1 0 1 ---------------------+----+-----------+---------------+------------- x. .. o.3o.3*a & ♦ 4 | 6 0 0 | 4 0 0 0 | N * * * * .. x.3o.3o. & ♦ 4 | 0 6 0 | 0 4 0 0 | * N * * * xo .. ox .. &#x & ♦ 4 | 1 1 4 | 0 0 2 2 | * * 6N * * xo .. .. ox3*a&#x ♦ 6 | 6 0 6 | 2 0 6 0 | * * * 2N * .. xo3ox .. &#x ♦ 6 | 0 6 6 | 0 2 0 6 | * * * * 2N
or o.3o.3o.3o.3*a & | 2N ♦ 6 6 | 6 18 | 2 6 6 ---------------------+----+-------+--------+-------- x. .. .. .. & | 2 | 6N * | 2 2 | 1 1 2 oo3oo3oo3oo3*a&#x | 2 | * 6N | 0 4 | 0 2 2 ---------------------+----+-------+--------+-------- x. .. .. o.3*a & | 3 | 3 0 | 4N * | 1 0 1 xo .. .. .. &#x & | 3 | 1 2 | * 12N | 0 1 1 ---------------------+----+-------+--------+-------- x. .. o.3o.3*a & ♦ 4 | 6 0 | 4 0 | N * * xo .. ox .. &#x & ♦ 4 | 2 4 | 0 4 | * 3N * xo .. .. ox3*a&#x & ♦ 6 | 6 6 | 2 6 | * * 2N
: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
:xo:4:oo:4:ox:&##x (N → ∞) → all heights = 1/sqrt(2) = 0.707107 o. 4 o. 4 o. | N * ♦ 4 4 4 0 | 8 4 8 4 | 4 4 4 2 .o 4 .o 4 .o | * N ♦ 0 4 4 4 | 4 8 4 8 | 2 4 4 4 -------------------+-----+-------------+-------------+---------- x. .. .. | 2 0 | 2N * * * | 2 0 2 0 | 2 1 1 0 oo 4 oo 4 oo &#x | 1 1 | * 4N * * | 2 2 0 0 | 1 2 0 1 :oo:4:oo:4:oo:&#x | 1 1 | * * 4N * | 0 0 2 2 | 1 0 2 1 .. .. .x | 0 2 | * * * 2N | 0 2 0 2 | 0 1 1 2 -------------------+-----+-------------+-------------+---------- xo .. .. &#x | 2 1 | 1 2 0 0 | 4N * * * | 1 1 0 0 .. .. ox &#x | 1 2 | 0 2 0 1 | * 4N * * | 0 1 0 1 :xo: .. .. &#x | 2 1 | 1 0 2 0 | * * 4N * | 1 0 1 0 .. .. :ox:&#x | 1 2 | 0 0 2 1 | * * * 4N | 0 0 1 1 -------------------+-----+-------------+-------------+---------- :xo:4:oo: .. &#xt ♦ 4 2 | 4 4 4 0 | 4 0 4 0 | N * * * xo .. ox &#x ♦ 2 2 | 1 4 0 1 | 2 2 0 0 | * 2N * * :xo: .. :ox:&#x ♦ 2 2 | 1 0 4 1 | 0 0 2 2 | * * 2N * .. :oo:4:ox:&#xt ♦ 2 4 | 0 4 4 4 | 0 4 0 4 | * * * N
4-colored( :xo:4:oo:4:ox:&##x ) (N → ∞) → all heights = 1/sqrt(2) = 0.707107
hemi_a( o. 4 o. 4 o. ) | N * * * ♦ 4 2 2 0 0 2 2 0 0 0 | 4 4 4 0 4 4 4 0 | 2 2 4 4 2 0
hemi_b( o. 4 o. 4 o. ) | * N * * ♦ 4 0 0 2 2 0 0 2 2 0 | 4 4 0 4 4 4 0 4 | 2 2 4 4 0 2
hemi_a( .o 4 .o 4 .o ) | * * N * ♦ 0 2 0 2 0 2 0 2 0 4 | 4 0 4 4 4 0 4 4 | 2 0 4 4 2 2
hemi_b( .o 4 .o 4 .o ) | * * * N ♦ 0 0 2 0 2 0 2 0 2 4 | 0 4 4 4 0 4 4 4 | 0 2 4 4 2 2
----------------------------+---------+-------------------------------+-------------------------+--------------
x. .. .. | 1 1 0 0 | 4N * * * * * * * * * | 1 1 0 0 1 1 0 0 | 1 1 1 1 0 0
qu._aa( oo 4 oo 4 oo &#x ) | 1 0 1 0 | * 2N * * * * * * * * | 2 0 2 0 0 0 0 0 | 1 0 2 0 1 0
qu._ab( oo 4 oo 4 oo &#x ) | 1 0 0 1 | * * 2N * * * * * * * | 0 2 2 0 0 0 0 0 | 0 1 2 0 1 0
qu._ba( oo 4 oo 4 oo &#x ) | 0 1 1 0 | * * * 2N * * * * * * | 2 0 0 2 0 0 0 0 | 1 0 2 0 0 1
qu._bb( oo 4 oo 4 oo &#x ) | 0 1 0 1 | * * * * 2N * * * * * | 0 2 0 2 0 0 0 0 | 0 1 2 0 0 1
qu._aa(:oo:4:oo:4:oo:&#x ) | 1 0 1 0 | * * * * * 2N * * * * | 0 0 0 0 2 0 2 0 | 1 0 0 2 1 0
qu._ab(:oo:4:oo:4:oo:&#x ) | 1 0 0 1 | * * * * * * 2N * * * | 0 0 0 0 0 2 2 0 | 0 1 0 2 1 0
qu._ba(:oo:4:oo:4:oo:&#x ) | 0 1 1 0 | * * * * * * * 2N * * | 0 0 0 0 2 0 0 2 | 1 0 0 2 0 1
qu._bb(:oo:4:oo:4:oo:&#x ) | 0 1 0 1 | * * * * * * * * 2N * | 0 0 0 0 0 2 0 2 | 0 1 0 2 0 1
.. .. .x | 0 0 1 1 | * * * * * * * * * 4N | 0 0 1 1 0 0 1 1 | 0 0 1 1 1 1
----------------------------+---------+-------------------------------+-------------------------+--------------
hemi_a( xo .. .. &#x ) | 1 1 1 0 | 1 1 0 1 0 0 0 0 0 0 | 4N * * * * * * * | 1 0 1 0 0 0
hemi_b( xo .. .. &#x ) | 1 1 0 1 | 1 0 1 0 1 0 0 0 0 0 | * 4N * * * * * * | 0 1 1 0 0 0
hemi_a( .. .. ox &#x ) | 1 0 1 1 | 0 1 1 0 0 0 0 0 0 1 | * * 4N * * * * * | 0 0 1 0 1 0
hemi_b( .. .. ox &#x ) | 0 1 1 1 | 0 0 0 1 1 0 0 0 0 1 | * * * 4N * * * * | 0 0 1 0 0 1
hemi_a(:xo: .. .. &#x ) | 1 1 1 0 | 1 0 0 0 0 1 0 1 0 0 | * * * * 4N * * * | 1 0 0 1 0 0
hemi_b(:xo: .. .. &#x ) | 1 1 0 1 | 1 0 0 0 0 0 1 0 1 0 | * * * * * 4N * * | 0 1 0 1 0 0
hemi_a( .. .. :ox:&#x ) | 1 0 1 1 | 0 0 0 0 0 1 1 0 0 1 | * * * * * * 4N * | 0 0 0 1 1 0
hemi_b( .. .. :ox:&#x ) | 0 1 1 1 | 0 0 0 0 0 0 0 1 1 1 | * * * * * * * 4N | 0 0 0 1 0 1
----------------------------+---------+-------------------------------+-------------------------+--------------
hemi_a(:xo:4:oo: .. &#xt ) ♦ 2 2 2 0 | 4 2 0 2 0 2 0 2 0 0 | 4 0 0 0 4 0 0 0 | N * * * * *
hemi_b(:xo:4:oo: .. &#xt ) ♦ 2 2 0 2 | 4 0 2 0 2 0 2 0 2 0 | 0 4 0 0 0 4 0 0 | * N * * * *
xo .. ox &#x ♦ 1 1 1 1 | 1 1 1 1 1 0 0 0 0 1 | 1 1 1 1 0 0 0 0 | * * 4N * * *
:xo: .. :ox:&#x ♦ 1 1 1 1 | 1 0 0 0 0 1 1 1 1 1 | 0 0 0 0 1 1 1 1 | * * * 4N * *
hemi_a( .. :oo:4:ox:&#xt ) ♦ 2 0 2 2 | 0 2 2 0 0 2 2 0 0 4 | 0 0 4 0 0 0 4 0 | * * * * N *
hemi_b( .. :oo:4:ox:&#xt ) ♦ 0 2 2 2 | 0 0 0 2 2 0 0 2 2 4 | 0 0 0 4 0 0 0 4 | * * * * * N
© 2004-2025 | top of page |