| Acronym | octet |
| Name |
alternated cubic honeycomb, tetrahedral-octahedral honeycomb, Delone complex of face-centered cubic (fcc) lattice |
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| VRML |
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| Vertex figure |
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| Coordinates | (i/sqrt(2), j/sqrt(2), k/sqrt(2)) for integers i,j,k with i+j+k even |
| Dual | radh |
| Confer |
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External links |
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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∞o))2((s4o4o)) (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∞o))2((o4s4o)) (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&##xt (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
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