The Pattern of Patterns

Structural combinations from 4 IVM (fcc) to Bcc lattices

from A, B, C, D to A+B+C+D

Last update 2-24-2003

(This is a Chart of the pattern of patterns)

I. Combinations of 4 IVMs (A, B, C, D)



[Combinations of 2, tetrahedral edges]
1a. 2 Cubic nets

A+B

AB+CD 2 interpenetrating cubic nets (bcc)

C+D





1b. 4 Diamond nets


A+C = AC

B+C = BC

A+D = AD

B+D = BD

[Combinations of 3, tetrahedral faces]

2. 4 Rhombic Dodecahedral nets ABC, ABD, ACD, BCD ivm (see II-1b below)


A+B+C = ABC

A+B+D = ABD

A+C+D = ACD

B+C+D = BCD


3. Combination of (I-1a), 2 cubic nets

AB+CD = 2 interpenetrating cubic nets = bcc lattice





II. Combinations of 4 Diamond nets (AC, BC, AD, BD) [I-1b]

[Combinations of 2]
1a. Two Interpenetrating Diamond nets

AC+BD

ACBD+BCAD 4 interpenetrating diamond nets (bcc)

BC+AD


1b. 4 Rhombic Dodecahedral nets (RD nets), (see I-2)

AC+BC = D’ RD net = ABC ivm

AD+BD = C’ RD net = ABD ivm

AC+AD = B’ RD net = ACD ivm

BC+BD = A’ RD net = BCD ivm


[Combinations of 3]
2. 4 Half-filled RD nets (of 3 diamond combinations)

AC+BC+BD = D’ RD net ½ filled by BD

AD+BD+AC = C’ RD net ½ filled by AC

AC+AD+BC = B’ RD net ½ filled by BC

BC+BD+AD = A’ RD net ½ filled by AD


3. Combination of (II-1a), 2 interpenetrating diamond nets

ACBD+BCAD = 4 interpenetrating diamond nets = bcc





III. Combinations of 4 RD nets (ACBC, ADBD, ACAD, BCBD) [II-1b]


[Combinations of 2]

1a. 2 Coupler nets


ACBC+ADBD = D’+C’= odd coupler

ACBC,ADBD+ACAD,BCBD =“Siamese Couplers”

=2 interpenetrating coupler nets = RITE net = (bcc)

ACAD+BCBD = B’+A’= even coupler





1b. 4 Half-filled RD nets (of 2 RD combinations), note the filling difference between III-1b and II-2

D’+B’ = ACBC+ACAD = D’ RD net ½ filled by AD

C’+A’ = ADBD+BCBD = C’ RD net ½ filled by BC

C’+B’ = ADBD+ACAD = B’ RD net ½ filled by BD

D’+A’ = ACBC+BCBD = A’ RD net ½ filled by AC




[Combinations of 3]
2. 4 Three RD nets (degenerated? into pairs of odd and even coupler nets)

A’B’C’ = BCBD+ACAD+ADBD = 2 interpenetrating RD, B’+A’= even coupler net (ADBD is redundant)

A’B’D’ = BCBD+ACAD+ACBC = 2 interpenetrating RD, B’+A’= even coupler net

A’C’D’ = BCBD+ADBD+ACBC = 2 interpenetrating RD, D’+C’= odd coupler net

B’C’D’ = ACAD+ADBD+ACBC = 2 interpenetrating RD, D’+C’= odd coupler net





3. Combination of (III-1a), 2 coupler nets

ACBC,ADBD+ACAD,BCBD = 2 interpenetrating coupler nets = RITE net = bcc





IV. Combinations of 4 Half Filled RD nets [III-1b]


1a.

1b.

2.

3.


I. Combinations of 4 IVMs (A, B, C, D)



[Combinations of 4]
3. Combination of all 4 IVMs

A+B+C+D = 4 IVM = bcc, including all the above pattern combinations.


Notes on main patterns:

• The various patterns obtained through the possible combinations of the 4 IVMs or 4 fcc lattices are limited between the fcc lattices and the bcc lattice.
• With 4 different lattices we can have a total combination of 6 pairs. And they are differentiated into two groups, group 1a with 2 combinations and 1b with 4 combinations.
• The 2 pairs in I-1a, II-1a, III-1a, are always complementary, and their combination lead directly to the final stage the bcc lattice.
• The 4 pairs in I-1b, II-1b, III-1b, are the new 4 lattices that can combine in the same way as the previous 4 lattices.
• I do not know if this repeating pattern goes on indefinitely, it appears that at III-2 we start seeing a redundancy where it suggest an end to the 4 pairs pattern. The 4 pairs become in essence 2 pairs due to redundancy.