Pretty Patterns

Contents:

NOTATION AND TERMINOLOGY

Here is Mirek's classification of two-color patterns which he introduced in 1982.

1. Ornaments

A typical example of an ornament is "cube in cube". Colors of the faces are mixed: (U->R->F)(D->L->B). Axis of each pattern goes through the corners URF and DLB. In order to desribe ornaments in a short, symbolic, form the following notation is introduced:

D is an exchange of the centers (F,U,R) (L,D,B)

V are twists of the corners FUR+ and LDB-

S is a permutation of the edges (FU,UR,RF)

S' is a permutation of the edges (LB,BD,DL)

L is permutations of the edges (FU,FR,UR) (LB,LD,BD)

O is a permutation of the corners (FUL,URB,RFD)

O' is a permutation of the corners (LBU,BDR,DLF)

H is a permutation of the edges (UL,RB,FD)

H' is a permutation of the edges (UB,RD,FL)

2. Semi-ornaments

Every face has at most two colors which are mixed: (F<->R)(B<->L)(U<->D)

D is an exchange of the centers (FR)(BL)(UD)

W are flips of the edges FR and BL.

O is an exchange of the edges FL and RB.

R is an exchange of the corners FRU and RFD

R' is an exchange of the corners BLU and LBD

K is an exchange of the corners FLU and RBD

K' is an exchange of the corners FLD and RBU

S are permutations of the edges FU, FD, RU, RD

S' are permutations of the edges BU, BD, LU, LD

D is used to shorten the notation. [D] = [WORR'KK'S(14)(23)S'(14)(23)]

The alphabetical order of letters is: D,W,O,R,K,S.

Patterns containing all of R, R', K, K', S, S' are mostly denoted as [D...]. You should use mirroring (F<->R) to get K rather then K' and S(1.. rather then S(3.. or S(2.. .

3. Cyclic patterns

Every face has at most two colors which are mixed: (F->R->B->L)(U<->D)

4. Simple patterns

Every face is a simple two-color combination of colors from opposite faces of the Cube: (U<->D)(R<->L)(B<->F)

5. Specials

Other permutations of colors, for ex. (F->L->U->B->R->D) or (F->B->L->R) (U<->D)

Letters on all faces

The U's - ornaments

The U's - semi-ornaments

Supplements of the 'U's

The H's

The J's - ornaments

Supplements of the 'J's

The I's - semi-ornaments

The K's

Supplements of the 'K's

The L's - simple patterns

The first of them has the most interesting symmetries.

UsBsR²FaD'aF'aD²Fa	16,14,12	L1	MG.1982
F²L²B²U²L²D²L²U²B²U²B²R²	24,12		p176 6 L
FaL²F'aR²UsBsR²F'aUs	18,15,12	L2
L²D²B²D²B²L²B²L²B²D²R²U²	24,12		MG.1997
F'aD²BsLsD²Ra	12,10,8	L3	MG.1982
R²D²R²F²R²F²R²F²D²F²R²s	24,12		MG.1997
F²sL²D²L²B²L²F²L²F²D²B²	24,12		MG.1997
F'aUaL²DsBsR²	12,10,8	L4	MG.1982
R²D²R²F²R²F²R²F²D²F²	20,10		MG.1997
R²U²L²B²L²B²L²B²U²F²	20,10		MG.1997
FaD²BsLsU²R'a	12,10,8	L5	MG.1982
R²D²R²F²R²F²R²F²D²B²	20,10		MG.1997
R²U²L²B²L²B²L²B²U²B²	20,10		MG.1997
FaUaR²DsBsL²	12,10,8	L6	MG.1982
F²sL²D²L²B²L²F²L²F²D²F²	24,12		MG.1997
R²D²R²F²R²F²R²F²D²B²R²s	24,12		MG.1997

{L3} + R²s = {L4}, {L3} + F²s = {L6}

{L4} + F²s = {L5}, {L4} + R²s = {L3}

{L5} + R²s = {L6}, {L5} + F²s = {L4}

{L6} + F²s = {L3}, {L6} + R²s = {L5}

FaD²L²Fa	8,6	L7	MG.1982
U²B²R²B²U²B²D²F²	16,8		MG.1997
D²F²L²B²U²B²U²B²	16,8		MG.1997
FaD²L²FaD²R²	12,8	L8	MG.1997
U²B²R²B²U²B²D²F²D²R²	20,10		MG.1997
B²L²U²L²BsD²Fa	14,9,8	L9	MG.1997
D²F²R²D²F²D²F²D²F²D²	20,10		MG.1997
FaD²L²FaU²L²	12,8	L10	MG.1997
U²B²R²B²U²B²D²F²U²L²	20,10		MG.1997
D²F²R²D²F²D²F²D²F²R²	20,10		MG.1997
DsR²sDsF'aU²R²Fa	16,12,9	L11	MG.1997
U²B²D²sR²F²D²F²D²F²	20,10,9		MG.1997
DsL²UsFsR²F'aD² = L²UsB²RaF²R'aD'a	14,11,8	L12	MG.1997
U²B²D²sR²F²D²F²D²F²D²R²	24,12,9		MG.1997
U²B²R²B²U²B²D²F²D²R²U²s	24,12,9		MG.1997
LsU²sLsF'aU²R²Fa = BsD²RsF²U²F²R'aFa	16,12,9	L13	MG.1997
B²L²B²L²B²D²R²sF²R²	20,10,9		MG.1997
DsR²UsBsR²F'aU² = R²UsF²RaF²R'aDa	14,11,8	L14	MG.1997
= FaR²F'aR²UsF²Ua = FaR²F'aUaR²UsF²	14,11,10		MG.1997
D²F²R²D²F²D²F²D²F²R²U²s	24,12,9		MG.1997

  {L7} +D²R²={L8}, {L7} +U²L²={L10}
  {L8} +R²D²={L7}, {L8} +L²U²={L9}, {L8} +U²D²={L12}, {L8} +F²B²={L8}^
  {L9} +D²R²={L10},{L9} +U²L²={L8}
  {L10}+R²D²={L9}, {L10}+L²U²={L7}, {L10}+U²D²={L14}, {L10}+F²B²={L10}^
  {L11}+D²R²={L12},{L11}+U²L²={L14}
  {L12}+R²D²={L11},{L12}+L²U²={L13},{L12}+U²D²={L8},  {L12}+F²B²={L12}^
  {L13}+D²R²={L14},{L13}+U²L²={L12}
  {L14}+R²D²={L13},{L14}+L²U²={L11},{L14}+U²D²={L10}, {L14}+F²B²={L14}^

A schema of short bridges:

Connections between L3, L4, L5 and L6 are of the type F²B².
Connections between L7,L8,L9,L10,L11,L12 and L13 are of the type F²R² and F²B².

                                                L7
            R²s                               /    \
        L3  ---  L4                      = L8 - L9 -L10 =
   F²s   |        |  F²s            U²D²    |        |   U²D²
        L6  ---  L5                      =L12 -L11 -L14 =
            R²s                              \     /
                                               L13

  = ... F²B² bridge to its own mirroring

  {L1} + U²R²sD² = {L2}
  {L1} + F²U²sB² = ru{L2}
  {L1} + R²F²sL² = dl{L2}
  {L1} + D²R²sU² = {L2}
  {L1} + B²U²sF² = ru{L2}
  {L1} + L²F²sR² = dl{L2}
  ru.ru=dl,   ru.ru.ru=identity

          L1
        /  |  \
      L2  L2  L2
       | \ | \ | \
      L2^ L2^ L2^
         \ | /
          L1°

  ° ... center of the cube symmetry (FB)(RL)(UD)

  {L3} + D²R²sU² = {L5}^  (F<->B mirroring)
  {L4} + D²R²sU² = {L6}^  (F<->B mirroring)
  {L5} + D²R²sU² = {L3}^  (F<->B mirroring)
  {L6} + D²R²sU² = {L4}^  (F<->B mirroring)
  {L3} + F²R²sB² = {L6}^  (R<->L mirroring)        L3   L4
  {L4} + F²R²sB² = {L5}^  (R<->L mirroring)         | X  |
  {L5} + F²R²sB² = {L4}^  (R<->L mirroring)        L6   L5
  {L6} + F²R²sB² = {L3}^  (R<->L mirroring)

  {L7} + L²F²sR²  = {L9}^  (F<->B mirroring)
  {L11}+ L²F²sR²  = {L13}^ (F<->B mirroring)
  {L7} + F²U²sB²  = rr {L9}         {L9} + F²U²sB²      = rr {L7}
  {L8} + F²U²sB²  = rr {L8}         {L10}+ F²U²sB²      = rr {L10}
  {L11}+ F²U²sB²  = rr {L13}        {L13}+ F²U²sB²      = rr {L11}
  {L12}+ F²U²sB²  = rr {L12}        {L14}+ F²U²sB²      = rr {L14}

  About inversions:
  {L1}' = {L1}^   any mirroring along axis FRU--BLD
  {L2}' = {L2}^ (B<->U)(F<->D)
  {L4}' = {L4}^ (F<->R)(B<->L)               {L3}'   is not 6 L's
  {L5}' = f²u{L5}                            {L6}'   is not 6 L's
  {L7}' = {L7}^ (R<->U)(L<->D)               {L8}'   is not 6 L's
  {L9}' = {L9}^ (R<->D)(L<->U)               {L10}'  is not 6 L's
  {L11}' = {L13}^ (F<->B)(R->U->L->D)        {L12}'  is not 6 L's
  {L13}' = {L11}^ (F<->B)(R->D->L->U)        {L14}'  is not 6 L's

Regular ornaments - supplements

Specials

SEMI-REGULAR PATTERNS

Semi-regular ornaments

Semi-regular ornaments - supplements

Ornaments composed from U2, U3 or U4

Semiregular semi-ornaments

Even positions

Odd positions

Cyclic patterns

Three-color regular patterns

Three-color regular ornaments

More-colored regular patterns

More-colored nonregular patterns

Flips and twists

Patterns possible by disassembly

Ornaments

Regular semi-ornaments

Semiregular semi-ornaments

Cyclic patterns

Simple patterns

Two corner little squares of opposite color

Odd simple patterns

Algorithms with (K) as the author were probably produced by a computer program based on Kociemba's 2-step algorithm. The authors of those algorithms are not known to us. Please, if you know who obtained those algorithms for the first time, let us know. Thank you.

Algorithms signed (MG.) are Mirek Goljan's results, JF. denotes Jessica Fridrich.