§13

Famous patterns — concrete faces of group elements

Group elements are abstract, but every cube state is visible. Each celebrated pattern below is a specific element of G — with its order, defining alg, cycle structure, and place in G's architecture. Visual symmetry of a pattern usually corresponds to algebraic symmetry of its group element — that is the theme of §13.

Superflip

all 12 edges flipped
order 2

U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2

Checkerboard

all 6 axes half-turned
order 2

U2 D2 F2 B2 L2 R2

4 dots

4-face centre swap
order 2

U R2 L2 U2 R2 L2 U' D R2 L2 D2 R2 L2 D'

Cube in cube

classic Escher-style visual illusion
order 4

F L F U' R U F2 L2 U' L' B D' B' L2 U

Cross pattern

a cross on every face
order 2

U F B' L2 U2 L2 F' B U2 L2 U

Anaconda

a winding band of colour
order 6

L U B' U' R L' B R' F B' D R D' F'

Six spots

each face centre swapped with opposite
order 4

U D' R L' F B' U D'

Plus minus

a 6-move classic
order 2

U2 R2 L2 U2 R2 L2

Pons Asinorum (6X)

all six faces half-turned; one of three antipode candidates
order 2

R2 L2 F2 B2 U2 D2

Six H-bars

three orthogonal H-bars on the equators
order 2

U2 B2 R2 D2 U2 R2 F2 U2

Stairs

colours staircase across the cube
order 6

F D2 B R B' L' F D' L2 F2 R F' R' F2 L' F'

Tetris

short 8-move medium-order pattern
order 4

L R F B U' D' L' R'

Order-1260

Singmaster's classic: this 5-move alg has order 1260 = lcm(3,4,5,7); repeat 1260× to return
order 1260

R U2 D' B D'

4 spots (90°)

4 face centres rotated 90° (≠ 6-spot 180°)
order 4

R F' L' U2 B' D' R B U2 L U F'

13.1 Order + cycle structure table

Translated into algebra, each pattern's order (how many applications return to e) and cycle structure (corner / edge permutation decomposition) lays bare:

pattern	order	corner cycles	edge cycles	character
Superflip	2	identity	12 edges flipped	centre Z(G)
Checkerboard	2	4 transpositions	6 transpositions	Abelian 6-tuple
4 dots	2	identity	4 transpositions	edges only
Cube in cube	4	one 8-cycle	mixed	non-Abelian
Cross	2	identity	6 transpositions	symmetric
Anaconda	6	3-cycle + twists	6-cycle	order = lcm(2,3)
Six spots	4	2 four-cycles	4 transpositions	90° type
Plus minus	2	identity	6 transpositions	6-move minimum

Note: order = lcm of the corner-cycle order and the edge-cycle order (orientations multiplied in). Anaconda's 6 = lcm(2, 3) — edges loop in 2, corners in 3.

13.2 The special status of superflip

Superflip is an order-2 element, with $c_{p} = e, c_{o} = 0$ (corners untouched) and $e_{p} = e$ (edges home), and only $e_{o} = (1, 1, \dots, 1)$ — all 12 edges flipped. It is uniquely distinguished in G by three facts:

The unique non-identity element of Z(G) — commutes with every g ∈ G (§9.4): $Z (G) = {e, superflip} ≅ Z /2$ . The most concrete reason G is not simple.
HTM distance exactly 20 — the first lower bound nailed down in Rokicki et al.'s 2010 proof that God's number = 20. Of 4.3 × 10¹⁹ states, only three require the full 20 moves: superflip, the superflip ∘ 4-spot family, and Reid's dual.
Invariant under all 48 outer cube symmetries — when projected to G/Sym, superflip is a singleton orbit. Group-theoretically the "most symmetric" non-identity state.

Z (G) = {g \in G : \forall h \in G, g h = h g} = {e, superflip} ≅ Z /2

"Superflip is, group-theoretically, the most singular position among 43 quintillion. Its uniqueness is not coincidence — it is the algebraic consequence of its geometric place in G."

— Tomas Rokicki, God's Number is 20 (2010)

13.3 Algebra of generating simple patterns

Some patterns have a clean algebraic origin. Example: checkerboard = U² D² F² B² L² R². These six half-turn generators mutually commute (same-axis half-turns commute, different-axis pairs act on disjoint cubies), so the subgroup they generate is Abelian:

⟨ U^{2}, D^{2}, F^{2}, B^{2}, L^{2}, R^{2} ⟩ ≅ (Z /2)^{3} (after the 3 axis-fold relations)

(Three axes give three pairs U²/D² etc.; same-axis half-turns invert each other, kicking 3 relations, leaving 6 − 3 = 3 independent ℤ/2 factors.) Checkerboard's order is therefore ≤ 2 — and direct check gives 2. The "Pons Asinorum" (M² E² S²) and superflip both live in this Abelian subgroup.

Observation 13.1 — visual ↔ algebraic symmetry

A pattern fixed by all 48 outer cube symmetries ⇔ it is a fixed point of conjugation ⇔ it lies in Z(G). That is why superflip is simultaneously the most visually symmetric and the algebraically unique non-trivial element. A pattern fixed by a subset of symmetries lies in the corresponding symmetrizing subgroup — e.g. the three-axis-symmetric checkerboard sits in a 24-element rotational stabilizer, not in Z(G).

∎

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