Reticulated 3x3 Cube Variants
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Reticulated 3x3 Cube Variants

All the different graphs are variants arising out of the original graph by a morphing procedure.

standard Rubik's Cube™

Not only puzzle kids are familiar with this graph. Our layout is not completely standard though, but somehow it resembles a ball.

Figure 7.25. Cube 3x3

here a snapshot should be displayed

an (interestingly rendered) 3×3 Rubik's cube™


vertices: 56
faces: 54
edges: 108
genus: 0

characteristic polynomial: (1 - 3x + x2)2 (-1 - x + x2)6 (-1 + x + x2)6 (1 + 3x + x2)2 (6 - 5x - 3x2 + x3) (-6 - 5x + 3x2 + x3) (2 - 9x - x2 + x3)3 (-2 - 9x + x2 + x3)3

dual's characteristic polynomial: (-4 + x) (-2 + x) (2 + x) (-1 + x) x (1 + x) (-1 + 2x + x2) (2 - 3x - 2x2 + x3) (-1 - 3x + x2 + x3) (18 + 9x - 13x2 - x3 + x4) (12 + 111x - 306x2 - 2002x3 + 1484x4 + 10231x5 - 83x6 - 19231x7 - 5886x8 + 14263x9 + 5906x10 - 4936x11 - 2269x12 + 837x13 + 403x14 - 67x15 - 33x16 + 2x17 + x18)2

chromatic number: 3

lattice group (lossy): Z(2) 21 × Z(3) 11

rotated corner

You may guess (and your eyes/brain are actually not sure about it either) that there should be two allowed layouts for the corner, going outwards or into the inside. Compare the following polynomials with those of the original graph.

Figure 7.26. 3x3 rotated corner

here a snapshot should be displayed

a 3×3 Rubik's cube™ with rotated corner


vertices: 56
faces: 54
edges: 108
genus: 0

characteristic polynomial: x2 (1 - 3x + x2)2 (-1 - x + x2)5 (-1 + x + x2)5 (1 + 3x + x2)2 (2 - 9x - x2 + x3)2 (-2 - 9x + x2 + x3)2 (-3244 + 14696x2 - 22543x4 + 14369x6 - 4301x8 + 619x10 - 41x12 + x14)

dual's characteristic polynomial: (-4 + x) (-2 + x) (2 + x) (-1 + x) x3 (1 + x)3 (-1 + 2x + x2)3 (2 - 3x - 2x2 + x3)3 (-1 - 3x + x2 + x3)3 (18 + 9x - 13x2 - x3 + x4)3 (2 - x - 9x2 + x3 + x4)2

chromatic number: 4

lattice group (lossy): Z(2) 2 × Z(3) 70 × A(36) 2

rotated 2x2 area

As you can see, here two faces are connected along two neighboured edges, but never mind (in what can follow, there arises even the possibility of a pentagon with 2 edges glued together thereby becoming a triangle).

Figure 7.27. 3x3 rotated 2x2

here a snapshot should be displayed

a 3×3 Rubik's cube™ with two faces connected via neighboured edges (after rotating 4 faces)


vertices: 56
faces: 54
edges: 108
genus: 0

characteristic polynomial:

dual's characteristic polynomial:

chromatic number: 4

lattice group (lossy): Z(2) 2 × A(108) 2

rotated 3x3 area

This is the result of rotating an 3×3 area. Doing a 1/3 standard Rubik's cube™ move instead (affecting 9 + 12 faces) will give a graph with the same polynomials and group.

Figure 7.28. 3x3 rotated 3x3 area

here a snapshot should be displayed

a 3×3 Rubik's cube™ after rotating an 3×3 area


vertices: 56
faces: 54
edges: 108
genus: 0

characteristic polynomial: (-1 - x + x2)2 (-1 + x + x2)2 (11 - 10x - 44x2 + 36x3 + 27x4 - 12x5 - 3x6 + x7) (1 - 6x - 8x2 + 28x3 + 5x4 - 12x5 - x6 + x7) (-1 - 6x + 8x2 + 28x3 - 5x4 - 12x5 + x6 + x7) (-11 - 10x + 44x2 + 36x3 - 27x4 - 12x5 + 3x6 + x7) (-1 + 97x2 - 278x4 + 143x6 - 22x8 + x10)2

dual's characteristic polynomial: (-4 + x) (-3 + x + 15x2 + 4x3 - 8x4 - x5 + x6) (-8 - 19x + 35x2 + 47x3 - 16x4 - 16x5 + x6 + x7) (-2 - 7x + 13x2 + 25x3 - 6x4 - 12x5 + x6 + x7) (-2 - 3x + 19x2 + 9x3 - 22x4 - 8x5 + 3x6 + x7) (4 + 35x - 28x2 - 387x3 + 160x4 + 971x5 - 164x6 - 714x7 + 52x8 + 210x9 - 4x10 -25x11 + x13)2

chromatic number: 4

lattice group (lossy): Z(2) 53 × A(27)

a follow up to variant 3

After having rotated the top 3×3 area counter-clockwise, then the (freshly formed) right-hand 3×3 area has been moved one step counter-clockwise. Now how would you define new areas to be moved (in any case one should seek to get a subgroup of the full group)?

Figure 7.29. 3x3 follow up

here a snapshot should be displayed

subsequently to variant 3, another move of same sort has been done


vertices: 56
faces: 54
edges: 108
genus: 0

characteristic polynomial: x2 (14 + 500x - 8834x2 + 9004x3 + 108312x4 - 152919x5 - 544248x6 + 808010x7 + 1459295x8 - 2109449x9 - 2299002x10 + 3068713x11 + 2223753x12 - 2625825x13 - 1353956x14 + 1385850x15 + 525805x16 - 464902x16 - 130168x18 + 100110x19 + 20181x20 - 13673x21 - 1877x22 + 1136x23 + 95x24 - 52x25 - 2x26 + x27) (-14 + 500x + 8834x2 + 9004x3 - 108312x4 - 152919x5 + 544248x6 + 808010x7 - 1459295x8 - 2109449x9 + 2299002x10 + 3068713x11 - 2223753x12 - 2625825x13 + 1353956x14 + 1385850x15 - 525805x16 - 464902x16 + 130168x18 + 100110x19 - 20181x20 - 13673x21 + 1877x22 + 1136x23 - 95x24 - 52x25 + 2x26 + x27)

dual's characteristic polynomial: (-4 + x) (56 + 540x - 198x2 - ... + 2x25 + x26) (112 + 896x - ... + 2x26 + x27)

chromatic number: 4

lattice group (lossy): Z(2) × A(108)

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