Reticulated 3x3 Cube Variants
All the different graphs are variants arising out of the original graph by a morphing procedure.
Not only puzzle kids are familiar with this graph. Our layout is not completely standard though, but somehow it resembles a ball.
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
lattice group (lossy):
Z(2) 21 × Z(3) 11
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.
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
lattice group (lossy):
Z(2) 2 × Z(3) 70 × A(36) 2
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).
vertices: 56
faces: 54
edges: 108
genus: 0
characteristic polynomial:
dual's characteristic polynomial:
lattice group (lossy):
Z(2) 2 × A(108) 2
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.
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
lattice group (lossy):
Z(2) 53 × A(27)
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)?
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)
lattice group (lossy):
Z(2) × A(108)