Moves and thoughts about the Megaminx
-------------------------------------
The megaminx is a platonic dodecahedron with a rubik type
mechanism. I.e. it has 12 sides which rotate. There are two
types: The Hungarian SuperNova & The Megaminx manufactured by
Tomy. The edge pieces are a bit thicker on the Megaminx than
the SuperNova.
The megaminx has a slice group, analagous to the cube slice group.
All the possible spot patterns are in the megaminx's slice group,
e.g. the 10 spot and the 12 spot patterns. With process M1 we may
easily generate any spot pattern, although there is much room
for improvement.
During slice moves 10 corners and 10 edges move and 10 corners and
20 edges remain stationary.
The slice group of the megaminx is generated by turning the faces
opposite to each other in the opposite direction (i.e. opposite
looking at a face head-on!)
We aren't going to be exhaustively searching the slice group of the
megaminx any time soon as GAP shows it contains
9,627,755,206,121,277,812,101,663,948,800,000 elements
or approximately 9.6 * 10^33
The GAP has also shown that:
Size (megaminx) = Size (DerivedSubgroup (megaminx)), so that the
anti-slice group of the megaminx is isomorphic to the
whole megaminx group.
After doing a Monte Carlo analysis of the various orders of the
megaminx group, the largest order known is length 720,720.
Other known orders include:
540,540
471,240
360,360
353,430
240,240
102,102
10,080
5
In the case of process M1, L is opposite to R, not just separated
by a face F, as in processes M2, M3a and M3b.
My original diagram is rather limited, but it does illustrate the
idea of L & R separated by F only (as opposed to real opposites
but I have no satisfactory notation).
/\
/ \
/ \
\ U /
L \ / R
\____/
F
Moves for the Magic Dodecahedron (Megaminx)
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C_U = Rotate entire dodecahedron clockwise via the U face
suffix notation: f = face turns
u = unit turns
M1a 10 spot (shift 1) (L+ R- C_U-)^36 (slice group) (72f)
M1b 10 spot (shift 2) (L++ R-- C_U+)^48 (slice group) (96f)
M2 3 cycle of edges (uf, lf, rf)
R+ F+ U+ F- U- R- L- U- F- U+ F+ L+ (12f)
M3a 2 flip L-- R++ F+ U- R+ U+ L++ R++ U+ (18f, 26u)
R-- L-- U- R- U+ F- R-- L++ U-
M3a 2 flip (ul, uf)
R- F- U+ L- U- L+ F+ R+ (16f, 16u)
L+ F+ U- R+ U+ R- F- L-