DoDot M12
The DoDot M12 is a twisty puzzle that mechanically demonstrates Mathieu group M12. It’s a modification of my original vertex-turning dodecahedron where sets of three turning axes are geared together internally. This gearing was chosen to greatly reduce the number of possible permutations of the ‘dot’ pieces, which behave according to the rules of the 12-element M12 permutation group. The edge pieces of the puzzle behave as 6 disjoint orbits of 5 pieces, which makes them easy to solve. I greyed out the petals to avoid making the puzzle too complicated.
Mechanical Implementation
The mechanism uses a 4-layer gearbox where three axis gears mesh with a circular gear rack so that they move simultaneously in the same direction. The gearbox was designed to minimize play as much as possible, and so I used rigid stainless steel gear racks made by DMLS printing.
12 of the turning axis are connected to the gearing mechanism and the remaining 8 don’t move at all.
Group Theory
I independently discovered this generator while searching for weird groups over various deep geometries, then decided to take a detour into shallow-cut puzzles. I want on to fully catalog all of the groups that can be generated using this puzzle as a basis, with different gearing: Google Drive Link
I’m working on an online JavaScript program to visualize these results, but in the meantime DM me for a viewer written in Java that can preview them. Some familiar things show up in the results like Ree(3) (Trapentrix group), as well as some new things. Most of the symmetries are interesting and beautiful. My favorite is the 4x3 M11 generator which has this bizarre 5-fold symmetry - I plan on prototyping it next.
The DoDot M12 expresses M12 using the following generator set:
[(2,12,1)(10,6,3)(5,4,7),(8,1,5)(2,3,12)(10,9,11),
(8,9,1)(5,12,4)(6,11,7),(10,2,9)(4,3,6)(8,7,11)]
If the edges didn’t exist then God’s number would be 9. Including the edges it should have 95040 × 60^6 permutations = 4,434,186,240,000,000.
I have solved the puzzle without a computer and it’s not terribly hard. The friendly behavior of the edges and 4 choices of axes makes it easier than biaxe-style puzzles. My longest sequence, the dual 4 cycle, was 40 moves - I imagine this can be optimized with a computer.
This implementation demonstrates snub tetrahedral symmetry, with distinct chiral forms.
Historical Context
Mathieu groups have been expressed in these puzzles previously:
- M12 - Oskar van Deventer’s Topsy Turvey
- M12 - Oskar van Deventer’s Number Planet
- M12 - Alexander Holroyd’s Croc-o-dial
- M24 - Casey Mihaloew’s M24 Cube
- M12 - Alexander Holroyd’s Magic Cogra
The idea of M12 over a vertex-turning dodecahedron is not new - it was previously discussed on the TwistyPuzzles forum in 2018. This thread contains the first reference to the idea and reads like this:
- Initial inquiry by Bram Cohen
- Katelyn Neily suggests a 4x3 gearing exists, which is likely the same gearing that I used
- William Kretschmer confirms and describes a full 20-axis representation by using 5 sets of 4-axis groups
- Possible gearing systems proposed by Oskar van Deventer and Bram Cohen
To my knowledge, despite further discussion occurring later on, no physical prototypes were made.
Video
The YouTube Video contains more details and a demonstration of the puzzle.
