The Crammed Cube is an internally geared twisty puzzle that expresses the Mathieu M11 group over its 12 edge pieces. The puzzle features a modified Compy Cube mechanism with 6 axes, which are mechanically linked into two sets using universal joints and gears. It is called the ‘Crammed Cube’ because, while the axis system would normally have 11 moving edges, I’ve crammed an extra one into the red/white edge and lifted the corner angle to make space. This edge duplication is necessary to make the group actions fit over cubic geometry.
The Fano Gem is a twisty puzzle that I designed to showcase the symmetries of the Fano Plane, which is a familiar object in combinatorics, coding theory, and algebra. The Fano Plane is the smallest possible finite projective plane, constructed from 7 points and 7 lines, where each line contains exactly 3 points.
From the red face (pictured left), the numbers on the corners of the puzzle match up with the numbers on the plane (with number 7 on the back of the puzzle, out of view).
A comment on my last YouTube video pointed out that the SL(2,3) group expressed by the Quaternion Cube is isomorphic to the group formed by multiplication of the unit Hurwitz quaternions.
Simulator
‘Order’
There are five different orders of unit Hurwitz quaternions: 1, 2, 3, 4, and 6; On the puzzle, the order represents how many times a sequence must be repeated to return to where you started.
The Quaternion Cube is a rather easy-to-solve puzzle that I designed to visually demonstrate the Quaternion Group Q₈.
The puzzle turns over the top 4 vertices of a cube, with 1:2 gearing connecting the opposing corners. There are linear slides under the edge pieces that prevent the static corners from blocking the overall movement.
The puzzle has 24 possible positions, which express the group SL(2,3). SL(2,3) contains the Quaternion Group (Q₈) as a subgroup, and these Q₈ states are expressed over the 8 edges whenever the corners are solved.
Updated 2025-03-16
STL Files (v2): https://quirkycubes.com/public/QuaternionCube_STL_v2.zip
Sticker Template (SVG, Silhouette Studio): https://quirkycubes.com/public/QuaternionCube_Stickers_v2.zip
Hardware required:
- 5x M3 Springs (~10mm uncompressed)
- 5x M3 Washers (6.9mm dia)
- 5x M3 x 20mm Screws (button or socket)
- 12x M2 x 8 Button Head Screws
- 8x M1.6 x 16 Socket Head Screws
- 7/64" drill bit
I printed my copy out of ABS and vapor-smoothed it to post process it. I don’t see any reason that it wouldn’t work in PLA, but you will need to sand the edge hooks a little to get smooth motion. Here are the settings I use on a Bambu P1S:
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.
