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).
There are two possible moves on the puzzle with different gearing.
- Move ‘A’ is geared such that turning the red face clockwise moves the orange face clockwise.
- Move ‘B’ is different — turning the blue side causes the green side to move in the opposite direction.
Both of these moves perform dual 3-cycles of the numbers, which correspond to two
Fano Plane collineations
that can be combined to scramble the puzzle and generate all 168 states of the GL(3,2)
symmetry group.
YouTube Video
Simulator
Matrix
Fano Plane arrow graphics by T. Piesk, licensed under CC BY 4.0.
Collineations
There are 168 possible collineations, and they fall into 5 categories,
- 56: dual 3-cycles
- 42: 2-cycle & 4-cycle
- 21: dual 2-cycle
- 48: 7-cycle
- 1: identity
Some interesting sequences:
- A̅ B̅ A̅ B A̅ B̅ A̅ - Fano Plane rotation about the center
- B A B̅ A̅ B A B̅ - Quarter turn around the 5,2,7 axis
GL(3,2)
The set of all invertible binary 3x3 matrices and their multiplications also
generate the GL(3,2) group, and correspond to the 168 collineations of the
Fano Plane. The simulator above shows the binary matrix and corresponding octal number for each state
of the puzzle.
Construction
There were two big challenges in designing this puzzle. The first is that the geometry is really odd – it’s an octahedron warped so that there’s an extra vertex. To make it a viable puzzle mechanism I curled each cutting plane by several degrees so they could all fit. The pieces don’t move perfectly smoothly because of the irregular intersection angles.
The other challenge was fitting the gearing into the core. I came up with a way to combine a universal joint with a screw / spring mechanism to implement the ‘B’ axes which spin in the same direction through the core. Then, 4 gears are added to implement the ‘A’ axes which turn opposite each other.
Watch the YouTube Video for a demonstration and explanation of the puzzle.
Gallery
