Tumbling Dice

I was walking a generic cubical die around a sheet of graph paper one day, contemplating mapping the surface of the die to the plane. Place a die on a square and assign the number from the top face of the die to that square. Then roll the die forward one square, right one square, back one square, and left one square onto the square on which you started. If you assign the numbers from the top face of the die to each square covered in turn, the number of the final square will be different than the number assigned to it at the start. To make the numbers come out consistently you have to rotate forward two squares, then right two squares, back two, then left two. By taking steps two at a time in a given direction, the numbers 1-6 can be distributed evenly and consistently across the plane. Only 3/4 of the squares in the plane are mapped to the faces of the cube in this manner and 1/4 are left blank.

diceplane1

The opposite sides of a die traditionally add up to seven, so the 1, 2 and 3 faces share a vertex. You can cut the edges and flatten out the 6-faced cube into a crucifix-shaped hexomino which can tile the plane by a series of rotations/translations to match the tumbling die pattern described above. First, rotate a copy of the hexomino 180° about the eccentric red point. (The rotated copy is shaded to make it clearly distinct.) This 2-hexomino cell can then be stepped across the plane in increments of 4 units, horizontally and vertically. Again, only 3/4 of the squares in the plane are mapped to the faces of the cube in this manner.

crux1

Note that walking a tetrahedral die on the plane can be done consistently such that the entire plane is covered, with no blank spaces.

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