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.
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.
Note that walking a tetrahedral die on the plane can be done consistently such that the entire plane is covered, with no blank spaces.
Any tromino (3 squares, joined edgewise) yields two ‘wedge’ shapes if you halve it by cutting the middle square diagonally. The wedges can be viewed as half-trominoes or as three 45-45-90 triangles aligned edgewise. Like the P-Pentomino, these wedges come in right- and left-handed forms which can tile themselves asymmetrically at a scale of 1/4.
Side lengths are 1, 1, √2 and 2. So for full plane tiling the non-integral side must align with the non-integral side of an adjacent wedge. Above are the shapes possible when two identical wedges are arranged edgewise. The chart below shows the ways two enantiomorphic wedges can be arranged edgewise.
Had a great time at ShortRun this past weekend. Hope you made it.
I premiered my latest book, “Three-Color Tiling,” at the Portland Zine Symposium on Saturday, July 9. This is another book from the analytical corner of my brain. The design is similar to “P-Pentomino Solutions” but the 46 patterns inside are in full color. The objective was to come up with a set of regular 3-color patterns in which the density of the three colors is equal. The patterns in this book are built on simple square or equilateral triangle units, but these generally combine with adjacent units as stripes, zigzags, chevrons, lozenges, and so forth. Some patterns are recognizable from common use in folk art and heraldry, but some of them seem novel. I applied a set of ship names to label the different designs, in part to memorialize the many hospital ships that were attacked and sunk as acts of war.