I originally skimmed the article about the mono tiles too hard and, missed that it is not a single specific shape but, rather, a range of shapes. H/t to forum alum @m_raynsford, who mentioned it on his blog:
A reflection is just a rotation around the other axis.
I’m well aware of that, but when the rotation takes the piece out of the plane, that’s cheating. If it were a 3D puzzle, then it would be OK, but absolutely not with 2D.
Not true in planar geometry. Think about a print of your left hand and right hand. The right is a reflection of the left, but there is no rotation in the plane that can transform one into the other.
thanks for the file!! I am spearheading my son’s 8th grade class art “gift” to the school. Planning to cut a bunch and have the kids paint them for a modular art piece for the entryway.
That sounds like fun, would you please take pictures and post the results? I’d love to see it.
Improved Einstein Tile
The “hat” aperiodic monotile resolves the question of whether a single shape can force aperiodicity in the plane. However, all tilings by the hat require reflections; that is, they must incorporate both left- and right-handed hats. Mathematically, this leaves open the question of whether a single shape can force aperiodicity using only translations and rotations. (It also complicates the practical application of the hat in some decorative contexts, where extra work would be needed to manufacture both a shape and its reflection.)
Got a link to a generator for that? Would make an awesome puzzle for my daughter when she gets home. Finally a good use for some of the colored PG acrylic I have. I like the ones already shared but this one is much more appealing (to me, she’ll hate me!)
Edit… duh… all the pieces are identical so I can just enlarge and trace one, then repeat…
what we really need is someone who can code a site that generates that as non-overlapping lines.
Curious about this. You want a simple cut path. I have already started laying out pieces with an engraved image to complete the “puzzle” but using different color acrylic sheets.
when i make regular puzzles, i have two sets of lines, horizontal and vertical. i want to mount a photo, and then cut it out in these shapes instead of regular puzzle shapes. if it cuts over the same spot twice, it’s likely to affect the kerf a little. not to mention take at least 2x the time to cut.
I understand what you are saying, but I have always broken up and combined cut lines for puzzle pieces so there’s only one pass. I let GF figure out the directions but all mine have been one-offs so time is not much of an issue. It usually does pretty good at what I would consider an “optimal” path.