Penrose
A few years back, I picked up a cool set of magnetic shapes. They were rhombuses, each about an inch long, of various widths and colors, printed on magnetic vinyl; they could be placed next to each other in various pleasing patterns.
I bought them thinking they were Penrose tiles, but was sad to discover that they had the wrong angles. Still pretty patterns, but not aperiodic tilings of the plane. Or of the refrigerator, in this case. (SPSU has more on aperiodic tilings, plus a nifty page showing Penrose's earlier and more complex snowflake-like tiles.)
So I determined to make my own set of magnetic Penrose tiles. I bought a sheet of magnetic vinyl and an X-Acto knife (or equivalent), and figured some day I would measure out the angles, draw the lines, and cut the pieces.
Fast-forward several years. I've still got the sheet of vinyl and the knife, and I still haven't done anything with them. Fortunately for me, Kam heard me being sad about not having magnetic Penrose tiles but didn't hear me saying I would make them myself, and so she made me some. She cleverly used a computer to do the layout (making the angles much more accurate than I would've by hand), then printed the pattern onto a sheet of thin computer-printable magnetic vinyl, then painstakingly cut them apart with scissors. (We've discussed various ways of streamlining the cutting process for next time.) She also made them in a variety of pretty shades of blue (the colors in this case don't have anything to do with the tiles; they're just pretty).
Last night we started playing with them. I knew that there were rules governing which tiles could be placed near each other (which is why the tiling is aperiodic; if there weren't such rules, it would be easy to make periodic tilings with these pieces), but I didn't know what those rules were. I knew about the rules for the other equivalent form of Penrose tiles (called "kites and darts"), but I hadn't ever seen rules for the "Penrose rhombs." So we poked around on the web for quite a while.
Eventually, after many fascinating digressions, including lots of pages containing cool pictures and math that made my head spin, we found the very nice and reasonably clear Science U Penrose Tilings page. The rules for the rhombs are kinda complicated, and they have some very non-obvious consequences—there are patterns that obey the rules locally, but result (as you add tiles to them) in blank spots where no tile can legally be placed. We spent quite a while adding little dots and arrows (in pencil) to some of the tiles and experimenting with what was and wasn't legal. We came up with half a dozen legal vertex patterns (such as the 3D cube, the other 3D cube, the crown, the jack-in-the-box, the flower, the octopus-in-the-box, and the inward-pointing and outward-pointing stars), and several illegal ones, and discussed how to design the next generation of magnetic tiles to best represent the rules on the tiles. Math geekery 'til past midnight. Much fun.
One thing I'm still a little vague on is the exact definition of aperiodic. Pretty much every page I've seen claims that no piece of an aperiodic tiling is ever repeated; this is obviously untrue on the face of it. Every picture of Penrose tilings shows several elements that appear more than once, unrotated, even if you don't consider the tiles themselves. The Science U Nonperiodic Tilings page is a little more informative, but relies on circular logic: it says the tiling is aperiodic because the star isn't repeated, then says if the star were repeated, it would be repeated, and since it isn't repeated, it must not be repeated, and therefore the tiling is aperiodic.
If you want your very own magnetic-vinyl Penrose tiles but you don't want to make them yourself, you can order them from Kadon Enterprises in the UK. They have not only the rhombs but kites and darts (much harder to cut from the magnetic vinyl); you can get 100 kites and 60 darts for $40. It appears that none of their tiles have markings to represent the rules, though, so if you want to tile aperiodically you'll have to either memorize which shapes are legal and which illegal, or put markings on the tiles.
Oh, and if you don't want a physical set but do want to play with the tiles, try ShuXiang Zeng's Java applet to play with Penrose tiles. (There are other applets out there too, including one that will automatically place all forced tiles for you, but this entry is long enough already.)
