Baby Felix Klein’s family probably didn’t give him lots of useless presents around the holidays, and he turned out okay (aka brilliant mathematician/physicist). All of the Holiday Sales are already starting, trying to entice all of us poor people to dish out dollars for useless junk. So I thought I’d start a list of mathy gifts under $15 that are sure to please. Everyone knows that mathematicians love to be disoriented — or at least they know that being disoriented is nothing to be ashamed of! I suppose these might also make good extra credit projects to give your students as well (maybe not the first one).

**Make a Glass Klein Bottle for your favorite mathematician:**What’s a Klein Bottle? Take a cylinder and glue the ends together with opposite orientation. Can’t do it? That’s because it can’t be done in three dimensions without letting the cylinder intersect with itself. Acme Klein Bottles is a company that specializes in beautiful glass Klein Bottles that are, in general, pretty expensive. BUT, the good news is Acme Klein Bottles, there is a $10 option that may appeal to those conservationists/penny-pinchers among us. This “Jigsaw Puzzle” comes with a free band-aid!**Knit a Klein Bottle Hat to keep those precious brains warm:**Seeing as yarn is so much softer than glass, this option may be preferable. I have not tried this next one, but judging from my minimal knitting skillz, I may stick to the “Mobius Scarf”, which I saw at Oiyi’s Crafts Blog. Mathematician Sarah-Marie Belcastro generously provides instructions at her website for knitting these self-intersecting representations of the Klein Bottle. There is also a link to making hyperbolic baby pants.**Make a Mobius Music Box for your loved one**: I ordered my very own DIY Music Box Kit, and made my own little music box that plays music upside down! (Just twist the music sheet once first after threading it through the music box as done at This artists site ) It was fun and you can do it too. Take a gander at Think Geek’s ;. AND Note that you can punch your own holes to compose an original backwardsupside down masterpiece. By the way, August Mobius is the namesake of this strip, which was also discovered by a man whose last name was Listing. But which sounds cooler “Listing Strip” or “Mobius Strip”? Yeah, that’s what I thought.**Make Your Boy (or Girl) a Boy’s Surface: Really, Real Projective Space deserves more recognition here. I mean, it’s disorienting too! Two dimensional real projective space is what you get by taking a sphere and identifying (gluing together) antipodal points (e.g. the north and south pole). Since this space cannot be embedded in three dimensions (try it), we will make do with “Boy’s Surface”, which is an immersion (has self-intersection) just as our models of the Klein Bottle are. This immersion was part of Werner Boy’s 1901 Thesis written under the Famous Mathematician Hilbert. The surface was discovered as a result of Hilbert’s request for Boy to prove that no such immersion existed.***Lesson to be learned: Even famously intelligent people can be wrong! Courtesy of Joe Field’s website, you can use just good old fashioned paper, scissors, and tape to make Boy’s Surface. This is a truly unusual gift– not nearly as pedestrian as Klein Bottles or Mobius Strips.**Okay, I’d love to see some other inexpensive gift ideas — what mathy things have you made?*

Those are some very creative math gift ideas. Thanks for posting!