Looking for a cool gift for someone mathematically inclined? An unusual, conversation-generating token of appreciation? The World Wide Web offers a number of intriguing possibilities–if you know where to stop and shop.

A little ad in a mathematics newsletter recently alerted me to the Math Hatter Web site (http://www.mathhatter.com/), which offers “mathematical softwear for your head.” It specializes in winter headgear: Klein bottle caps and Möbius earbands, each available in several different colors (one side fits all).

Although I haven’t actually tried these particular items, the notion of real-world products based on exotic mathematical forms–in this case, the Möbius strip and the Klein bottle–appeals to me.

Since its discovery in the 19th century in a purely mathematical context, the Möbius strip, with its one-sided surface, has achieved a life of its own independent of mathematics–in magic, science, engineering, literature, music, film, and art. It has shown up in all sorts of unexpected settings: as monumental sculptures, knitted scarves, and glittering pendants and in designs on postage stamps and greeting cards.

The Klein bottle, also discovered in the 19th century, has equally intriguing features. This bizarre, mind-bending object has no lip or edge and only one surface. What appears to be the bottle’s inside is smoothly connected with its outside.

Cliff Stoll’s Web site (http://www.kleinbottle.com/) offers Klein bottles fashioned from glass–along with Klein steins, knitted Klein bottle hats, and more. In Great Britain, Village Games (http://www.cix.co.uk/~villagegames/) sells its own brand of glass Klein bottle. I know from personal experience that a glass Klein bottle on display invites attention and, inevitably, questions about whether it can actually hold liquid and, if so, how you would drink from it.

Logic puzzles and strategy games represent another category of possible interest to the mathematically inclined. Lately, I’ve particularly enjoyed two puzzles offered by Binary Arts (http://www.puzzles.com/). Cleverly packaged, *Lunar Lockout* requires you to figure out a set of moves, given certain constraints, that brings your peg safely to its mother ship on a five-by-five grid. *Metro* involves sliding two colored “cars” along tracks and through switches so that each car is backed into the station that matches its color. Both puzzles present interesting and varied challenges.

The serious puzzler can find additional recommendations and commentary from puzzle master Ed Pegg Jr. at http://www.mathpuzzle.com/.

If you happen to like word games such as *Scrabble*, you might also enjoy Binary Arts’ *Smart Mouth*, which features an ingenious device for delivering random pairs of letters. Competitors race to come up with the longest word for which the two letters serve as bookends. It makes for a lively pastime, with a tantalizing word-association element.

Speaking of speediness and time, I’ve appreciated having a sleek digital clock that provides the time and date with exquisite precision (http://www.atomic-clocks.com/). Every night, the clock automatically checks the radio-delivered official time signal and makes whatever adjustments are necessary to stay synchronized. Now I know what time it *really* is, though that doesn’t always help me meet my deadlines.

The clock reminds me of another timepiece that I obtained nearly two decades ago–one that tackles the passage of time in a considerably more contemplative manner. It’s the “Stonehenge Watch,” advertised as a “great leap backward in time.” When you open the watchcase, you see a tiny replica of the major components that make up this ancient megalithic monument and astronomical observatory.

My Stonehenge watch came with a droll instruction booklet, written and autographed by the watch’s inventor, poet Peter Payack (http://www.peterpayack.com/). To my great delight, I recently found that the Stonehenge watch is still available and now has its own Web site (http://www.stonehengewatch.com/).

For the more forward-looking, a binary digital timepiece might do the trick. A small company in Norway (http://www.rsi-digital.com/) now offers such a watch, advertising it as a “modern watch for modern people.” I don’t actually have one, but its 1s and 0s look just as cryptic as Stonehenge’s pillars.

For people interested in the artistic side of mathematics or the mathematical side of art, the Web site http://www.mathartfun.com/ offers a variety of items, including puzzles based on tessellations, polyhedra, and anamorphic illustrations, for exploring such interactions. The gallery of mathematical art at http://mathartfun.com/shopsite_sc/store/html/Gallery.html displays the work of several contemporary artists.

If you’re looking for note cards bearing attractive geometric designs, you might want to check out Robert J. Krawczyk’s spirolateral Web site. At http://home.netcom.com/~bitart/, you can view a wide variety of spirolateral designs, try out your own variants of the rules for creating such patterns, and order personalized cards with the chosen images.

I’ve mentioned just a sampling of the items that I’ve enjoyed learning about during the past year. Given the range and reach of human ingenuity and creativity, I’m sure there are many more possibilities out there for teasing the mind. Happy hunting!