# October 30, 2000

9:41 PM Subscribe

After getting the inside story (ha?) on the inventor of everyone's favorite non-orientable surface, the Klein Bottle; and perhaps playing a few games inside of one, you can check out a few 3-dimensional immersions of klein bottles: in Lego, knitted fabric, paper, or glass.

Why not buy one for the home?. The maker of these Klein Bottles is none other than Cliff Stoll, author of

posted by peterme at 7:10 AM on October 31, 2000

*The Cuckoo's Egg*and*Silicon Snake Oil*, erstwhile commentator on*The Site*, and overall delightful nutcase.posted by peterme at 7:10 AM on October 31, 2000

Huh. Looking for a good link, I have discovered that MathWorld is apparently defunct. Shame, that. I did stumble across an amusing page at kleinbottle.com, though.

Anyhow, I'm not providing links, but the Klein bottle has an interesting relationship with the well-known four-color (mapping) theorem, which states that any map with no discontinuous areas (a state like Michigan could potentially screw things up) can be colored with four colors so no two adjacent areas have the same color. (Try it! It's a fun exercise for recreational math fans.) The proof has been steadily improved on over the years, but originally was the first non-human-readable mathematical proof; the mathematicians who proved it showed that all maps can be reduced to some large (in the multiple thousands) of possible cases, all of which can be four-colored.

The Heawood conjecture gives a general formula for how many colors are required (referred to, IIRC, as the chromatic number) to color a map on surfaces in general -- such as, say, a three-holed torus (think of drawing on the surface of a pretzel). Unfortunately, the proof of the Heawood conjecture (at least the one I had to walk through as a seminar presentation in my college topology class) doesn't work for surfaces of Euler characteristic zero, which includes the sphere, the plane, and the Klein bottle.

Strangely, the Klein bottle is the one surface for which Heawood's formula doesn't hold up -- unlike any other surface, the Klein bottle can be colored with one color less than the chromatic number predicted with the Heawood formula. Why? Because it's just that cool!

posted by snarkout at 7:54 AM on October 31, 2000

Anyhow, I'm not providing links, but the Klein bottle has an interesting relationship with the well-known four-color (mapping) theorem, which states that any map with no discontinuous areas (a state like Michigan could potentially screw things up) can be colored with four colors so no two adjacent areas have the same color. (Try it! It's a fun exercise for recreational math fans.) The proof has been steadily improved on over the years, but originally was the first non-human-readable mathematical proof; the mathematicians who proved it showed that all maps can be reduced to some large (in the multiple thousands) of possible cases, all of which can be four-colored.

The Heawood conjecture gives a general formula for how many colors are required (referred to, IIRC, as the chromatic number) to color a map on surfaces in general -- such as, say, a three-holed torus (think of drawing on the surface of a pretzel). Unfortunately, the proof of the Heawood conjecture (at least the one I had to walk through as a seminar presentation in my college topology class) doesn't work for surfaces of Euler characteristic zero, which includes the sphere, the plane, and the Klein bottle.

Strangely, the Klein bottle is the one surface for which Heawood's formula doesn't hold up -- unlike any other surface, the Klein bottle can be colored with one color less than the chromatic number predicted with the Heawood formula. Why? Because it's just that cool!

posted by snarkout at 7:54 AM on October 31, 2000

Don't mean to nitpick peterme, but I already had that link in the original post :).

posted by kidsplateusa at 2:07 PM on October 31, 2000

posted by kidsplateusa at 2:07 PM on October 31, 2000

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posted by Kikkoman at 12:04 AM on October 31, 2000