A Klein Bottle is the equivalent of a Mobius strip, rasied another dimension. A Mobius strip is a two dimensional strip, wrapped into a cylinder, with a twist in the third dimension. So a Mobius strip thus has only one surface and one edge. A Klein bottle is a three dimensional tube, with the ends connected to make a torus (donut), with the mathematically plausible (but physically impossible) "twist" through the fourth dimension. For a "real" Klein bottle, the tubes' piercings of the surface would actually bypass from one side to the other in the fourth dimension, and would not pass through the wall. So A Klein Bottle has no inside or outside, and only one surface. This model would be the equivalent of a Mobius strip with three twists instead of one.
Tuesday, September 2, 2008
Nested Klein Bottles
From Boing Boing, a beautiful picture of a really cool construct. (Larger pic at the Boing Boing link, more info on the piece here)A Klein Bottle is the equivalent of a Mobius strip, rasied another dimension. A Mobius strip is a two dimensional strip, wrapped into a cylinder, with a twist in the third dimension. So a Mobius strip thus has only one surface and one edge. A Klein bottle is a three dimensional tube, with the ends connected to make a torus (donut), with the mathematically plausible (but physically impossible) "twist" through the fourth dimension. For a "real" Klein bottle, the tubes' piercings of the surface would actually bypass from one side to the other in the fourth dimension, and would not pass through the wall. So A Klein Bottle has no inside or outside, and only one surface. This model would be the equivalent of a Mobius strip with three twists instead of one.
A Klein Bottle is the equivalent of a Mobius strip, rasied another dimension. A Mobius strip is a two dimensional strip, wrapped into a cylinder, with a twist in the third dimension. So a Mobius strip thus has only one surface and one edge. A Klein bottle is a three dimensional tube, with the ends connected to make a torus (donut), with the mathematically plausible (but physically impossible) "twist" through the fourth dimension. For a "real" Klein bottle, the tubes' piercings of the surface would actually bypass from one side to the other in the fourth dimension, and would not pass through the wall. So A Klein Bottle has no inside or outside, and only one surface. This model would be the equivalent of a Mobius strip with three twists instead of one.
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