Sure, sure, we all saw that haunting illustration of a bat’s wings that won first place for informational graphic in Science magazine’s 2007 Science and Engineering Visualization Challenge. And we saw the other winners, too, including the arresting photo of “What’s Behind our Nose?” and that Irish moss, pulled from the sea, that looks like an alien made entirely out of hands and fingers. (Click here to check out the winners’ gallery.)
But I’ve seen little mention of a video that explains, with elegance and simplicity, a nifty utensil from the geometer’s toolbox. (Th
The Möbius transformation, which is really four functions in one, is used by geometers to map points to points and shapes onto shapes. The most remarkable part of the transformation is the way it effortlessly illustrates unusual geometries. You can map a two-dimensional shape onto the surface of a sphere.
And, as an added bonus, by showing that the transform is one-to-one (a bijection), you can prove that an infinite two-dimensional plane and the sphere (ie, the surface of the sphere) have the same number of points!
You can watch a low-res version of the video here, and a high res (130MB) version here. (And yes, the Möbius transformation is named for that Möbius, August Ferdinand Möbius, who came up with the twisted strip that has two dimensions but only one side.)
