http://www.thebigquestions.com/2012/02/23/who-expected-that/ Published by <http://www.thebigquestions.com/author/landsburg/>Steve Landsburg on February 23, 2012 in <http://www.thebigquestions.com/category/coolstuff/>Cool Stuff, <http://www.thebigquestions.com/category/empirical-puzzles/>Empirical Puzzles and <http://www.thebigquestions.com/category/math/>Math . <http://www.thebigquestions.com/2012/02/23/who-expected-that/#comments>14 Comments Suppose you go around taking extremely close-up black-and-white pictures of randomly chosen natural and unnatural objects (rocks, trees, streams, buildings, etc.). What do they look like? Well, each one looks like a patch of varying shades of gray, of course. But do some patches arise more than others? If each of your close-ups is, say, three pixels by three pixels, Which would you expect to see more of: This? [] Or this? [] Well, each 3 x 3 square of pixels is essentially a list of nine numbers (one for the darkness of each pixel, so that a pure black pixel is a “1 , a pure white pixel is a “0 , a nearly-black pixel is, say, a “.9 , etc.). A list of nine numbers specifies a location in nine-dimensional euclidean space (just as a list of three numbers specifies a location in the three-dimensional space in which we appear to live). So the question becomes: Where in nine-dimensional space do close-up patches of photos tend to live? A natural expectation is that they’re scattered randomly ­ and at first blush that appears to be accurate. But it turns out that if you take a closer look, using a mathematical “lens” that lets you see more clearly into nine dimensions, they’re not scattered randomly at all. Instead, they’re clustered around (of all things!) a <http://en.wikipedia.org/wiki/Klein_bottle>Klein bottle, which is a two-dimensional surface that can’t be squeezed into three dimensions, but fits perfectly well in nine (or for that matter in four). (Actually it’s a somewhat thickened Klein bottle, and hence four-dimensional instead of two-dimensional, much as a solid hula hoop is a somewhat thickened circle and hence three-dimensional instead of one-dimensional.) In other words, if you randomly photograph 10,000 objects, randomly choose 3-by-3 pixel patches, and plot the corresponding points in nine-dimensional space, what you’ll “see” (insofar as you can see in nine dimensions) is a somewhat blurry Klein bottle. Who’da thunk? <http://www.dam.brown.edu/people/mumford/Papers/DigitizedVisionPapers--forNonCommercialUse/x03a--Stats-LeePedersen.pdf>Here is some of the original research and <http://www.math.upenn.edu/%7Eghrist/preprints/barcodes.pdf>here is an expository (but pretty high-tech) paper on this and related matters. Edited to add: The original computational work is <http://www.springerlink.com/content/pm1718435n685294/>here. A survey article by Gunnar Carlsson, one of the original researchers, is <http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/>here<http://comptop.stanford.edu/>. The research group’s webpage is --- co-chair http://ocjug.org/