I had the idea of doing short videos about mathematical objects I’ve got lying around. First up is a very unconventional group theory textbook.
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Walking on π in the NSF Visualization Challenge
Fran Aragón Artacho has emailed to tell us that he and Jon Borwein have entered their image of a walk on the first 100 billion digits of π in the National Science Foundation (of the USA)’s International Science & Engineering Visualization Challenge. Fran says:
Jon Borwein and I have submitted our picture of a walk based on 100 billion digits of pi to a visualisation contest from the NSF (National Science Foundation). The winners will appear in Science (one will be selected for the front cover!). And we have good news: we are one of the 10 finalists in the Illustration category!
Flat tori in three-dimensional space and convex integration
French researchers Vincent Borrelli, Saïd Jabrane, Francis Lazarus and Boris Thibert have described an isometric embedding of the flat torus in 3D space, using the convex integration theory developed by Gromov in the 1970s. That means they’ve produced a surface which is topologically a torus – it has a single hole — which preserves distances between points in the 4D flat torus. Interestingly, the tangent plane is defined everywhere — the surface is in a sense smooth — but the normal vector is not defined, so it’s also a fractal. This is impossible in higher dimensions
I’ve recorded a short video explaining in a handwavey fashion, with a few props made from things I had lying around, just what has been done.
Conformal Models of Hyperbolic Geometry by Vladimir Bulatov
Vladimir Bulatov makes art, including metal sculptures and jewellery, based on tilings of non-Euclidean spaces.
He has posted online some slides he made to go with a talk he gave at the JMM in 2010, about the many ways conformal mappings of the hyperbolic plane can produce interesting images. Quite a few of the diagrams are animated if you click on them, which I missed first time round.
There are a few other slideshows on similar topics on his site.
In addition to all that, Vladimir shares some very cool things on Google+.