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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

© Borrelli et al / PNAS

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.

Life imitates Life

A while ago somebody created a simulation of Conway’s Game of Life inside a bigger version of the Game of Life. Now, YouTube user Phillip Bradbury has created a very simple — and aurally pleasing — video showing it in action.

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Apparently this is made possible by the Outer Totalistic Cellular Automata Meta-Pixel (OTCAMP), a “two state programmable unit cell which allows Conway’s Life to simulate any outer totalistic rule. OTCAMP is a meta-cell which is also a meta-pixel. OTCAMP meta-pixels display evolving meta-patterns on-screen in meta-realtime.”

An outer totalistic rule is a rule for a cellular automaton which defines the transitions between cell states based on the total number of switched-on surrounding cells surrounding them. The Game of Life is one such rule.

Source: Richard Elwes on Google+.

Top N Facts about the Klein Bottle

In honour of Felix Klein’s birthday, Matt Parker and Katie Steckles investigate the amazing surface which bears his name.

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Using a zero-knowledge protocol to prove you can solve a sudoku

I’ve just uploaded to youtube a video I made with Katie Steckles to demonstrate why zero-knowledge protocols exist and how one works.

Katie is a habitual liar, so we followed the zero-knowledge protocol described in the paper, “Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles” which you can download from

By following this protocol, Katie can prove that she isn’t lying to me about being able to solve the puzzle, without revealing anything about how she solved it.

The paper I mentioned, “How to explain zero-knowledge protocols to your children” is an excellent explanation of the ideas behind zero-knowledge proof.

Click here to continue reading Using a zero-knowledge protocol to prove you can solve a sudoku on cp’s mathem-o-blog