Posts tagged maths

An aperiodic monotile, sometimes called an “einstein”, is a shape that tiles the plane, but never periodically. In this paper we present the first true aperiodic monotile, a shape that forces aperiodicity through geometry alone, with no additional constraints applied via matching conditions. We prove that this shape, a polykite that we call “the hat”, must assemble into tilings based on a substitution system. The drawing above shows a patch of hats produced using a few rounds of substitution.

via https://cs.uwaterloo.ca/~csk/hat/

Mathematical notation provides perhaps the best-known and best-developed example of language used consciously as a tool of thought. Recognition of the important role of notation in mathematics is clear from the quotations from mathematicians given in Cajori’s A History of Mathematical Notations [2, pp.332,331]. Nevertheless, mathematical notation has serious deficiencies. In particular, it lacks universality, and must be interpreted differently according to the topic, according to the author, and even according to the immediate context. Programming languages, because they were designed for the purpose of directing computers, offer important advantages as tools of thought. Not only are they universal (general-purpose), but they are also executable and unambiguous. Executability makes it possible to use computers to perform extensive experiments on ideas expressed in a programming language, and the lack of ambiguity makes possible precise thought experiments. In other respects, however, most programming languages are decidedly inferior to mathematical notation and are little used as tools of thought in ways that would be considered significant by, say, an applied mathematician.

via http://www.jsoftware.com/papers/tot.htm

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“And I promise you,” said the big green bat. “You don’t need to factor the number. You just need to GET OUT OF THE CAR.”

“I can’t get out of the car until you factor the number.”

“I won’t factor the number until you get out of the car.”

“Please, I’m begging you, factor the number!”

“Yes, well, I’m begging you, please get out of the car!”

“FOR THE LOVE OF GOD JUST FACTOR THE FUCKING NUMBER!”

“FOR THE LOVE OF GOD JUST GET OUT OF THE FUCKING CAR!”

“FACTOR THE FUCKING NUMBER!”

“GET OUT OF THE FUCKING CAR!”

“Universal love,” said the cactus person.

–SCOTT ALEXANDER in http://slatestarcodex.com/2015/04/21/universal-love-said-the-cactus-person/

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a paper by Marko Rodriguez called A Methodology For Studying Various Interpretations of the N,N-dimethyltryptamine-Induced Alternate Reality, [suggesting] among other things that you could prove DMT entities were real by taking the drug and then asking the entities you meet to factor large numbers which you were sure you couldn’t factor yourself. So to that end, could you do me a big favor and tell me the factors of 1,522,605,027, 922,533,360, 535,618,378, 132,637,429, 718,068,114, 961,380,688, 657,908,494, 580,122,963, 258,952,897, 654,000,350, 692,006,139?

“Universal love,” said the cactus person.

“Transcendent joy,” said the big green bat.

The sea turned hot and geysers shot up from the floor below. First one of wine, then one of brine, then one more yet of turpentine, and we three stared at the show.

–SCOTT ALEXANDER in http://slatestarcodex.com/2015/04/21/universal-love-said-the-cactus-person/

“It’s not the subject of calculus as formally taught in college,” Droujkova notes. “But before we get there, we want to have hands-on, grounded, metaphoric play. At the free play level, you are learning in a very fundamental way—you really own your concept, mentally, physically, emotionally, culturally.” This approach “gives you deep roots, so the canopy of the high abstraction does not wither. What is learned without play is qualitatively different. It helps with test taking and mundane exercises, but it does nothing for logical thinking and problem solving. These things are separate, and you can’t get here from there.”

http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/

I very much enjoy reading the “What Is…?” column in the Notices of the AMS. Unfortunately, there seemed to be no index to this column. I have therefore created this one in the hope that it’ll be helpful to others as well.

http://arminstraub.com/math/what-is-column

Excellence comes from qualitative changes in behavior, not just quantitative ones. More time practicing is not good enough. Nor is simply moving your arms faster! A low-level breaststroke swimmer does very different things than a top-ranked one. The low-level swimmer tends to pull her arms far back beneath her, kick the legs out very wide without bringing them together at the finish, lift herself high out of the water on the turn, and fail to go underwater for a long ways after the turn. The top-ranked one sculls her arms out to the side and sweeps back in, kicks narrowly with the feet finishing together, stays low on the turns, and goes underwater for a long distance after the turn. They’re completely different!

http://johncarlosbaez.wordpress.com/2013/09/29/levels-of-excellence/

Primecoin is the first scientific computing cryptocurrency, a project derived from Bitcoin but with novel non-hashcash proof-of-work that mines prime numbers.

https://github.com/primecoin/primecoin/wiki