Essence of linear algebra - "[Grant Sanderson of 3Blue1Brown (now at Khan Academy) animates] the geometric intuitions underlying linear algebra, making the many matrix and vector operations feel less arbitrary." [more inside]
Compass-and-straightedge construction (aka Euclidean construction) is a method of drawing precise geometric figures using only a compass and a straightedge (like a ruler without the markings). MathOpenRef maintains a catalog of many common constructions, each with an explanatory animation and a proof. This YouTube video demonstrates how to construct almost every polygon that can be constructed using these methods. [more inside]
Nearly four years after Shinichi Mochizuki (previously, previously, previously) unveiled an imposing set of papers (1, 2, 3, 4) that could revolutionize the theory of numbers, other mathematicians have yet to understand his work or agree on its validity — although they have made modest progress. [more inside]
Logic hacking - "Writing shorter and shorter computer programs for which it's unknowable whether these programs run forever, or stop... the winner of the Busy Beaver Game for N-state Turing machines becomes unknowable using ordinary math - somewhere between N = 5 and N = 1919." [more inside]
Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.
Scientists find evidence of mathematical structures in classic books. [The Guardian] James Joyce’s Finnegans Wake has been described as many things, from a masterpiece to unreadable nonsense. But it is also, according to scientists at the Institute of Nuclear Physics in Poland, almost indistinguishable in its structure from a purely mathematical multifractal.
“The absolute record in terms of multifractality turned out to be Finnegans Wake by James Joyce. The results of our analysis of this text are virtually indistinguishable from ideal, purely mathematical multifractals,” said Professor Stanisław Drożdż, another author of the paper, which has just been published in the computer science journal Information Sciences.
Shinichi Mochizuki and the impenetrable proof - "Fesenko has studied Mochizuki's work in detail over the past year, visited him at RIMS again in the autumn of 2014 and says that he has now verified the proof. (The other three mathematicians who say they have corroborated it have also spent considerable time working alongside Mochizuki in Japan.) The overarching theme of inter-universal geometry, as Fesenko describes it, is that one must look at whole numbers in a different light — leaving addition aside and seeing the multiplication structure as something malleable and deformable. Standard multiplication would then be just one particular case of a family of structures, just as a circle is a special case of an ellipse." (previously: 1,2; via) [more inside]
When the father of a second grader got annoyed by common core math tools (namely, ten frame cards), his annoyance went viral when he wrote a check to his student's school using common core numbers. Then the Friendly Athiest on Patheos used that check to teach how common core math works at the second grade level.
The Singular Mind of Terry Tao - "Imagine, he said, that someone awfully clever could construct a machine out of pure water. It would be built not of rods and gears but from a pattern of interacting currents." (via) [more inside]
Loop - Pool on an elliptical table. The ellipse has two significant points, called focuses, which have a remarkable geometrical property that is almost always explained using the example of an imaginary pool table. "If a pool table is the shape of an ellipse, then a ball shot from one focus will always rebound to the other focus no matter in which direction the ball is shot." That sounded interesting! Wouldn’t it be fun, I thought, if I could build one of these imaginary tables? So I did.
Univalent Foundations Redefines Mathematics - "When a legendary mathematician found a mistake in his own work, he embarked on a computer-aided quest to eliminate human error. To succeed, he has to rewrite the century-old rules underlying all of mathematics." (previously) [more inside]
Understanding e to the pi i - "An intuitive explanation as to why e to the pi i equals -1 without a hint of calculus. This is not your usual Taylor series nonsense." (via via; reddit; previously) [more inside]
Nothing but an endless supply of mental arithmetic problems. Five levels of difficulty, from "10 - 6" to "√370881." [more inside]
Imperfect Congruence - It is a curious fact that no edge-to-edge regular polygon tiling of the plane can include a pentagon ... This website explains the basic mathematics of a particular class of tilings of the plane, those involving regular polygons such as triangles or hexagons. As will be shown, certain combinations of regular polygons cannot be extended to a full tiling of the plane without involving additional shapes, such as rhombs. The site contains some commentary on Renaissance research on this subject carried out by two renowned figures, the mathematician-astronomer Johannes Kepler and the artist Albrecht Dürer. [more inside]
Two enjoyable chapters [PDF, 33 pages] from the book Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers. "This book does not purport to show you how to create precocious high achievers. It is just one person's story about things he tried with a half-dozen young children."
At the Far Ends of a New Universal Law
The law appeared in full form two decades later, when the mathematicians Craig Tracy and Harold Widom proved that the critical point in the kind of model May used was the peak of a statistical distribution. Then, in 1999, Jinho Baik, Percy Deift and Kurt Johansson discovered that the same statistical distribution also describes variations in sequences of shuffled integers — a completely unrelated mathematical abstraction. Soon the distribution appeared in models of the wriggling perimeter of a bacterial colony and other kinds of random growth. Before long, it was showing up all over physics and mathematics. “The big question was why,” said Satya Majumdar, a statistical physicist at the University of Paris-Sud. “Why does it pop up everywhere?”
Privilege and oppression explained through math - specifically, matrices and Venn diagrams.
Hyperreal numbers: infinities and infinitesimals - "In 1976, Jerome Keisler, a student of the famous logician Tarski, published this elementary textbook that teaches calculus using hyperreal numbers. Now it's free, with a Creative Commons copyright!" (pdf—25mb :) [more inside]
Norbert Wiener: The Eccentric Genius Whose Time May Have Finally Come (Again) - "The most direct reason for Wiener's fall to relative obscurity was the breakthrough of a young mathematician and engineer named Claude Shannon." [more inside]
Math or Maths? A few minutes with Dr Lynne Murphy (an American linguist in England) should clear this right up. Via Numberphile.
Computers are providing solutions to math problems that we can't check - "A computer has solved the longstanding Erdős discrepancy problem! Trouble is, we have no idea what it's talking about — because the solution, which is as long as all of Wikipedia's pages combined, is far too voluminous for us puny humans to confirm." (via; previously ;)
Finite time blowup for an averaged three-dimensional Navier-Stokes equation - "[Terence Tao] has shown that in an alternative abstract universe closely related to the one described by the Navier-Stokes equations, it is possible for a body of fluid to form a sort of computer, which can build a self-replicating fluid robot that, like the Cat in the Hat, keeps transferring its energy to smaller and smaller copies of itself until the fluid 'blows up.' " [1,2,3] (previously)
Network Theory Overview - "The idea: nature and the world of human technology are full of networks! People like to draw diagrams of networks. Mathematical physicists know that in principle these diagrams can be understood using category theory. But why should physicists have all the fun? This is the century of understanding living systems and adapting to life on a finite planet. Math isn't the main thing we need, but it's got to be part of the solution... so one thing we should do is develop a unified and powerful theory of networks." (via ;)
It's a bit late for the holiday, but math(s) comedian Helen Arney sings about her Christmas wish -- the largest known Mersenne Prime, Mersenne 48. [more inside]
Closing in on the twin prime conjecture (Quanta) - "Just months after Zhang announced his result, Maynard has presented an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, to try to combine the collaboration's techniques with Maynard's approach to push this bound even lower." [more inside]
"It's just one of those days where you wake up thinking that if you jazzed up Stravinsky's Owl And The Pussycat it'd be awesome..." [SLYT] [more inside]
In August of last year, mathematician Shinichi Mochizuki reported that he had solved one of the great puzzles of number theory: the ABC conjecture (previously on Metafilter). Almost a year later, no one else knows whether he has succeeded. No one can understand his proof.
Using computer systems for doing mathematical proofs - "With the proliferation of computer-assisted proofs that are all but impossible to check by hand, Hales thinks computers must become the judge." [more inside]
"Draw some random points on a piece of paper and join them up to make a random polygon. Find all the midpoints and connecting them up to give a new shape, and repeat. The resulting shape will get smaller and smaller, and will tend towards an ellipse!" [code to make this in Mathematica] [a version which allows you to watch the process step by step, with 10 vertices or 100]
The Nature of Computation - Intellects Vast and Warm and Sympathetic: "I hand you a network or graph, and ask whether there is a path through the network that crosses each edge exactly once, returning to its starting point. (That is, I ask whether there is a 'Eulerian' cycle.) Then I hand you another network, and ask whether there is a path which visits each node exactly once. (That is, I ask whether there is a 'Hamiltonian' cycle.) How hard is it to answer me?" (via) [more inside]
Morton and Vicary on the Categorified Heisenberg Algebra - "In quantum mechanics, position times momentum does not equal momentum times position! This sounds weird, but it's connected to a very simple fact. Suppose you have a box with some balls in it, and you have the magical ability to create and annihilate balls. Then there's one more way to create a ball and then annihilate one, than to annihilate one and then create one. Huh? Yes: if there are, say, 3 balls in the box to start with, there are 4 balls you can choose to annihilate after you've created one but only 3 before you create one..." [more inside]
FatFonts creates numerical fonts where the amount of ink/pixels for each number is in direct proportion to its value.
In Russian roulette, is it best to go first? | The Mathematics of Tetris | What is the result of infinity minus infinity? [more inside]
Geometrically the irrationality of the square root of 2 means that there is no integer-by-integer square whose area is twice the area of another integer-by-integer square. A visual proof that the square root of 2 is irrational (not found in previous visual proof post.)
Science through yarn: Wooly Thoughts. The Home of Mathematical Knitting, including knitted klein bottles and hyperbolic planes. The Museum of Scientifically Accurate Fabric Brain Art (previously). Much, much, more on knitting, crochet and quilting used to visualize complex theories in topology, probability, chaos and fractals. [more inside]
Beaded Polyhedra ❂ More beadwork (mathematical and otherwise) by Gwen Fisher ❂ Still more beadwork galleries at beAdinfinitum ❂ Three-dimensional finite point groups and the symmetry of beaded beads [pdf - some algebra, but lots of illustrations]
A thread full of proofs without words at MathOverflow and quite a lot more of them courtesy of Google Books.
New math theories reveal the nature of numbers [1,2] - "We prove that partition numbers are 'fractal' for every prime. These numbers, in a way we make precise, are self-similar in a shocking way. Our 'zooming' procedure resolves several open conjectures, and it will change how mathematicians study partitions." (/.|via) [more inside]
The OEIS Movie is simply a slideshow of one thousand plots from the Online Encyclopedia of Integer Sequences, at two plots per second with sequence-generated music. [more inside]
A Brief History of Mathematics is a BBC series of ten fifteen-minute podcasts by Professor Marcus du Sautoy about the history of mathematics from Newton and Leibniz to Nicolas Bourbaki, the pseudonym of a group of French 20th Century mathematicians. Among those covered by Professor du Sautoy are Euler, Fourier and Poincaré. The podcasts also include short interviews with people such as Brian Eno and Roger Penrose.
How do you calculate Pi? Build a supercomputer. The Mountains of Pi, a New Yorker profile of the mathematician (sic) the Chudnovsky brothers. Warning: the article is from 1992, and internet is missing its definite article. (Previously)
Plus magazine has compiled all their articles on mathematics and the arts into one handy-dandy page full of highly enjoyable articles ranging from limericks and screeching violins to the restoration of frescoes.
Since its first printing in 1964, Abramowitz and Stegun's Handbook of Mathematical Functions has been a standard (and public domain) reference manual for special functions and applied mathematics. This week, NIST released its successor, the Digital Library of Mathematical Functions, online to the public.
Mathematics Illuminated is a set of thirteen surveys in varied topics in mathematics, nicely produced with video, text, and interactive Flash gadgets for each of the topics.
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