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]
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]
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]
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 ;)
Discovering Free Will (Part II, Part III) - a nice discussion of the Conway-Kochen "Free Will Theorem". [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.
"Maybe it's a sore point: your field should have an answer (people think you do) but there isn't one yet. Perhaps it's simple to pose but hard to answer. Or it's a question that belies a deep misunderstanding: the best answer is to question the question."
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]
The BBC reports that "Beer drinkers in the US have filed a $5m (£3.3m) lawsuit accusing Anheuser-Busch of watering down its beer." The lawsuits are based on information from former employees at breweries owned by the multinational. [more inside]
It's Saturday; why not think about the pigeonhole principle? Here are problems and more problems and what you might call a problem with the principle itself as it is often stated.
The Royal Society of Chemistry is offering £1000 to the person or team producing the best and most creative explanation of the phenomenon, known today as The Mpemba Effect [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.)
The Angel Problem. The Angel and the Devil play a game on an infinite chess board...
A thread full of proofs without words at MathOverflow and quite a lot more of them courtesy of Google Books.
The First Zombie-Proof House: "The Safe House" was designed by KWK Promes, and completely folds in on itself to become an impenetrable concrete bunker, keeping you safe from the undead.
"Surely you can question my policies without questioning my faith. Or for that matter my citizenship." - President Obama
Handwritten 1961 memo in father's immigration file notes Obama born in Hawaii. "Documents obtained from the United States Citizenship and Immigration Service through a Freedom of Information Act request offer evidence that President Barack Obama was born in Hawaii. A memo dated Aug. 31, 1961 from William Wood of Immigration and Naturalization Services indicates that Barack Obama, Sr., was attending the University of Hawaii on a student visa and that a son, Barack Obama, II, was born in Honolulu on Aug, 8, 1961." [Image of Memo]
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]
Futurama has always been a haven for geek humor, but last week's episode "The Prisoner of Benda" pushed things to the next level. First hinted at in an American Physical Society interview with showrunner David X. Cohen (previously), staff writer and mathematics Ph.D. Ken Keeler devised a novel mathematical proof based on group theory to resolve the logic puzzle spawned by the episode's brain-swapping (but no backsies!) conceit. Curious how it works? Read the proof (in the show or in plain text), then see it in action using this handy chart. Too much math for a lazy Sunday? Then entertain your brain with lengthy clips from the episode -- including two of the funniest moments in the series in the span of two minutes.
I am pleased to announce a proof that P is not equal to NP. In this paper, Vinay Deolalikar (HP Labs) proposes a proof to answer the most important problem in its field of mathematics. [more inside]
Our results open a fascinating new direction for position-based security in cryptography where security of protocols is solely based on the laws of physics and proofs of security do not require any pre-existing infrastructure.
Did you know that you can create a simple set of directions to your house that works no matter where the recipient starts from? After 38 years this remarkable conjecture has now been proved by a 63-year old former security guard.
The Duke lacrosse rape case hurtled toward perhaps sinister motives last week with testimony from the head of the private DNA lab prosecutor Nifong hired to test the rape kit samples taken from the accuser. Brian Meehan revealed that not only had his lab found DNA samples from five unknown men, none of whom were Duke lacrosse players, Meehan had also agreed with Nifong not to put that info in the DNA Security's final report. Were it not for the fact that the three defendants have counsel capable of pouring over thousands of pages of technical documents, this vital, exculpatory evidence would have gone unnoticed. Previous opinions in MeFi.
Proofs and Pictures: The Role of Visualization in Mathematical and Scientific Reasoning [video] "The picture is a telescope for looking into Plato's heaven." -- James Brown [cached]
Mercury News D-12's Proof, shot dead at 32. Proof stayed close to give back what he owed - and paid with his life.
There is a website that successfully argues for the existence of God.Here is the URL.Therefore, God exists.