Building a Computer 1: Numerals - recently my kids have been asking me about how computers work. I like to give in-depth answers to such questions, so we set out on a quest to understand how they work... Follow-up parts 2 3 4 5 6 7 8 9 10 11 12 13 14 15.
The best known packings of equal circles within a circle. Best packings with 1-12 circles. Best packings with 49-60 circles. Best packings with 1093-1104 circles. Also, circles whose areas form a harmonic series. Circles in an isosceles right triangle. Or generate your own circle packings. (Background for beginners: circle packing. Background for experts: circle packing.)
One way to measure corporate fraud is look at reported numbers and see if they follow Benford's law - number sets that are manipulated usually deviate from Benford's law. A recent analysis of all public companies over the past 50 years has shown a steady upward deviation, strongly suggesting there is more corporate fraud now than ever before (peaked in 2008). [more inside]
This is a special collection of problems that were given to select applicants during oral entrance exams to the math department of Moscow State University. These problems were designed to prevent Jews and other undesirables from getting a passing grade. (via Hacker News)
Science! (autoplaying video) The 42nd season of "Sesame Street," which premiered today, will be including a few new educational categories for preschoolers in its usual mix of lessons and parodies: STEM skills — Science, Technology, Engineering and Mathematics. In addition to more scientifically accurate slapstick, characters will try experiments, build bridges and boats, launch rockets and think through problems that require trial and error, observation and data -- all problem areas for America's students. [more inside]
Who's Afraid of the Seven Times Table? Ernst Kummer, one of the great mathematicians of the late 1800s, was hopeless at arithmetic. He was giving an advanced maths lecture and in the middle of a complicated calculation he needed to know what six times seven was. “Um ... six times seven is ... six times seven . . .” A student put up his hand: “41, Professor.” Kummer chalked 41 on the blackboard. “No, no, Professor!” shouted another. “It’s 44!” Kummer gave the students a quizzical look. “Come, come, gentlemen. It can’t be both. It must be either one or the other!” [more inside]
"Perhaps twenty or thirty people in England may be expected to read this book." G.H. Hardy's review of Whitehead and Russell's Principia Mathematica, published in the Times Literary Supplement 100 years ago last week. "The time has passed when a philosopher can afford to be ignorant of mathematics, and a little perseverance will be well rewarded. It will be something to learn how many of the spectres that have haunted philosophers modern mathematics has finally laid to rest."
Systems, networks, and strategies is a math course being developed and taught this semester at the San Francisco Art Institute, by Lee Worden. The course-outline-in-progress is online at the linked wiki, including links to course materials like "the two-in-one-out game," "Places to intervene in a system," on-line flocking simulations, and "street math in graffiti art."
iTex2Img converts LaTeX equations into images online.
Donald in Mathmagic Land is a 27-minute Donald Duck featurette released on June 26, 1959. As Walt Disney said, "We have recently explained mathematics in a film and in that way excited public interest in this very important subject." (Wiki)
Strongly Connected Components is a podcast of interviews with mathematicians. Hear complexity theorist Scott Aaronson (of Shtetl-Optimized), Tom Henderson (of Punk Mathematics) algebraist Olga Holtz of UC-Berkeley, master combinatorist Richard Stanley of MIT, and many more.
Old Theories As Limits of New Ones -- Theoretical physicist, Lubos Motl, takes a brief tour through the history of physics, and explains the simple mathematical relationship of old theories to the theories that replace them.
This short computer graphics animation presents the regular 120-cell: a four dimensional polytope composed of 120 dodecahedra and also known as the hyperdodecahedron or hecatonicosachoron. [more inside]
Lisa Sauermann of Germany has won her fourth gold medal at the International Mathematical Olympiad, making her the top performer in the high school math competition's history. The IMO has been has been run continuously since 1959. Sauermann scored a perfect 42 on this year's exam, the only contestant in the world to do so. Not impressed yet? Here are this year's problems: day 1 and day 2. Watch a bunch of mathematicians wrestle with problem 2 in real time at the polymath blog.
Bret Victor on WorryDream The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols. When most people speak of Math, what they have in mind is more its mechanism than its essence. This "Math" consists of assigning meaning to a set of symbols, blindly shuffling around these symbols according to arcane rules, and then interpreting a meaning from the shuffled result. The process is not unlike casting lots.
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.
Larry Gonick is a veteran American cartoonist best known for his delightful comic-book guides to science and history, many of which have previews online. Chief among them is his long-running Cartoon History of the Universe (later The Cartoon History of the Modern World), a sprawling multi-volume opus documenting everything from the Big Bang to the Bush administration. Published over the course of three decades, it takes a truly global view -- its time-traveling Professor thoroughly explores not only familiar topics like Rome and World War II but the oft-neglected stories of Asia and Africa, blending caricature and myth with careful scholarship (cited by fun illustrated bibliographies) and tackling even the most obscure events with intelligence and wit. This savvy satire carried over to Gonick's Zinn-by-way-of-Pogo chronicle The Cartoon History of the United States, along with a bevy of Cartoon Guides to other topics, including Genetics, Computer Science, Chemistry, Physics, Statistics, The Environment, and (yes!) Sex. Gonick has also maintained a few sideprojects, such as a webcomic look at Chinese invention, assorted math comics (previously), the Muse magazine mainstay Kokopelli & Co. (featuring the shenanigans of his "New Muses"), and more. See also these lengthy interview snippets, linked previously. Want more? Amazon links to the complete oeuvre inside! [more inside]
"As recently as a year ago, there were many publishers, librarians, and scholars who thought that electronic publishing was just a passing fad." In 1996, the number theorist Andrew Odlyzko, a pioneer in the development of "experimental mathematics" via large-scale computation, wrote a article, prescient in many respects, about the effect the Internet would have on the economics of scholarly publication, and on commerce more generally.
"so you can imagine, here I was, an analyst at a hedge fund; it was very strange for me do to something of social value"
Salman Khan: The Messiah of Math - "His free website, dubbed the Khan Academy, may well be the most popular educational site in the world. Last month about 2 million students visited. MIT's OpenCourseWare site, by comparison, has been around since 2001 and averages 1 million visits each month... [more inside]
Calculus Rhapsody Like it says on the tin.
Do you like integer sequences? Do you like poking around in the The On-Line Encyclopedia of Integer Sequences? Do you think, whoa, wait, okay, actually I like integer sequences but the OEIS is a goddam intractable maze of numbers? Do you think, man, what I wish is that someone would make an accessible blog that discusses some of the interesting entries in the OEIS for the casual fan of integer sequences? Well, that's an amazing coincidence; you should take a look at The On-Line Blog of Integer Sequences, by our very own Plutor.
In the afternoon of May 4, 1971, in the Stouffer's Somerset Inn in Shaker Heights, Ohio, Steve Cook presented his STOC paper proving that Satisfiability is NP-complete and Tautology is NP-hard. [more inside]
"Value-added modeling is promoted because it has the right pedigree -- because it is based on "sophisticated mathematics." As a consequence, mathematics that ought to be used to illuminate ends up being used to intimidate." John Ewing, president of Math for America and former executive director of the American Mathematical Society, criticizes the "value-added modeling" approach used as a proxy for teacher quality, most famously in a Los Angeles Times story that called out low-scoring teachers by name. A Brookings Institution paper says value-added modeling is flawed but the best measure we have of teacher value, arguing that the metric's wide fluctuations from year to year are no worse than those of batting averages in baseball. (Though the weakness of that correlation is mostly a BABIP issue.) Can we assign a numerical value to teacher quality? If so, how?
Are graphical calculators pointless? Graphical calculators are required by many college-level math courses, but they don't perform as well as mobile phones. Pedagogically, they may be less useful than a slide rule. [more inside]
This is what pi sounds like. At least, that's one person's interpretation. There are certainly plenty of others, including touchtone pi, hammered dulcimer pi, violin pi, smooth techno pi, crazy awesome pi, vaguely unsettling pi (sounds best with headphones), and lots of piano pi. Pi has even done a duet with its buddy e. Nothing here that tickles your fancy? Think you could do better? Why not make your own pi song? Hell, make two! If you're having trouble remembering all those pesky digits, don't worry: there's a song for that, too.
(pi as music previously on metafilter)
(pi as music previously on metafilter)
So you're me and you're in math class and you're learning about graph theory, a subject too interesting to be included in most grade school's curricula so maybe you're in some special program or maybe you're in college and were somehow not scarred for life by your grade school math teachers. [more inside]
Rediscovering WWII's female "computers". While researching a documentary in Philadelphia, filmmaker LeAnn Erickson came across two women with a story she'd never heard before: thousands of women with advanced mathematical skills employed as "computers", working day and night during WWII to supply soldiers in the field with precise ballistics algorithms. Some of those women also went on to program ENIAC, the first general-purpose computer (previously). Erickson turned their stories into Top Secret Rosies, a documentary released to theaters last year and to DVD this month. One of those programmers, Betty Jean Jennings Bartik, spoke at length to the Computing History Museum in 2008. [youtube, 1:07:19] [via]
Snowdecahedron. When life hands you a blizzard, make a Platonic solid. "Temporary public art" from Dan Sternof Beyer.
Horizon asks "What is reality?" -- youtube for links for those outside the UK: 1, 2, 3, 4. It's a hard question. To help you answer it, Stanford has a set of free courses available on line by Leonard Susskind: General Relativity, Cosmology, New Revolutions in Particle Physics, Quantum Entanglement, Special Relativity, Classical Mechanics, Statistical Mechanics, The Standard Model. (Each link is to lecture 1 of a full college course of a dozen or so lectures.) If you need help with the math, the Khan Academy should help get you up to speed.
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]
An open source, html5 based graphing and computation engine does in your browser what is usually outsourced to the cloud. It graphs, solves, simplifies, integrates and differentiates expressions, and needs no internet connection once you load the page in your browser (or save it on your computer). RTFM.
Google is known to ask the following question in job interviews: In a country in which people only want boys every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the proportion of boys to girls in the country? Think you know the answer? If so, Steve Landsburg may be willing to bet you up to $5000. [more inside]
"Normal" human pregnancies last 40 weeks, right? Well, no; they can vary quite a bit by the mother's race, age, number of previous children, family history of delivering early or late, home state, work habits, and even the fetus' HLA type. So where does that "40 week" thing come from? Oh, dear. So check out this super-nerdy pregnancy statistics website, from an engineer mom who is collecting data from the public (see the raw data and auto-generated graphs, and read the FAQ about the survey, with more cool graphs). Looking for day-by-day probabilities on when that baby's due? This would be your stats table with daily prediction (adjust dates at top of page as needed). Of course, you could always shut up your constantly inquiring relatives and friends another way.
"A few weeks ago, my algebra class was assigned a project called “Mathematic Karaoke,” for which were told to pick a song, make it about numbers (and stuff), and record ourselves singing it. [....] Of course, Single Ladies was my tune of choice."
Measure-theoretic probability: Why it should be learnt and how to get started. The clickable chart of distribution relationships. Just two of the interesting and informative probability resources I've learned about, along with countless other tidbits of information, from statistician John D. Cook's blog and his probability fact-of-the-day Twitter feed ProbFact. John also has daily tip and fact Twitter feeds for Windows keyboard shortcuts, regular expressions, TeX and LaTeX, algebra and number theory, topology and geometry, real and complex analysis, and beginning tomorrow, computer science and statistics.
Let's say you're me and you're in math class, and you're supposed to be learning about factoring. Trouble is, your teacher is too busy trying to convince you that factoring is a useful skill for the average person to know with real-world applications ranging from passing your state exams all the way to getting a higher SAT score and unfortunately does not have the time to show you why factoring is actually interesting. It's perfectly reasonable for you to get bored in this situation. So like any reasonable person, you start doodling.[more inside]
In today's example of kids smarter than you and I, Wired follows the exploits of two teens competing at the International Olympiad in Informatics.
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.
The Geometry of the Snail Ball [pdf] - an interesting article (with some DIY advice at the end) about a toy shop curiosity you may have encountered.
Kaggle hosts competitions to glean information from massive data sets, a la the Netflix Prize. Competitors can enter free, while companies with vast stores of impenetrable data pay Kaggle to outsource their difficulties to the world population of freelance data-miners. Kaggle contestants have already developed dozens of chess rating systems which outperform the Elo rating currently in use, and identified genetic markers in HIV associated with a rise in viral load. Right now, you can compete to forecast tourism statistics or predict unknown edges in a social network. Teachers who want to pit their students against each other can host a Kaggle contest free of charge.
If you look around, you'll see that the ratio of 1.618:1 appears in architecture, nature, and artistic works (such as music, previously). Studied by the Greeks, the Golden Ratio is pretty much everywhere and is common accepted as aesthetically pleasing, and now it has been found to exist down into the nanoscale level, as a byproduct of investigating the Heisenberg Uncertainty Principle. We may not be able to nail down both position and speed, but it appears the macro ratio is an echo of the micro one. [more inside]