Halloween Math Lecture
Professor Matthew Weathers went the extra mile for his math lecture Wednesday at Biola University. Who says math isn’t fun? -via Cynical-C
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Things Mathematicians See at the Movies
Most moviegoers don’t notice the math in popular films, but it’s there if you know what to look for. For example, one mathematician compared the spread of zombies to that of infectious diseases.
The problem of zombies intrigued Philip Munz of Carleton University and his colleagues at the University of Ottawa, who recently wrote a scientific paper quantifying various properties of zombie epidemics. Standard modeling techniques for disease outbreaks weren’t quite sufficient, the authors found. “The key difference between the models presented here and other models of infectious disease,” they wrote, “is that the dead can come back to life.”
After a thorough, if tongue-in-cheek, analysis, the authors found that the optimal method for halting such epidemics involves killing zombies early and often – the rare scientific paper that satisfies both the splatter-film aficionado and the Centers for Disease Control.
Other math questions come up in The Dark Knight, Harry Potter and the Half-Blood Prince, and other films you are familiar with. Link -via Buzzfeed
(image credit: Flickr user joelf)
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The Math Book: Milestones in the History of Math
I love math (though it's debatable whether math loves me back, I suspect not) so it's a pleasure to read Cliff Pickover's newest creation, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics.
Don't let the title fool you - The Math Book is a thoroughly enjoyable "walk" through the history of mathematics with each milestone narrated by Pickover in a short and sweet fashion (and surprisingly, with very little equations) that even non-mathemagicians like myself can enjoy. If you've ever heard the terms Bessel functions, Transcendental numbers, and Riemann hypothesis, and want to know more, then this is the book for you.
Below is an excerpt from the book (selecting which ones to show was a hard thing to do - there were just so many interesting articles!):
Cicada-Generated Prime Numbers
Cicadas
are winged insects that evolved around 1.8 million years ago during the
Pleistocene epoch, when glaciers advanced and retreated across North America.
Cicadas of the genus Magicicada spend most of their lives below
the ground, feeding on the juices of plant roots, and then emerge, mate,
and die quickly. These creatures display a startling behavior: Their emergence
is synchronized with periods of years that are usually the prime numbers
13 and 17. (A prime number is an integer such as 11, 13, and 17 that has
only two integer divisors: 1 and itself.) During the spring of their 13th
or 17th year, these periodical cicadas construct an exit tunnel. Sometimes
more than 1.5 million individuals emerge in a single acre; this abundance
of bodies may have survival value as they overwhelm predators such as
birds that cannot possibly eat them all at once. (Photo: Joelmills [Wikipedia])
Some researchers have speculated that the evolution of prime-number life cycles occurred so that the creatures increased their chances of evading shorter-lived predators and parasites. For example, if these cicadas had 12-year life cycles, all predators with life cycles of 2, 3, 4, or 6 years might more easily find the insects. Mario Markus of the Max Planck Institute for Molecular Physiology in Dortmund, Germany, and his coworkers discovered that these kinds of prime-number cycles arise naturally from evolutionary mathematical models of interactions between predator and prey. In order to experiment, they first assigned random life-cycle durations to their computer-simulated populations. After some time, a sequence of mutations always locked the synthetic cicadas into a stable prime-number cycle.
Of course, this research is still in its infancy and many questions remain. What is special about 13 and 17? What predators or parasites have actually existed to drive the cicadas to these periods? Also, a mystery remain as to why, of the 1,500 cicada species worldwide, only a small number of the genus Magicicada are known to be periodical.
Borromean Rings
(L) Borromean Rings;
(M) Valknut, or three
interlocked triangles, on the Stora Hammar Stone; (R) Molecular
Borromean Rings by J. Fraser SToddart
Peter Guthrie Tait (1831 - 1901) - A simple yet intriguing set of interlocking objects of interest to mathematicians and chemists is formed by Borromean rings - three mutually interlocked rings named after the Italian Renaissance family who used them on its coat of arms in the fifteenth century. (Image: Theon [Wikipedia])
Notice that Borromean rings have no two rings that are linked, so if we cut any one of the rings, all three rings come apart. Some historians speculate that the ancient ring configurations once represented the three families of Visconti, Sforza, and Borromeo, who formed a tenuous union through intermarriages. The rings also appear in 1467 in the Church of San Pancrazio in Florence. Even older, triangular versions were used by the Vikings, one famous example of which was found on a bedpost of a prominent woman who died in 834.
The rings appear in mathematical context in the 1876 paper on knots by Scottish mathematical physicist Peter Tait. Because two choices (over or under) are possible for each ring crossing, 26 = 64 possible interlaced patterns exist. If we take symmetry into account, only 10 of these patterns are geometrically distinct.
Mathematicians now know that we cannot actually construct a true set of Borromean rings with flat circles, and in fact, you can see this for yourself if you try to create the interlocked rings out of wire, which requires some deformation or kinks in the wires. In 1987, Michael Freedman and Richard Skora proved the theorem stating that Borromean rings are impossible to construct with flat circles.
In 2004, UCLA chemists created a molecular Borromean ring compound that was 2.5 nanometers across and that included six metal ions. Researchers are currently contemplating ways in which they may use molecular Borromean rings in such diverse fields as spintronics (a technology that exploits electron spin and charge) and medical imaging.
Golden Ratio
Fra Luca Bartolomeo de Pacioli (1445 - 1517) - In 1509, Italian mathematician
Luca Pacioli, a close friend of Leonardo da Vinci, published Divina
Proportione, a treatise on a number that is now widely known as the
"Golden Ratio." This ratio, symbolized by 
,
appears with amazing frequency in mathematics and nature. We can understand
the proportion most easily by dividing a line into two segments so that
the ratio of the whole segment to the longest part is the same as the
ratio of the longer part to the shorter part, or (a+b)/b = b/a = 1.61803
...
If the lengths of the sides of a rectangle are in the golden ratio, then the rectangle is a "golden rectangle." It's possible to divide a golden rectangle into a square and a golden rectangle. Next, we can cut the smaller golden rectangle into a smaller square and golden rectangle. We may continue this process indefinitely, producing smaller and smaller golden rectangles.
If we draw a diagonal from the top right of the original rectangle to the bottom left, then from the bottom right of the baby (that is, the next smaller) golden rectangle to the top left, the intersection point shows the point to which all the baby golden rectangles converge. Moreover, the lengths of the diagonals are in golden ratio to each other. The point to which all the golden rectangles converge is sometimes called the "Eye of God."
The golden rectangle is the only rectangle from which a square can be cut so that the remaining rectangle will always be similar to the original rectangle. If we connect the vertices in the diagram, we approximate a logarithmic spiral that "envelops" the Eye of God. Logarithmic spirals are everywhere - seashells, animal horns, the cochlea of the ear - anywhere that nature needs to fill space economically and regularly. A spiral is strong and uses a minimum of materials. While expanding, it alters its size but never its shape.
Benford's Law
Simon
Newcomb (1835 - 1909), Frank Benford (1883 - 1948) - Benford's Law, also
called the first-digit law or leading-digit phenomenon, asserts that in
various number lists, the digit 1 tends to occur in the leftmost position
with probability of roughly 30 percent, much greater than the expected
11.1 percent that would result if each digit occurred with a 1 to 9 probability.
Benford's law can be observed, for instance, in tables that list populations,
death rates, stock prices, baseball statistics, and the area of rivers
and lakes. Explanations for this phenomenon are very recent. (Photo from
Mark J. Nigrini)
Benford's law is named after Dr. Frank Benford, a physicist at the General Electric Company who publicized his work in 1938, although it had been previously discovered by mathematician and astronomer Simon Newcomb in 1881. Pages of logarithms, with numbers starting with the numerals 1 are said to be dirtier and more worn by other pages, because the number 1 occurs as the first digit about 30 percent more often than any other. In numerous kinds of data, Benford determined that the probability of any number n from 1 through 0 being the first digit is log10 (1 + 1/n). Even the Fibonacci sequence - 1, 1, 2, 3, 5, 8, 13 - follows Benford's law. Fibonacci numbers are far more likely to start with "1" than any other digit. It appears that Benford's law applies to any data that follows a "power law." For example, large lakes are rare, medium-size lakes are more common, and small lakes are even more common. Similarly, 11 Fibonacci numbers exist in the range 1 - 100, but only one in the next three ranges of 100 (101 - 200, 201- 300, 301- 400)
Benford's law has often been used to detect fraud. For example, accounting consultants can sometimes use the law to detect fraudulent tax returns in which the occurrence of digits does not follow what would be expected according to Benford's law.
Menger Sponge
Menger Sponge by Jeannine Mosely, at the Institute
for Figuring. Photo: Ravi Apte
Karl Menger (1902 - 1985) - The Menger sponge is a fractal object with an infinite number of cavities - a nightmarish object for any dentist to contemplate. The object was first described by Austrian mathematician Karl Menger in 1926. To construct the sponge, we begin with a "mother cube" and subdivide it into 27 identical smaller cubes. Next, we remove the cube in the center and the six cubes that share faces with it. This leaves behind 20 cubes. We continue to repeat the process forever. The number of cubes increases by 20n, where n is the number of iterations performed on the mother cube. The second iteration gives us 400 cubes, and by the time we get to the sixth iteration, we have 64,000,000 cubes.
Each face of the Menger sponge is called a Sierpinski carpet. Fractal antennae based on the Sierpinski carpet are sometimes used as efficient receivers of electromagnetic signals. Both the carpets and the entire cube have fascinating geometrical properties. For example, the sponge has an infinite surface area while enclosing zero volume.
According to the Institute for Figuring, with each iteration, the Sierpinski carpet face "dissolves into a foam whose final structure has no area whatever yet possesses a perimeter that is infinitely long. Like the skeleton of a beast whose flesh has vanished, the concluding form is without substance - it occupies a planar surface, but no longer fills it." This porous remnant hovers between a line and a plane. Whereas a line is one-dimensional and a plane two-dimensional, the Sierpinski carpet has a "fractional" dimension of 1.89. The Menger sponge has a fractional dimension (technically referred to as the Hausdorff Dimension) between a plane and a solid, approximately 2.73, and it has been used to visualize certain models of a foam-like space-time. Dr. Jeannine Mosely has constructed a Menger sponge model from more than 65,000 business cards that weights about 150 pounds (70 kilograms).
The Quest for Lie Group E8
E8 graph as a 2-dimensional projection, by Peter McMullen
(image by Claudio Rocchini [wikipedia])
Marius Sophus Lie (1842 - 1899), Wilhelm Karl Joseph Killing (1847 - 1923) - For more than a century, mathematicians have sought to understand a vast, 248-dimensional entity, known to them only as E8. Finally, in 2007, an international team of mathematicians and computer scientists made use of a supercomputer to tame the intricate beast.
As background, consider the Mysterium Cosmographicum (The Sacred Mystery of the Cosmos) of Johannes Kepler (1571 - 1630), who was so enthralled with symmetry that he suggested the entire solar system and planetary orbits could be modeled by Platonic Solids, such as the cube and dodecahedron, nestled in each other forming layers as if in a gigantic crystalline onion. These kinds of Keplerian symmetries were limited in scope and number; however, symmetries that Kepler could have hardly imagined may indeed rule the universe.
In
the late nineteenth century, the Norwegian mathematician Sophus Lie (pronounced
"Lee") studied objects with smooth rotational symmetries, like
the sphere or doughnut in our ordinary three-dimensional space. In three
and higher dimensions, these kinds of symmetries are expressed by Lie
groups. The German mathematician Wilhelm Killing suggested the existence
of the E8 group in 1887. Simpler Lie groups control the shape
of electron orbital and symmetries of subatomic quarks. Larger groups,
like E8, may someday hold the key to a unified theory of physics
and help scientist understand string theory and gravity.
Fokko du Cloux, a Dutch mathematician and computer scientist who was one of the E8 team members, wrote the software for the supercomputer and pondered the ramifications of E8 while he was dying of amyotrophic lateral sclerosis and breathing with a respirator. He died in November 2006, never living to see the end of the quest for E8.
On January 8, 2007, a supercomputer computed the last entry in the table for E8, which describes the symmetries of a 57-dimensional object that can be imagined as rotating in 248 ways without changing its appearance. The work is significant as an advance in mathematical knowledge and in the use of large-scale computing to solve profound mathematical problems.
Mathematical Universe Hypothesis
Max
Tegmark (b. 1967) - In this book, we have encountered various geometries
that have been thought to hold the keys to the universe. Johannes Kepler
modeled the solar system with Platonic Solids such as the dodecahedron.
Large Lie groups, like E8, may someday help us create a unified
theory of physics. Even Galileo in the seventeenth century suggested that
"nature's great book is written in mathematical symbols." In
the 1960s, physicist Eugene Wigner was impressed with the "unreasonable
effectiveness of mathematics in the natural sciences." (Photo: MIT
Physics Faculty website)
In 2007, Swedish-American cosmologist Max Tegmark published scientific and popular articles on the Mathematical Universe Hypothesis (MUH) that states that our physical reality is a mathematical structure and that our universe is not just described by mathematics - it is mathematics. Tegmark is a professor of physics at the Massachusetts Institute of Technology and scientific director of the Foundational Questions Institute. He notes that when we consider equations like 1 + 1 = 2, the notations for the numbers are relatively unimportant when compared to the relationship that are being described. He believes that "we don't invent mathematical structures - we discover them, and invent only the notation for describing them."
Tegmark's hypothesis implies that "we all live in a gigantic mathematical object - one that is more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names like Calabi-Yau manifolds, tensor bundles, and Hilbert spaces, which appear in today's most advanced theories. Everything in our world is purely mathematical - including you." If this idea seems counterintuitive, this shouldn't be surprising, because many modern theories, like quantum theory and relativity, can defy intuition. As mathematician Ronald Graham once said, "Our brain have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions."
__________
Cliff
Pickover is a prolific author, having published more than 40 books, translated
into over a dozen languages, on topics ranging from science and mathematics
to religion, art, history, computers and creativity, human intelligence,
higher dimensions, time travel, and science fiction. He received his Ph.D.
from Yale University's Department of Molecular Biophysics and Biochemistry,
holds over 50 U.S. patents, and is an associate editor for several scientific
journals. His computer graphics have appeared on the cover of numerous
magazines, and his research has received considerable attention from media
outlets ranging from CNN and WIRED to The New York Times. His website,
pickover.com, receives millions
of visits.
Links: The
Math Book website | The
Math Book on Amazon
__________
Previously on Neatorama: 5 Scientific Laws and the Scientists Behind Them
Math T-shirts from the Neatorama Store:
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Why It’s Called Pi
It all makes perfect sense now. I don’t know the original source for this image. If I find who did it, I’ll celebrate by calculating the area of a blackberry pie. -via Digg
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Researchers Break Pi Calculation Record
Researchers at the University of Tsukuba in Japan have broken the record for the number of calculated digits of the constant pi:
The T2K Tsukuba System is a 640-computer cluster with a processing speed of 95 trillion floating-point operations per second. The T2K calculated a total of 2,576,980,377,524 decimal places in 73 hours 36 minutes, which is a small fraction of the 600 hours taken by the previous record holders—Hitachi and the University of Tokyo—who calculated only 1.2 trillion places.
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How Long Does Bill Murray Spend in Groundhog Day?
The burning question everyone (or at least those who have seen the movie) asks: how long does Bill Murray spend trapped in a time loop in the movie Groundhog Day? Wolf Gnards took clues from the film to come up with a definitive answer of 8 years, 8 months, and 16 days. I’m glad they figured that one out. Follow the link to find out how they did it. Link -via Digg
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Math Mom: If I've Told You N Times, I've Told You N+1 Times
Katie would make one great math mom!
Math Mom: If I’ve Told You N Times, I’ve Told You N+1 Times
I can practically hear my mother’s voice saying "if I told you once, I’ve told you a thousand times." Well, for this shirt, we’ve boiled the adage down to its mathematical terms.
Now buy the shirt, stand up straight and go clean your room! Link
More math and geektastic science T-shirts from the newly spruced up Neatorama Online Store (work in progress, mmkay?):
- Integral of 1/Cabin = Log Cabin
- Geometry is For Squares
- Math Puns are the First Sine of Madness
- I Love Math (in Queen’s English)
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How Do You Pronounce Psi?
Brown Sharpie is an excellent mathematical cartoon series by Courtney Gibbons – it’s like xkcd, but much brainier 
This one above is a classic about how to pronounce the Greek letter psi: Link
And by the way, my graduate thesis was on a yeast prion called PSI+, and I say "psigh" which would make me tainted by a pretentious physicist.
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Odd Day
Ron Gordon, the California teacher who founded and promoted Square Root Day reminds us that Thursday is another math holiday, Odd Day! The calendar date (as written by people in the US) will be 5-7-9, which only happens once a century. Odd Days happen six times a century. It’s a day to take the opportunity to do something odd. In celebration, there’s another contest, with $579 up for prizes. Get all the details at the Odd Day website. Link -Thanks, Ron!
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Chicks Are No Dummies: They Can Do Math!
Photo: Rugani et al.
They may be just 3 or 4 days old, but chicks can already do simple arithmetic. Inspired by experiments with human babies, Rosa Rugani of the University of Trento Center for Mind/Brain Science in Italy and colleagues decided to test chick’s mathematic abilities:
… Rugani and her colleagues worked out tests based on adding objects to and taking them away from little piles behind screens. With no special math coaching, the chicks did a decent job of keeping track of object shifts representing such problems as 4 – 2 = 2 and 1 + 2 = 3, she and her colleagues report online March 31 in Proceedings of the Royal Society B.
“This is the first demonstration of adding and subtracting in young animals” other than humans, Rugani says. Other animals, including some primates and dogs, have demonstrated numerical powers as adults.
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Chocolate Helps You Do Better in Math
Bad at math? Try chocolate! At least that’s what researchers Emma Wightman, David Kennedy and colleagues found in the latest (and yummiest) scientific study:
Mental arithmetic became easier after volunteers had been given large amounts of compounds found in chocolate, called flavanols, in a hot cocoa drink.
They were also less likely to feel tired or mentally drained, the findings, presented at the British Psychological Society annual conference in Brighton show.
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More New Math
Craig Damrauer uses math equations to illustrate concepts that aren’t math. For example:
Cleanliness = Godliness – 1
Crazy = Talking to oneself – ( cell phone + ear piece )
Nagging = reminding + reminding + reminding
Link -via Metafilter
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Sierpinkski Lace
Math meets home decor in this tatted Sierpinski triangle by mathematician Ted Ashton! The tiny lace triangles became fractals as they are connected. Link -via Evil Mad Scientist Laboratories
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Happy Pi Day!
March 14th (3/14) is National Pi Day, as officially designated by the US congress this year. Traditional festivities are to study the mathematical constant pi in school (which took place yesterday since 3/14 fell on Saturday this year), bake and eat a pie, and sing Pi Day Carols. Evil Mad Scientist Laboraties constructed a Pi Pie Trivet for the occasion, and posted instructions for making your own. Link
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3/3/09: Celebrate Square Root Day!
When the month and day form the square root of the year, math geeks celebrate a special holiday. Well, presumably. I never made it past precalculus, so I’m only going by what the Associated Press says:
The math-buffs’ holiday, which only occurs nine times each century, falls on Tuesday — 3/3/09 (for the mathematically challenged, three is the square root of nine).
“These days are like calendar comets, you wait and wait and wait for them, then they brighten up your day — and poof — they’re gone,” said Ron Gordon, a Redwood City teacher who started a contest meant to get people excited about the event.
The winner gets, of course, $339 for having the biggest Square Root Day event.
Gordon’s daughter even set up a Facebook page — one of a half-dozen or so dedicated to the holiday — and hundreds of people had signed up with plans to celebrate in some way. Celebrations are as varied: Some cut root vegetables into squares, others make food in the shape of a square root symbol.
How will you celebrate Square Root Day?
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Why Learning Math is Important
Milk at the this Wal*Mart store is $2.25 a gallon, or 2 for $5. A half-gallon of milk is $2.47. What would your purchase be? Link -via Bits and Pieces
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Why You Should Learn Math: Mathematicians Have the Best Job
Sarah Needleman of The Wall Street Journal wrote an interesting article about a new CareerCast.com study from Les Krantz, author of Jobs Rated Almanac, about the best and worst jobs in the U.S.
The study evaluated 200 jobs according to environment, income, employment outlook, physical demand and stress. The data are from the US Bureau of Labor Statistics and the Census Bureau, amongst others:
According to the study, mathematicians fared best in part because they typically work in favorable conditions — indoors and in places free of toxic fumes or noise — unlike those toward the bottom of the list like sewage-plant operator, painter and bricklayer. They also aren’t expected to do any heavy lifting, crawling or crouching — attributes associated with occupations such as firefighter, auto mechanic and plumber.
The study also considers pay, which was determined by measuring each job’s median income and growth potential. Mathematicians’ annual income was pegged at $94,160, but Ms. Courter, 38, says her salary exceeds that amount.
Neatorama, of course, already know that Math rocks. Check out our I Love Math T-shirt.
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The Holy Grail of Rock, "A Hard Day's Night" Chord, Solved by Math
Take that, sweet mystery of rock ‘n roll. Math has just solved the Holy Grail of Rock: the mysterious "A Hard Day’s Night" chord.
Dalhousie University math professor Jason Brown applied Fourier transform to solve the Beatles’ riddle: there was a mystery piano!
… the frequencies he found didn’t match the known instrumentation on the song. “George played a 12-string Rickenbacker, Lennon had his six string, Paul had his bass…none of them quite fit what I found,” he explains. “Then the solution hit me: it wasn’t just those instruments. There was a piano in there as well, and that accounted for the problematic frequencies.”
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The 13 Most Famous Numbers and Their Stories
When I saw the title, I thought this might be a comedy posting, but it really is the stories of famous numbers. The pictured number is called the “golden section”.
This number [represented by the symbol at left] is also known as the golden section and is commonly accepted as an expression that describes the perfect proportions in architecture or anatomy. In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is a mathematical constant, approximately 1.6180339887.
Of course, not all of them are this serious, but they are fascinating. Link -via the Presurfer
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