by Joseph Cliburn, Dept. of Institutional Research/Planning, Mississippi Gulf Coast Community College,

Starting from a statement brought home by Bob Dylan [1965aL we estimate the value of Love using basic algebra of need [Mottram, 1965], perhaps some calculus, maybe a bit of the geometry of innocence [Dylan, 1965f], and a lot of wishful thinking.

The Limits of Love

We begin with the following assertion by Dylan [1965a]:

(Love – 0) / No Limit (1)

using the expression on the record label in preference to the statement on the back cover [1965b], and taking a cue from the author’s statement that it is a fraction [1965c]. Setting aside the question of whether the use of an expression here marks Dylan as an Expressionist, we set the expression equal to X, which is unspecified for the moment, and solve for Love:

x = (Love – 0) / No Limit (2)

Thus:

(No Limit) X = Love – 0 = Love (3)

where we’ve made use of the fact that for any A, A – 0 = A. Thus Love = something times “No Limit.” The traditional quantity that has no limit is infinite, thus we get Love is infinite, assuming that X is finite. If X is 0, we have 0 times infinity, which is indefinite.

Signs of Love

However, if X is negative, or “Less than Zero” [Costello, 1977], we get the result that Love is infinitely negative. This is perhaps enough negativity to succeed when gravity fails you [Dylan, 1965dL and will probably get the reader down. We may allow (no limit) to be negative, in which case we'll want either both X and (no limit) to be positive at the same time or both negative.

Other than the sign of X [Dylan, 1967aL however, there is nothing specified about it. If X is complex, then it has a real part that acts as above and an imaginary part, in which case (No Limit) times X is also complex, which makes Love both complex and partly imaginary [Whitfield-Strong, 196?]. Dylan himself has explored this idea extensively in later investigations [1975a, 1975bL with extensive revisions [1984, 1974/1993, various public presentations since 1975].

At any rate, we can conclude definitely [Anderson, 1982] that:

X=X

We thus sum up by offering the following observations:

1. Love is infinite if X is finite.
2. Love is indefinite if X is zero.
3. Love is infinitely negative if X is negative.
4. Love is imaginary if X is imaginary.

Fractal Love is Problematic

There remain some questions regarding the appropriateness of using fractal mathematics to resolve these problems, e.g., “i accept chaos. i am not sure whether it accepts me” [Dylan, 1965e]. But we should also clarify that we are not putting infinity up on trial [Dylan, 1966] here. Love is, after all, just a four-letter word [Dylan, 1967b].

References

Anderson, L., 1982, “Let X = X,” Big Science, (Warner Brothers, Burbank CA).
Costello, E., 1977, “Less Than Zero,” My Aim Is True, 2nd ed., (Columbia, New York NY).
Dylan’s 1975 research reDylan, B., 1965a, “(Love – 0) /No Limit,” Subterranean Homesick Blues, (Columbia, New port Blood On the Tracks. New York NY).
Dylan, B., 1965b, “Love – O/No Limit,” Subterranean Homesick Blues, back cover, (Columbia’, New York NY).
Dylan, B., 1965c, broadcast communication.
Dylan, B., 1965d, “Just Like Tom Thumb’s Blues,” Highway 61 Revisited, (Columbia, New York NY).
Dylan, B., 1965e, liner not~s, Highway 61 Revisited, (Columbia, New York NY).
Dylan, B., 1965f, “Tombstone Blues,” Highway 61 Revisited, (Columbia, New York NY).
Dylan, B., 1966, “Visions of Johanna,” Blonde on Blonde, (Columbia, New York NY).
Dylan, B., 1967a, “Sign on the Cross,” Writings and Drawings, (Random House, New York NY).
Dylan, B., 1967b, “Love Is Just A Four-Letter Word,” Writings and Drawings, (Random House, New York NY).
Dylan, B., 1974/1993, “Tangled Up In Blue,” The Bootleg Series, vol. 2, (Columbia, New York NY).
Dylan, B., 1975a, “Simple Twist of Fate,” Blood On the Tracks, (Columbia, New York NY).
Dylan, B., 1975b, “Tangled Up In Blue,” Blood On the Tracks, (Columbia, New York NY).
Dylan, B., 1978,” ,” Street Legal, (Columbia, New York NY).
Dylan, B., 1984, “Tangled Up In Blue,” Real Live, (Columbia, New York NY).
Mottram, E., 1965, William Burroughs: The Algebra of Need.
Whitfield-Strong, 196?, “Just My Imagination,” as reviewed in R. Stones, 1978, Some Girls, (Atlantic, New York NY).

_____________________

This article is republished with permission from the September-October 1995 issue of the Annals of Improbable Research. You can download or purchase back issues of the magazine, or subscribe to receive future issues. Or get a subscription for someone as a gift!

Visit their website for more research that makes people LAUGH and then THINK.

Ok, maybe not hell per se, but Silent Hill is pretty darn close. Anyway, I sure wouldn’t want to learn anything from Pyramid Head. Who knows what he would do when you get a wrong answer?

Link Via Kotaku

Vi Hart has a bone to pick with Nickelodeon, in that the show Spongebob Squarepants does not represent the world the way it really is to children. Does she complain about the talking kitchen sponge who wears pants? The squid who runs an underwater hamburger stand? The squirrel in scuba gear? No, it’s the pattern drawn in the pineapple that Spongebob lives in. -via Metafilter

With this equation, of course:

Graph that and you'll get this:

via Krulwich Wonders

See also: I Heart Math T-Shirt over at the NeatoShop

(Video Link)

Wanna play Skyrim at school without getting in trouble? Well now you can play on your TI-84 graphing calculator. Sure it might not be as fun as the original, but hey, how often can you play your favorite game in the middle of class?

Link Via Kotaku

When designing their newest hotel to be built in downtown Kansas City, the fine people at Hyatt Regency wanted all the bells and whistles in it. The architectural firm in charge of the building design came up with a series of aerial walkways suspended from the ceiling so that guests could people-watch from a heightened vantage point. All in all, it was a pretty nifty feature. Until it suddenly collapsed and killed more than a hundred people.

Now they know what design flaw caused it, and my mouth dropped open to see how simple it was. Read the rest of the story and others at Cracked. Link -via Digg

Who ya callin' birdbrained? Researchers at the University of Otago in New Zealand found that the lowly pigeon is actually quite brilliant at math:

Scarf and her colleagues began their search for mathematical ability in pigeons by training subject birds to recognize groups of one, two or three objects on a screen and peck at them in proper numerical sequence. This, admittedly, was not an easy lesson to get across. It took about a year of practice and rewards before the pigeons could be said with certainty to have gotten the idea. The birds may have actually understood earlier, but the researchers had to make sure they were indeed responding to the number of items, as opposed to their color, shape or relative size. As a result the pigeons had to be trained with random selections of ovals, triangles, rectangles and even computer clip art before it was clear they had their counting skills down cold.

Link (Photo: William van der Vliet)

According to the British Egg Information Service, one in every thousand eggs on average is a double-yolker. They’re not sure how they’ve come to this figure but you would like to think that the British Egg Information Service was able to supply useful information about British Eggs, so let’s give them the benefit of the doubt.

So, if the probability of finding an egg with two yolks is 1/1000 – then to find the likelihood of discovering four in a row you simply multiply the probabilities together four times. One thousand to the power of four brings us to the grand total of one trillion – that’s the new-school US-style trillion with 12 zeroes.

If true that would mean the event that occurred in Jen’s kitchen was a trillion-to-one event. But is it true? No is the short answer.

Many factors can affect these odds, like the possibility that a certain chicken or flock laying several eggs that ended up in one carton, or the sorting of eggs by size. There are other factors as well, explained in this BBC article. Link -via Metafilter

You can understand why someone would want to make a giant pie, but why make a fractal pie? Because the bigger a pie gets, the more the crust becomes overwhelmed by the filling. It’s simple math. Instructables member turkey tek solved that problem by using the fractal shape of the Koch snowflake. At this size, the pie had to be baked in a custom-made outdoor oven. The cake pie is 50 inches in diameter at its widest point, but could have been bigger if the materials for the cooking process were more readily available. Yes, I will have a slice, thank you very much. Link -via Everlasting Blort

In 2002, a reclusive Russian genius named Grigori Perelman put an end to more than 100 years of suffering in the mathematical community. He solved the most difficult math problem of the 20th century -the Poincaré Conjecture. Its siren call had lured generations of mathematicians to intellectual graves. It first, its simplicity would seduce them, and they’d become convinced the answer was near. But as years passed, they’d be left with nothing to show for their lives’ toil but dead ends. By the time Grigori Perelman proved the Conjecture, the solution was worth $1 million.

THE MAN BEHIND THE MADNESS

Henri Poincaré

In 1885, all of Europe was talking about Henri Poincaré, a 30-year-old genius who’d mathematically proven why the solar system holds together. When a hole appeared in his calculations, he plugged it up by essentially inventing chaos theory: Kings were tripping over themselves to make him a knight· and Sweden gave him a small fortune in prize money. To this day; Poincare holds the record for the most physics Nobel Prize nominations, though he never actually won one.

But his most legendary achievement was something no one noticed until much, much later. At the turn of the century: Poincaré invented an entirely new field called algebraic topology; and today, it’s one of the most complicated and vibrant branches of mathematics. Think of it as a twisted version of geometry, in which shapes stretch, bend, and fold inside out. Poincaré’s goal was to classify objects by identifying their basic form, much the same way botanists classify new species of plants. In the process of creating topology, Poincaré tossed out a conjecture that seemed to be true. It was a side note to a larger problem, and he figured he’d work out the details later. Little did he know; his side note would become one of the greatest challenges in the mathematical world.

THE VICTIMS

Poincaré’s conjecture seemed simple enough. It claimed that any object without a loop is essentially a sphere. Think of a knife made out of Play-Doh. Without punching a hole in it or closing a loop, can you squish it into a ball? Yes, of course. Now picture a pair of Play-Doh scissors. No matter how hard you try, you can’t crush it into a ball without closing up the finger holes. It’s impossible. Poincare believed that objects like the knife were related to spheres, while objects with holes and loops in them were not.

Poincaré thought the conjecture would be easy to prove, and he even published a solution. But then, he saw a flaw in his work and retracted it. After his death in 1912, the question lay dormant for decades, until an Oxford professor named J.H.C. Whitehead rediscovered it in the late 1930s. J,H.C. (known to his students as “Jesus, he’s confusing”) also published a solution. But he, too, found a mistake and retracted it. However, his work sparked interest in the problem. By the 1950s, the Poincaré Conjecture was one of the best-known challenges in the math community:

Christos Papakyriakopoulos

That’s when two Princeton students, Edwin Moise and Christos Papakyriakopoulos (commonly known as Papa), decided to try their hands at it. Moise in particular looked like the guy to do it. Young and brash, he liked to announce his next big problem like a batter calling his shot. Twice that included one of the toughest problems in topology; and twice he returned with the solution. Then, he set his sights on Poincaré.

Papa was vastly different. A self-taught political refugee from Greece, he was famous for his odd, obsessive nature. Legend has it that when he came to Princeton, he checked into a motel and never checked out. He never even unpacked his bags. He simply fell into a routine that he followed every day; down to the minute, which always included a midday nap on top of his desk.

Throughout the 1950s, the two geniuses dueled with each other over Poincaré. Papa would announce a proof, and Moise would shoot it down. Then Moise would announce a proof, and Papa would shoot it down. This went on for years, while neither man worked on almost anything else.

Eventually, Moise cracked. One day he simply turned away from math altogether. Michael Freedman, a topologist who works for Microsoft, describes this phenomenon as getting “wrecked” by a problem. He says many mathematicians thrive on the knowledge that they are smart enough to solve almost anything, given enough time. When Moise realized his best efforts were never going to solve Poincaré, his spirit broke. He never did serious math research again and spent his last few years critiquing poetry. Papa kept working on the problem for 25 years, swearing he wouldn’t marry until he’d solved the Conjecture. In 1976, he died of stomach cancer, still a bachelor.

HOPE IN A HIGHER DIMENSION

Steven Smale

After dozens of mathematicians had devoted their careers to the Poincaré Conjecture, a breakthrough came in 1960, when a young hotshot named Steven Smale made the first tangible headway into the problem. Smale decided not to worry about objects in the three-dimensional or even the four-dimensional universe. Instead, he proved that the Conjecture was true in the fifth dimension and higher. Until then, it had always been assumed that problems were easier to solve in dimensions that we can visualize. Smale broke new ground by solving a problem in higher dimensions before the lower ones, and today; it’s common practice. Mathematicians say the extra dimensions give them room to manipulate imaginary objects.

Smale’s discovery inspired the math world, and a new generation of Don Quixotes started sharpening their lances. Another ray of hope came in 1982, when mathematician Michael Freedman managed to scoop up Poincaré in the fourth dimension. Both he and Smale received the Fields Medal, math’s equivalent to the Nobel Prize, just for their partial proofs. And yet, the question of the third dimension-the only one that had actually interested Henri Poincaré-still remained. Technically speaking, without the third dimension, mathematicians were no closer to the answer than they’d been in 1904.

TO THE VICTOR GO NONE OF THE SPOILS

Through all of this, some mathematicians devoted themselves to disproving Poincaré. In fact, during the 1950s, a man named R.H. Bing would spend two weeks trying to prove the Conjecture was true and then two weeks trying to prove it was false. Neither effort panned out. Regardless, most mathematicians believed the solution to Poincare was out there-somewhere.

Grigori Perelman

Finally, in 2002, Grigori Perelman, a recluse living with his mother in St. Petersburg, posted a short paper on a math web site. The reticent Perelman never once mentioned Poincaré in his essay; but the few people who read it understood its implications. The paper addressed one of the biggest obstacles that had blocked mathematicians from proving the Conjecture. Whenever they’d tried reducing certain shapes to their most basic forms, little irregularities kept popping up like painful burrs. To smooth out the rough spots, Perelman applied a type of mathematical sandpaper called a Ricci flow. The math community started to buzz that he’d resolved the underlying issues of the problem. Eventually; one mathematician asked Perelman directly if his paper answered the Poincaré Conjecture. Never one to be long-winded, Perelman wrote back, “That is correct.”

Because Perelman hadn’t taken the normal steps of running his ideas past colleagues and publishing in a referenced journal, it took some time to verify this claim. It didn’t help that his paper only included the barest essentials to prove the Conjecture. It took six experts two years to fill in the gaps that Perelman had seen as self-evident.

By 2006, his proof had held up to every attack, and Perelman was offered a Fields Medal. But by that time, Perelman was so disillusioned with the field of mathematics that without a flourish, he turned it down, becoming the first person in history to refuse the award. Also without explanation, Perelman never claimed the $1 million prize from the Clay Mathematics Institute in Cambridge, Massachusetts.

Today, Poincaré’s side note may only be applicable to the most obscure physics problems. But mathematicians expect that, like most theoretical breakthroughs, the effects will eventually diffuse to the rest of science. After all, when Newton first unveiled calculus, experts said only a few people on Earth could possibly understand it. Now it’s taught to teenagers and used in everything from engineering to statistics. Whether Grigori Perelman likes it or not, someday he may find himself honored in high school textbooks, and the Poincaré Conjecture may become as easy to understand as gravity.

__________________________

The above article by Erik Vance is reprinted with permission from the July-August 2008 issue of mental_floss magazine.

Be sure to visit mental_floss‘ entertaining website and blog for more fun stuff!

Biola University professor Matthew Weathers does it again! He p[resented entertaining lectures for Halloween and April Fool’s Day that we’ve seen, and now Thanksgiving gets the treatment. -via reddit

Dr. Levin's golden grid.

A mathematical gauging of a smile

by Alice Shirrell Kaswell, Improbable Research staff

Dr. Eddy Levin of Harley Street puts a golden ratio, not just golden teeth, into his patients’ mouths. Dr. Levin has been at this for a while. It was he who in 1978 wrote a study called “Dental Esthetics and the Golden Proportion,” which graced pages 244–52 of that year’s September issue of The Journal of Prosthetic Dentistry.¹

The golden ratio is a special number that has caught the eye and imagination of mathematicians, of artists, and now, thanks to Dr. Levin, of dentists. Some call it the “golden mean” (philosophers, though, use that phrase to mean something else). Some call it the “golden section.” Some Germans call it, evocatively, the “goldener Schnitt.” Almost everyone calls it beautiful.

The golden ratio is the number you get when you compare the lengths of certain parts of certain perfectly beautiful things (among them: snail shell spirals, the Parthenon in Athens, and Da Vinci’s painting “The Last Supper”). You’ll find that the ratio of the bigger part to the smaller equals the ratio of the combined length to the bigger. That ratio, that number, is always the same, ever so slightly bigger than 1.6180339.

If doing sums causes you pain, just go find someone who has perfect teeth and who won’t mind you staring into his or her mouth.

Dr. Levin explains on his website² that many years ago he was both studying math and trying to find out what made teeth look beautiful. “It was at a moment,” he writes, “like when Archimedes got into his bath, that I suddenly realized that the two were connected — the Golden Proportion and the beauty of teeth. I began to put this into practise and started testing my ideas on my patients. My first case was a young girl in a hospital, where I was teaching, whose front teeth were in a terrible state and needed crowning. Despite the scepticism of the other members of staff and the unenthusiastic technicians with whom I had to work and whose co-operation I depended upon, I crowned all her front teeth, using the principles of the Golden Proportion. Everybody, including the young lady herself, agreed that her teeth now looked magnificent.”

Most important, in Dr. Levin’s reckoning, is the simple tooth-to-tooth ratio: “The four front teeth, from central incisor to premolar are the most significant part of the smile and they are in the Golden Proportion to each other.”

Dr. Levin created an instrument called the “golden mean gauge.” Made of stainless steel 1.5 millimeters thick, and retailing for £85, it shows whether the numerous major dental landmarks “are in the Golden Proportion,” and it is suitable for autoclaving.

Dr. Levin also offers a larger version that is “useful for full face measurements” and “useful to measure larger objects or bigger pictures of furniture etc.”

(Thanks to Stanley Eigen for bringing this to our attention.)

References

1. “Dental Esthetics and the Golden Proportion,” E.I. Levin, Journal of Prosthetic Dentistry, vol. 40, no. 3, September 1978, pp. 244–52.

2. Golden Mean Gauge

_____________________

The article above is from the May-June 2009 issue of the Annals of Improbable Research. You can download or purchase back issues of the magazine, or subscribe to receive future issues. Or get a subscription for someone as a gift!

Visit their website for more research that makes people LAUGH and then THINK.

by Marc Abrahams, Improbable Research staff

The Ham Sandwich Theorem has been a treat and a spur to mathematicians for more than half a century. It first cropped up in a branch of mathematics called algebraic topology. The theorem describes a particular truth about certain shapes. Most published papers on the topic make a hash of explaining it to anyone who is not an algebraic topologist. But the authors of a 2001 paper called “Leftovers from the Ham Sandwich Theorem” wrapped up an important little leftover: they put the idea into clear language.

“Leftovers from the Ham Sandwich Theorem,” Graham Byrnes, Grant Cairns, and Barry Jessup, The American Mathematical Monthly, vol. 108, no. 3, March 2001, pp. 246-9.

The authors are at La Trobe University, Melbourne, Australia, and University of Ottawa, Ottawa, Canada. The Ham Sandwich Theorem, they wrote, “rescues the careless sandwich maker by guaranteeing that it is always possible to slice the sandwich with one cut so that the ham and both slices of bread are each divided into equal halves, no matter how haphazardly the ingredients are arranged.”

For a while, most ham sandwich theorizing dealt with simple cases. A paper called “Computing a Ham-Sandwich Cut in Two Dimensions,” published in 1986, is typical.

Detail from the Edelsbrunner/Waupotitsch study “Computing a Ham-Sandwich Cut in Two Dimensions.”

“Computing a Ham-Sandwich Cut in Two Dimensions,” H. Edelsbrunner and R. Waupotitsch, Journal of Symbolic Computation, vol. 2, no. 2, June 1986, pp. 171–8.

It considered only ham sandwiches that had been flattened flatter than even the chintziest cook would dare devise. Mathematicians often do things this way, first considering the extreme cases, digesting those thoroughly, and only then moving on to more substantial versions. Indeed, the “Computing a Ham-Sandwich Cut in Two Dimensions” paper itself contains a section called “Getting Rid of Degenerate Cases”.

People did solve the mystery of slicing a thick ham sandwich. And inevitably, they developed a hunger for more substantial problems.

In 1990, Yugoslavian theorists wrote in the Bulletin of the London Mathematical Society about “An Extension of the Ham Sandwich Theorem.”

“An Extension of the Ham Sandwich Theorem”, Rade T. Živaljevi? and Sinisa T. Vre?ica, Bulletin of the London Mathematical Society, vol. 22, no. 2, 1990, pp. 183–6.

The authors are at Mathematics Institute Knez Mihailova and at Faculty of Mathematics Studentski, both in Beograd, Yugoslavia.

The Ham Sandwich Theorem Revisited…
Partly from nostalgia, partly with an eye to the future, these two mathematicians paused in 1992 to take a refreshing look back at the ham sandwich question:

“The Ham Sandwich Theorem Revisited,” Sinisa T. Vre?ica and Rade T. Živaljevi?, Israel Journal of Mathematics, vol. 78, no. 1, February 1992, pp. 21–32.

(Image credit: Flickr user jeffreyw)

…and Generalized
Elsewhere in 1992, a theorist at Yaroslav State University in Russia published a paper called “A Generalization of the Ham Sandwich Theorem.”

“A Generalization of the Ham Sandwich Theorem,” V.L. Dol’nikov and P.G. Demidov, Matematicheskie Zametki [Mathematical Notes], vol. 52, no. 2, 1992, pp. 27–37.

The authors explain:

[We] consider a group of problems connected with the “three-layered sandwich problem,” posed by Ulam, and the Neumann-Rado theorem on division of measures. In the ham sandwich problem it is required to cut a ham sandwich of bread with butter and cheese such that each part contains exactly half of each of the three ingredients.

That same year, a team of hungry American, Czech, and German mathematicians assembled a master collection of recipes for slicing ham sandwiches. Mathematicians almost never use the word “recipe”, so they called their paper “Algorithms for Ham-Sandwich Cuts.”

“Algorithms for Ham-Sandwich Cuts,” Chi-Yuan Lo, J. Matoušek, and W. Steiger, Discrete and Computational Geometry, vol. 11, no. 1, December 1994, pp. 433–52.

Research then moved on to exotic, distantly-related questions, exemplified by a 1998 monograph called “Green Eggs and Ham.”

“Green Eggs and Ham,” M.J. Kaiser and S. Hossaien Cheraghi, Mathematical and Computer Modeling, vol. 28, no. 1, 1998, pp. 91–99.

The authors, at Wichita State University, Kansas, U.S.A., explain:

A dish of ham and eggs is required to be divided equally by a single straight knife cut. A constructive solution procedure is described through the introduction of a balance functional and a geometric optimization problem. The two- and three-, dimensional cases are illustrated by example.

Whence the Ham Sandwich Theorem?
And who started this? A 2004 paper called “The Early History of the Ham Sandwich Theorem” took care of another lingering leftover: it identified the inventor. Mathematico- historians W.A. Beyer and Andrew Zardecki of Los Alamos National Laboratory in New Mexico, U.S.A., say that it was a Jewish theorist who introduced the ham sandwich into mathematical theory. Beye and Zardecki trace the theorem back to a 1945 paper by the Polish mathematician Hugo Steinhaus that “represents work Steinhaus did in Poland on the ham sandwich problem in World War II while hiding out with a Polish farm family.”

“The Early History of the Ham Sandwich Theorem,”
W.A. Beyer and Andrew Zardecki, The American Mathematical Monthly, vol. 111, no. 1, January 2004, pp. 58–61.

The authors say:

The following theorem is the well-known ham sandwich theorem: for any three given sets in Euclidean space, each of finite outer Lebesgue measure, there exists a plane that bisects all three sets, i.e., separates each of the given sets into two sets of equal measure. The early history of this result seems not to be well known. Stone and Tukey attribute the theorem to Ulam. They say they got the information from a referee. Is this correct? The problem appears in The Scottish Book as problem 123. The problem is posed by Steinhaus. A reference is made to the pre-World War II journal Mathesis Polska (Latin for “Polish Mathematics”). This journal is not easy to locate…

The Miraculous Years 2009 and 2010
The years 2009 and 2010 were anni mirabili for Ham Sandwich Theorem research. Consider just a few of the papers published during that time.

“Dynamic Ham-Sandwich Cuts in the Plane,” Timothy G. Abbott, Michael A. Burr, Timothy M. Chan, Erik D. Demaine, Martin L. Demaine, John Hugg, Daniel Kane, Stefan Langerman, Jelani Nelson, Eynat Rafalin, Kathryn Seyboth, and  Vincent Yeung, Computational Geometry, vol. 42,  no. 5, July 2009, pp. 419–28. “Ham Sandwich with  Mayo: A Stronger Conclusion to the Classical Ham Sandwich Theorem,” John H. Elton and Theodore P.  Hill, arXiv:0901.2589v2, 2009.

The authors of the latter paper, at Georgia Institute of Technology, Georgia, U.S.A., report:

The conclusion of the classical ham sandwich theorem of Banach and Steinhaus may be strengthened: there always exists a common bisecting hyperplane that touches each of the sets, that is, intersects the closure of each set. Hence, if the knife is smeared with mayonnaise, a cut can always be made so that it will not only simultaneously bisect each of the ingredients, but it will also spread mayonnaise on each. A discrete analog of this theorem says that n finite nonempty sets in n-dimensional Euclidean space can always be simultaneously bisected by a single hyperplane that contains at least one point in each set.

Other discussions followed:

“Generalized Ham-Sandwich Cuts,” William Steiger and Jihui Zhao, Discrete and Computational Geometry, vol. 44, no. 3, 2010, 535-45. “Uneven Splitting of Ham Sandwiches,” Felix Breuer, Discrete and Computational Geometry, vol. 43, no. 4, 2010, pp. 876–92.

From the Ham Sandwich to the Pizza Pie
The future, which of course is not truly predictable (according to most current theories, anyway), appears to include the continued diversification of Ham Sandwich theorizing. We conclude with a mention of a possible harbinger of things to come:

“From the Ham Sandwich to the Pizza Pie: A Simultaneous Z m Equipartition of Complex Measures,” Steven Simon, arXiv:1006.4614v8, 2010.

Acknowledgement
Thanks to Stanley Eigen for bringing the Ham Sandwich Theorem to our attention.

_____________________

This article is republished with permission from the May-June 2011 issue of the Annals of Improbable Research. You can download or purchase back issues of the magazine, or subscribe to receive future issues. Or get a subscription for someone as a gift!

Visit their website for more research that makes people LAUGH and then THINK.

Link

I’ll take two! This was on Neatorama’s Facebook page. Link

The Oatmeal has a hilarious collection of things we should have learned in school, but my favorite is the math lesson above. After all, we’ve all been in that frustrating situation before.

Link

There’s not much to say about this infinity tentacle other than it was created by DeviantArt user Andrew Strauss and that it is so delightfully geeky that it had to be featured here on Neatorama.

Link Via Laughing Squid

Mathematician Pierre de Fermat was born 410 years ago today, as we learn from the Google doodle of the day. The doodle recreates Fermat’s Last Theorem, which he left scribbled in the margin of a book.

Fermat’s claim left mathematicians puzzled for over 350 years — as mathematicians proved it true for many sets of possible values of n — until the general case was finally proved by Andrew Wiles in 1995. The story about the proof is told in Simon Singh’s book Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem.

Did Fermat really have a proof? Most likely not, since the techniques Wiles used to prove it weren’t developed until several centuries after Fermat’s death — and since, in the 30 years he lived after writing the note, he never wrote about the general case of the proof again — but nobody will ever know for sure. I’ll leave that, as they say, as an exercise for the reader.

Link

Actually, string theory is something completely different, but it’s a cute title for this geometry problem at Futility Closet. A boy has his toy boat in the water, and he is pulling it to shore by a string. If he pulls in one yard of string, will the boat advance a yard, or less than a yard, or more than a yard? The answer may surprise you. Link -via TYWKIWDBI

A while back we posted a video of actual Miss USA contestants being asked “Should Evolution Be Taught in Schools?” Now someone has posed an even greater question to some would be “Miss USA contestants.” Should math be taught in schools? Their answers may shock you. Watch second video at bottom of link post.

Link

Etsy seller Nausicaa Distribution sells these precious statistical distribution graph plushies that are just perfect for any Neatorama reader. Of course, if you’re looking for cute nerdy plushes, the Neatoshop is also a great place to go shopping.

Link Via Laughing Squid

Josh Sundquist shares some charts and graphs about fireworks, pie, and other Independence Day traditions. -via Buzzfeed

Brain Candy Toys came up with a great advertising strategy by simplifying nursery rhymes and fairy tales into adorable little math equations. Check out the rest on the ad company’s site.

Link via Laughing Squid

The date today is 6/28, which is Tau Day. The number Tau is 2pi, or 6.28 (followed by many more decimals). Geek Are Sexy has an explanation of tau, which is kind of like pi, only more so. And since tau is 2pi, you should celebrate Tau Day by baking two pies. One for me, and one for you. And not to throw at each other! Link

See also: The Pi is a Lie

A schoolbook that both postdated and outlived its time.

by Stephen Drew, Improbable Research staff

Mathematics teaching has been cocked up — well and properly and officially  — for a good long while, thanks to Edward Cocker and his amply-titled textbook Cocker’s Arithmetick: Being a Plain and Familiar Method Suitable to the Meanest Capacity for the Full Understanding of  That Incomparable Art, As It Is Now Taught by the Ablest School Masters in City and Country.

Published in 1667, and later reprinted in more than 100 editions, the book was a standard in British grammar schools for several generations. Foreign schoolteachers also took Cocker to their bosom.

A Man of Words, Word, and More Words, Plus More Words
The 34-word title exemplifies the book’s approach to explaining things clearly. One could (although the author would probably not) sum it up in three words: don’t be terse.

Here, for example, is how the book takes the student in  hand — nearly in handcuffs, really — to explain the so- called “Rule of Three.” This passage appears on page 88 of the book’s 47th edition, published in the year 1736:

Observe, that of the three given numbers, those two that are of the same kind, one of them must be the first, and the other the third, and that which is of the same kind with the number sought, must be the second number in the rule of three; and that you may know which of the said numbers to make your first, and which your third, know this, that to one of those two numbers there is always affixed a demand, and that number upon which the demand lieth must always be reckoned the third number.

The book’s very first page accustoms the student to what lies ahead. You might enjoy reading this aloud:

Unit is number; for the part is of the same matter that is his whole, the unit is part of the multitude of units, therefore the unit is of the same matter, that is the multitude of units; but the matter of the multitude of units is number; therefore the matter of units is number; or else, if from a number given no number but subtracted, the number given remaineth; as suppose 3 the given number, if as some suppose, 1 be no number, then if you subtract 1 from 3, there must remain 3 still; which is very absurd.

Words After Death
Scholars now debate whether Edward Cocker actually wrote the book (the first edition    was published nine years after his death). Some suggest the whole thing is just a pastiche of other people’s writings, issued by a greedy publisher. No matter. Like many of today’s textbooks, authorities deemed it authoritative, and it came to enjoy widespread use. In that respect, as perhaps in others, this antique textbook is a very 21st-century piece of work.

References
Cocker’s Arithmetick: Being a Plain and Familiar Method Suitable to the Meanest Capacity for the Full Understanding of That Incomparable Art, As It Is Now Taught by the Ablest School-Masters in City and Country, Edward Cocker, 1677, John Hawkins [publisher], London.

Bonus
Cocker’s Life and productive death are the subject of an essay called “Who Was Cocker,” in the July 1884 issue of The Bibiliographer. You can read it online.

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This article is republished with permission from the July-August 2010 issue of the Annals of Improbable Research. You can download or purchase back issues of the magazine, or subscribe to receive future issues. Or get a subscription for someone as a gift!

Visit their website for more research that makes people LAUGH and then THINK.

Detail from the Darke/Holmes study

by Michael Berry
H.H. Wills Physics Laboratory,
University of Bristol, Bristol, UK

The applications of mathematics can be bizarre. Soon after I arrived in Bristol in the 1960s, a senior colleague called me, saying that someone in the veterinary school needed help with mathematics — or was it physics? — and I seemed just the person to help. Cursing inwardly, I agreed to see the fellow. He was Peter Darke, a graduate student near the end of a Ph.D. studying horses’ hearts.

He showed me a paper by Gabor (Dennis Gabor, who invented holography) and Nelson1 and asked me to explain it. It took a while to understand. The idea is that a heart is like a little battery, pushing weak electric currents in a three-dimensional pattern round the body. The battery has a strength and a direction: it acts as a current dipole, represented as a little arrow — the heart vector. During each heartbeat, the vector (tip of the arrow) draws a loop – the heart loop — whose shape is a powerful diagnostic of health. Therefore it is useful to measure this loop, in a way that doesn’t involve killing the horse. Gabor’s paper gave the theory of a way to do that, inferring the heart vector by measurements of the electric potential on the surface of the horse. It is an ingenious application of Gauss’s theorem.

The Darke/Holmes study, which used the Berry approach to integrate over the surface of a horse.

Peter had spent three years preparing to implement this idea. He enveloped his horse in a coat he had made, of several hundred potentiometers, with electronics to measure the potential at each of them, fifteen times during each heartbeat, and he had arrived at the point where he had a huge file of all these measurements. But there was a difficulty: he knew only the most elementary high-school mathematics and so had no way to understand the formulas in Gabor’s paper. His specific  question was: does the theory apply to a real horse, or only to an ideal cylindrical horse? Unlike the physicists’ mythical ‘spherical cow,’ this was real.

I learned that the formulas work for a horse of any shape, but they do assume uniform conductivity — a better approximation, apparently, for horses than for people. (Actually, it doesn’t have to be accurate: who cares whether the loop describes the real dipole inside the real horse? To be useful for diagnosis, it is necessary only that the loop be reproducible.)

The formulas involved integration, and Peter didn’t know what an integral was, so it was hard to explain how to add up all those measurements. A complication was that what had to be inferred was a vector, so he needed to know, at each point on the horse, the components of the perpendicular to the surface of the horse with respect to the three symmetry directions of the horse. After some discussion, we made a ‘cos-theta-meter,’ and I left him to it, and never saw him again.

Further detail from the Darke/Holmes study.

But a year later, I received two papers from him,2 reporting the outcome of all that arithmetic. To my surprise, he had indeed calculated fifteen vectors for each heartbeat, and thereby deduced the heart loops for several horses in different states of health. At the end of the paper were the usual acknowlegements to colleagues and funding agencies. For technical help, he thanked me; and for financial support, he thanked the Horserace Betting Levy Board (financed by racecourse gamblers).

The moral of this is that applications of mathematical knowledge can be unexpected; you may find yourself taking a surface integral over a horse.

References
1. “Determination of the Resultant Dipole of the Heart from Measurements on the Heart Surface,” D. Gabor and C.V. Nelson, Journal of Applied Physics, vol. 25, 1954, pp. 413-6.

2. “Studies on the Equine Cardiac Electric Field. I. Body Surface Potentials, II. The Integration of Body Surface Potentials to Derive Resultant Cardiac Dipole Moments,” P.G.G. Darke and J.R. Holmes, Journal of  Electrocardiology vol. 2, 1969, pp. 222-234 and 235-244.

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This article is republished with permission from the July-August 2010 issue of the Annals of Improbable Research. You can download or purchase back issues of the magazine, or subscribe to receive future issues. Or get a subscription for someone as a gift!

Visit their website for more research that makes people LAUGH and then THINK.

The Making of a Fly

The behavior of profnath is easy to deconstruct. They presumably have a new copy of the book, and want to make sure theirs is the lowest priced – but only by a tiny bit ($9.98 compared to $10.00). Why though would bordeebook want to make sure theirs is always more expensive? Since the prices of all the sellers are posted, this would seem to guarantee they would get no sales. But maybe this isn’t right – they have a huge volume of positive feedback – far more than most others. And some buyers might choose to pay a few extra dollars for the level of confidence in the transaction this might impart. Nonetheless this seems like a fairly risky thing to rely on – most people probably don’t behave that way – and meanwhile you’ve got a book sitting on the shelf collecting dust. Unless, of course, you don’t actually have the book….

My preferred explanation for bordeebook’s pricing is that they do not actually possess the book. Rather, they noticed that someone else listed a copy for sale, and so they put it up as well – relying on their better feedback record to attract buyers. But, of course, if someone actually orders the book, they have to get it – so they have to set their price significantly higher – say 1.27059 times higher – than the price they’d have to pay to get the book elsewhere.

The price went as high as $23,698,655.93 (plus $3.99 shipping) on April 18th when someone apparently noticed, and manually adjusted the price. Read the whole story at Eisen’s blog. Link -via reddit

This is good, but the moves get really explicit once you start into calculus. If you know who’s responsible for this great cartoon, let us know in the comments.

via Geekosystem | Previously: Math Dances and Other College Application Videos

Can you figure out what movie each of these mathematical equations represents? I couldn’t. There are a lot of good guesses in the comments at Spiked Math Comics, but no definitive answer list. Link -via Geeks Are Sexy

1) Thomas Dolby: She Blinded Me With Science

How is it nerdy? This is the song that inspired me to write this list. It’s an ultimate geek love song in that a woman is able to seduce her love interest not through her looks but through chemistry and other scientific fields.

Choice lyrics: It’s poetry in motion/And now she’s making love to me/The spheres are in commotion/The elements in harmony/She blinded me with science/”She blinded me with science!”/And hit me with technology

Video:

Video link

2) Marshall Gray: Critical Hit On My Heart

How is it nerdy? It might just be impossible to write a romantic song with more Dungeons and Dragons references.

Choice lyrics: I picked up spell resistance from the enchanted school/So I could bend up all these magic pretences/And though always use it as a general rule /This time I’m lowering all my defences

Video: There’s no official video for the song, but here’s a YouTube video with the song.

Video link

3) Mc Chris: Nerd Girl

How is it nerdy? MC Chris is one of the biggest stars of the nerdcore hip hop scene and this serenade to a nerd shows just how geeky he can be, even when discussing matters of the heart.

Choice lyrics: She’s romantic, known to panic/With anxiety attacks/Literary, it’s so scary/Reading Brontes back to back/She’s playing Ragnarok on her mom’s Magnavox/She’s underneath my skin like a million nanobots

Video:

Video link

4) Bad Religion: I Love My Computer

How is it nerdy? It’s not even about a girl, it really is about loving a computer and how the computer is far better than a real girlfriend.

Choice lyrics: I’ve never been quite so happy/all I need to do is click on you/and we’ll be joined/in the most soul-less way/and we’ll never/ever ruin each other’s day

Video:

Video link

5) The Aquabats: Martian Girl

How is it nerdy? It takes all the silly scifi conventions about aliens and turns them into a love song about a green-skinned girl in a silver bikini. Of course, their love was not meant to be because after kissing the narrator and drawing blood with her razor sharp teeth, she flies away.

Choice lyrics: She’s not a bird/She’s not a plane/She’s got green blood/Pumping through her veins.

Video:

Video link

6) Logan Whitehurst: Calculator Love

How is it nerdy? Aside from the bizarro chipmunk-sounding vocals, the lyrics revolve around solving math problems together using a brand new calculator.

Choice lyrics: If you really love me darling, count it up and see/And we can solve our problems with some trigonometry

Video: I couldn’t find any videos featuring the song, but if you want to check it out, here’s a link to the MP3.

7) Freezepop: Science Genius Girl

How is it nerdy? The song is about a girl who’s great at science and wants to clone a human being to be both her boyfriend and bandmate.

Choice lyrics: Measure out the chemicals/safety goggles on my eyes/turn the bunsen burner on/my creation comes alive

Video:

Video link

8) Your Favorite Martian: Zombie Love Song

How is it nerdy? Geeks love zombies, so what’s more apt than a love song from the zombie’s point of view.

Choice lyrics: I knew you’d be surprised. You can bet that I/May not be alive, but I sure as hell ain’t dead inside./What’s with the shotty? I ain’t wishing you harm./You see, I’d try to hold your hand but I’m missing an arm.

Video:

Video link

9) The Klein Four Group: Finite Simple Group

How is it nerdy? It’s all about math and is performed and written by students of  Northwestern University’s Mathematics Department.

Choice lyrics: I’m not the smoothest operator in my class/But we’re a mirror pair, me and you/So let’s apply forgetful functors to the past/And be a finite simple group, a finite simple group/Let’s be a finite simple group of order two

Video:

Video link

10) Mouldy Peaches: Anyone Else But You

How is it nerdy? Aside from being the official theme for the movie Juno, the lyrics are decidedly dorky, particularly the line listing off the Contra Code.

Choice lyrics: Up up down down left right left right B A start/Just because we use cheats doesn’t mean we’re not smart/I don’t see what anyone can see/in anyone else but you

Video: There is no official video for the song, but here’s the audio with the lyrics.

Video link

Have any more to add? Share your favorites in the comments.

Vi Hart, who knows more about math than I ever will, made a video and two pies especially for Pi Day, which she says we should call Half-Tau Day. She lost me when she said a pie is really 2pi, because I never took that class. I will take a slice of cherry, if you don’t mind. Link -via The Daily What

Taken from Clifford Pickover’s book, Computers and the Imagination, is this experiment that creates butterfly like curves.

The formula is expressed in polar coordinates as:

By changing the A, B, a, b and c parameters you can get some nice results.

It’s fun to change the parameters to see what you get: Link – via Cliff Pickover’s Reality Carnival

Vi Hart calls herself a “recreational mathemusician”, which sounds like fun! In this video, she teaches more about math than she missed by doodling during class. See more of this sort of thing at her website. Link -Thanks, David Israel!

Just imagine Homer Simpson’s response to this brilliant tee shirt by John Sumrow, “mmm…infinite bacon.”

Link via Laughing Squid

The electricity generated by a 9-volt battery might be all there is between you and the mathematical brilliance of a Newton or an Einstein.

OK, we can’t guarantee you’ll be that smart, but, amazingly, British scientists have now shown that low voltage current applied to the right part of the scalp can spark changes that boost the brain’s math abilities. What’s more, that mild jolt can lock in your improved mathematical prowess for as long as six month, according to new research published in this month’s issue of Current Biology.

The findings come too late for those of us who already suffered through middle school algebra, but maybe future generations will benefit.

The researchers, led by Roi Cohen Kadosh from the University of Oxford, suspected that a little electricity directed at the right parietal lobe – a brain region at the top of the head and known to play a role in numerical calculations – might juice up a person’s math ability.

Link

This is a perfect tool for teaching my kids their algebraic order of operations! Oh, they can do them, but I think this will help them understand the concept better. Find this “s’more formula” on a t-shirt at W00t. Link -via Laughing Squid

Hyperboloid

What mathematicians call a hyperboloid of one sheet is a really cool structure that is made up of many (actually an infinite number) of perfectly straight lines that look to us like a curved structure. First, imagine that you have a cube. Stand it on one of its corners and spin it like a top, then look at it from the side -the sides seem to be curved, but you know they aren’t. Now, take a handful of uncooked spaghetti noodles. Use two hands, and twist the strands loosely. It forms the shape of a hyperboloid structure, which looks like a cooling tower at a nuclear reactor. All the spaghetti noodles are still straight, but the shape of the handful is curved. In architecture, this idea enables builders to produce curved structures by using straight line supports.

Apollonian Gasket

An Apollonian gasket is a fractal generated when you mash as many round soap bubbles together as you can. At least, that’s what it looks like. The pattern is based on threes: every circle touches two other circles. As you add more circles in the smaller spaces, they also touch two existing circles (and eventually many smaller ones). The number of smaller circles that can be added is mathematically infinite. Frothing soap bubbles can help us picture the Appolonian gasket, but the analogy is flawed, because real world soap bubbles do not like empty spaces. There is a limit to the volume of soap, and surface tension will connect round bubbles and flatten them against each other. This fractal is named for the ancient Greek mathematician Apollonius of Perga. The 3-dimensional fractal of this sort is called the Apollonian sphere packing, which is a pretty descriptive name for a math term.

Möbius Strip

When I was very young, first or second grade, my father told me he could make a piece of paper with only one side. Then he took a strip of paper, gave one end a half-twist, and taped the ends together. Then he showed me how he could draw one line down the strip without stopping and it covered the whole strip, no matter which way you turned it! I couldn’t wait to show the Möbius strip to my friends at school. When I did, they just stared at me and told me I was weird.

Maybe “continuous plane” is a better description than “one sided paper”. The Möbius strip is named for August Ferdinand Möbius who discovered it in 1858. Johann Benedict Listing also came up with the concept around the same time and actually published his work, but maybe someone thought calling it a “Listing strip” would be confusing. Anyway, the Möbius strip does have some real-world applications. For example, conveyor belts and recording tapes with a half-twist last twice as long as they would otherwise because the entire surface is used instead of just one side of a two-sided strip. It’s also an attention-getter in art and even architecture.

Klein Bottle

The Klein bottle came about from an attempt to make a 3-dimensional Möbius strip. In 1882, mathematician Felix Klein theorized about a container that had no inside or outside -just one “side”- in this manner:

Take a rectangle and join one pair of opposite sides — you’ll now have a cylinder. Now join the other pair of sides with a half-twist. That last step isn’t possible in our universe, sad to say. A true Klein Bottle requires 4-dimensions because the surface has to pass through itself without a hole.

Here in the real (3-dimensional) world, we cheat by passing the neck of the bottle through itself using a real hole. You can see that process in sequence. You can see (and purchase) the finished product. You can put water into a Klein bottle (carefully), but you can’t see a defined lip, meaning there is no discrete border between what is the inside and what is the outside of the container.

The Heliospheric Current Sheet

The Heliospheric current sheet is the shape of the sun’s magnetic field. As the sun rotates, the magnetic field is forced into a spiral, called a Parker Spiral. The spiral has four arms with two different polar phases. As these rotate, the shape of the current sheet (which is the effect of the magnetic field on the plasma in the solar wind) forms a 3-dimensional spiral that resembles a twirling dancer with a voluminous skirt. Picture the bottom polarity as the dancer pushing her skirt down (with two hands) while the force of the twirl raises it like the upper polarity does. The two different polarities causes the vertical shape and the outward spiral causes the horizontal shape. You can reproduce this shape by twirling around while holding a gushing garden hose. Be sure to move the hose up and down as you turn -or better yet, watch while someone else does all this.

Vesica Piscis

The words vesica piscis literally translates to English as fish bladder. It is the name of the particular shape created when two circles of the same size overlap so that the edge of each circle touches the center of the other, such as the shape you see in the middle part of a two-circle Venn diagram. You could easily say this is just two identical curves placed together, but those curves have a lot of symbolism, going back to ancient times. The two curves with a bit of the crossover left on in the shape of a fish were used as a symbol of Christianity in its early days (and today as well). The shape is also the basis of the Gothic arch, which is stronger than a regular round arch in that it resists pressure from the sides as well as from above. It enabled architects to build huge structures with taller but stable doors and windows. The vesica piscis is also symbolic of the overlapping visual field of our two eyes -and, of course, some see it as a vagina.

Buddhabrot

Pareidolia is the tendency of humans to see meaningful shapes in everyday objects, like the image of the Virgin Mary in a grilled cheese sandwich. In studying the Mandelbrot set of fractals, which we looked at the in the previous post, several people noticed that if you turned the traditional rending of the fractal on its side, the shape looks like a sitting Buddha. This led to the search for the precise formula to generate a Buddhabrot. Turn your head to left while looking at this image and try to tell me you don’t see a golden statue. The number of iterations (how many fractal levels are rendered to be visible) has a lot to do with the shape you see.

As in the previous post, you can get a much more mathematical explanation of each of these shapes, including formulas, by clicking on the links in this post.

Superegg

(Image credit:  Sir48 at da.wikipedia)

A superegg is a mathematical shape constructed by rotating a superellipse around an axis to the formula of |x/a|2.5 + |y/b|2.5 = 1, where a/b = 4/3. (If you search for “superegg formula”, you are liable to find something completely different.) But you don’t want to bother with formulas, do you? Just look at it! From the side, the superegg looks a bit like a cylinder, but has no corners. If you cut one horizontally, the cross-section will be a circle. However, unlike a natural egg, you can stand the superegg on its end -either end, as a matter of fact, as it is vertically as well as horizontally symmetrical, although it has no straight lines that you can find -although the curvature is zero at the ends, the “ends” are actually quite small and appear to be rounded. The superegg was popularized by Danish mathematician and physicist Piet Hein, who used the shape in designs for household items such as furniture, ice cubes, and candles, as well as a novelty toy (sometimes referred to as a stress-reliever) by itself.

Torus

I learned about the torus from crossword puzzles. If the clue says “donut shape”, the answer is torus. The solid is produced by rotating a circle around an imaginary axis, but in this surface of revolution, the axis is outside the circle. The resulting shape is a ring torus. Other torus shapes are produced when the axis is touching or slightly inside the circle. Some really strange mathematical shapes are produced when the rotating plane of the circle is not quite round, or is itself rotating around a point in the plane.  A toroid is a ring or donut shaped solid produced by a surface of revolution not necessarily limited to a circle. For example, a square used in this manner will produce a ring that would be uncomfortable on your finger. A toroidal polyhedron is a torus constructed with or converted into flat surfaces, with the shape dependent on how many flat surfaces you use. Toroidal Polyhedron would be a cool name for a band.

Gömböc

You might remember Weebles -they wobble, but they don’t fall down. However, if the heavy weight in the bottom of the toy ever came loose, you had a Weeble that fell down. In 1995, Russian mathematician Vladimir Arnold questioned whether there could be a 3-dimensional shape that would always return to its original position without the help of internal weights. If a shape could be found that had as few as two points of equilibrium, one stable and one unstable, the shape would naturally return to balancing on the one stable point. For a long time, mathematicians thought the shape was impossible. But in 2006, Gábor Domokos and Péter Várkonyi developed the gömböc. This odd shape has only two points it could possibly balance upon, and the point on top is too “pointed” to be stable. So, if you roll a gömböc around, it will soon right itself, returning to an upright position because of its shape, not because of any internal irregularities. It’s a Weeble that doesn’t wear out! Objet Geometries made the first fabricated gömböcs. They were numbered as a limited series (inside, using transparent materials of the same density as the rest of the object) and professor Arnold was presented with number one. You can buy one of your own.

Spidron

A spidron sounds like something that spins, and that’s what it looks like as well. A spidron is defined as a two-dimensional figure “consisting of an alternating sequence of equilateral and isosceles (30°, 30°, 120°) triangles.” OK, you know an equilateral triangle has three sides all the same length. An isosceles triangle doesn’t (the hypotenuse is longer). So when the equilateral triangle adjoins the right side of an triangle isosceles on one side and the hypotenuse on the other side, you see that the size of the isosceles triangles get smaller as they go on -and therefore the size of the equilateral triangle decreases as well. So the shape peters out as it grows longer, and it shifts a bit to the side because of the isosceles triangle shape. Since the shape grows in at least two directions (or it’s only a semi-spidron), the shape looks like a twisty pinwheel blade of sorts. But what good is it? Just take a look at how they fit together!

There are more spidron designs than can be contained in one post. See, with enough spidrons, any math whiz can be another M.C. Escher!

Bernoulli’s Spiral

Jacob Bernoulli was a Swiss mathematician who studied spirals produced by logarithms, which he called Spira mirabilis, or “the marvelous spiral”. These are often called Bernoulli’s spirals now in honor of the man who popularized them. There is a mathematical formula, but the simple explanation is that the spiral grows larger as it progresses outward, but the curve does not. This gives it a particular shape you may recognize in nature, most commonly in snail shells and seashells. You can also see logarithmic spirals in weather systems, spiral galaxies, and in flowers, in which case there are often spirals growing in opposite directions.

Mandelbrot Fractals

(Image credit: Wolfgang Beyer)

The word fractal has the same root as fracture, meaning broken. Fractals are geometric shapes that, if broken into parts, fracture into parts that are a smaller version of the whole (the same shape, that is) which is called a self-similar image. So those parts themselves are copies of the whole, and can be further broken down into smaller parts that are also copies of the whole. We non-math folks sometimes call this recursion. True fractals don’t exist in nature, as there is a limit to how small finite units can go -as far as we know, atoms only come in certain shapes. However, in geometry they can be broken down infinitely. A Mandelbrot set (named after mathematician Benoît Mandelbrot, who coined the term fractal) is a complex set of points on a plane that has a fractal boundary. In other words, the shape of the set repeats itself along the edge, and each repeating shape also contains repeating shapes, no matter how closely you zoom into the plane. This is easier to understand by looking at it.

(YouTube link)

That’s pretty mind-blowing by itself. Could a Mandelbrot set be made into a 3-dimensional shape?

Mandelbulb

The Mandelbulb is a 3-dimensional representation of a Mandelbrot set. Daniel White set out to go beyond the idea of rotating a plane around an imaginary axis to create a 3-dimensional shape. He published an experimental formula to create a true 3D Mandelbrot set in 2007. Paul Nylander from Fractal Forums helped refine the formula. The resulting shape called a Mandelbulb is not quite what they are looking for -yet, but the two men are still working on perfecting the formula. See more pictures of the Mandelbulb here. No, I can’t even pretend to know how they accomplished that one, so with my mind adequately blown, I’ll end this list.

There are formulas for each of these shapes, which mean little if you aren’t at ease with higher geometry. If you are interested, you can find those formulas by following the links in this post.

Declaring war on limp fish, bone crusher, politician and other types of handshakes, Chevrolet UK commissioned a study to determine the perfect handshake. After all, there's nothing more important to closing the deal in selling cars than the handshake:

Professor Geoffrey Beattie, who worked on the project for Chevrolet, said: "The human handshake is one of the most crucial elements of impression formation and is used as a source of information for making a judgement about another person.

"The rules for men and women are the same: right hand, a complete grip and a firm squeeze (but not too strong) in a mid-point position between yourself and the other person, a cool and dry palm, approximately three shakes, with a medium level of vigour, held for no longer than two to three seconds, with eye contact kept throughout and a good natural smile with a slow offset with, of course, an appropriate accompanying verbal statement, make up the basic constituent parts for the perfect handshake."

Just so you know this is very serious business, Professor Beattie of the University of Manchester codified the perfect handshake into mathematical formula:

(e) is eye contact (1=none; 5=direct) 5; (ve) is verbal greeting (1=totally inappropriate; 5=totally appropriate) 5; (d) is Duchenne smile - smiling in eyes and mouth, plus symmetry on both sides of face, and slower offset (1=totally non-Duchenne smile (false smile); 5=totally Duchenne) 5; (cg) completeness of grip (1=very incomplete; 5=full) 5; (dr) is dryness of hand (1=damp; 5=dry) 4; (s) is strength (1= weak; 5=strong) 3; (p) is position of hand (1=back towards own body; 5=other person's bodily zone) 3; (vi) is vigour (1=too low/too high; 5=mid) 3; (t) is temperature of hands (1=too cold/too hot; 5=mid) 3; (te) is texture of hands (5=mid; 1=too rough/too smooth) 3; (c) is control (1=low; 5=high) 3; (du) is duration (1= brief; 5=long) 3.

Link

After the math department at the University of Texas noticed some of its Dell computers failing, Dell examined the machines. The company came up with an unusual reason for the computers’ demise: the school had overtaxed the machines by making them perform difficult math calculations.

And I’m sure you can predict what would happen next if you continue to ship faulty computers to your customers. Five words: Dell, you’re getting a lawsuit!

Link

See also: Dude, You’re Getting A Tequila! | Alphabet of Computing

Be still my geeky heart! Neatoramanaut Ana models this wonderfully geeky I Reverse Polish Notation Heart T-Shirt.

What’s a Reverse Polish notation? It’s a mathematical notation where the operator (e.g. +, -, x) follows the operands. Those of you who have the old HP-10C series calculator are probably nodding in nostalgia right now!

Link

More I Heart T-shirts | Science T-Shirts over at the NeatoShop

San Francisco-based artist/coder Robert Hodgin of Flight 404 Blog created some of the most mesmerizing sculptures using magnetized balls and cylinders. They’re part of the Gray Area Foundation for the Arts (GAFFTA) exhibition.

That has got to take some mad skillz because I can easily envision the whole thing collapsing into a pile of magnetized blob at the slightest touch.

MAKE Blog has the gallery: Link | Robert’s official webpage

Related: Buckyballs over at the NeatoShop

ScienceBlogs, together with Serious Eats, held a Pi Day Bake-Off to celebrate Pi Day on March 14th. They received 35 pie entries, which have been narrowed down to ten finalists. Not only are these “pi pies” decorated in a mathematically clever way, they look scrumptious! Shown is Claudette’s amazing One-Hundred-Digit pie made with cherries, raspberries, blueberries, blackberries, and strawberries. Sure it’s not round, but remember, pie are square! Link to photographs. Link to voting.

The Inverse Graphing Calculator takes words and converts them into equations that would express them graphically:

The Inverse Graphing Calculator (version beta-1) is like a backwards graphing calculator. Normally, you enter an equation into your calculator and then get a graph of the curve. The way the IGC works is, you type something you’d like as your curve, like ‘Hello World’ or ‘I love you’. The IGC produces an *equation* which has this phrase as its graph!

Link via Geekologie

It makes sense that a movie has to conform to our average attention span. This way we will not get bored during the film. Cornell University psychologist James Cutting has worked out the formula for delivering a blockbuster hit.

To find out whether the length of camera shots in films might follow 1/f too, Cutting measured the duration of every shot in 150 high-grossing Hollywood movies in various genres released between 1935 and 2005. He then turned these into a series of waves for each film. He found that later films were more likely to obey the 1/f law than earlier ones (Psychological Science, in press). But he stresses that it isn’t just fast-paced action films like Die Hard II that follow 1/f. Rather, the important thing is having shots of similar length that recur in a regular pattern throughout a film.

Link – via popsci

From the Upcoming ueue, submitted by mrmunchies.

An object or scene in the visual world is projected as a two-dimensional image on the retina of each eye, so what we see can also be treated as flat sheet: the visual field. Every point on this sheet can be pin-pointed by two coordinates, just like a point on a map, or a point on the flat model of V1. The alternating regions of light and dark that make up a geometric hallucination are caused by alternating regions of high and low neural activity in V1 — regions where the neurons are firing very rapidly and regions where they are not firing rapidly.

A closer look at the types of specialized neurons in the V1 field and how they interact with each other explains the geometric patterns.

Bressloff and his colleagues used a generalised version of the equations from the original model to let the system evolve. The result was a model that is not only more accurate in terms of the anatomy of V1, but can also generate geometric patterns in the visual field that the original model was unable to produce. These include lattice tunnels, honeycombs and cobwebs that are better characterised in terms of the orientation of contours within them, than in terms of contrasting regions of light and dark.

That’s about as simple as I can make it in a short blurb; the entire article explains it better. Yes, there is math involved. Link -via Metafilter

Found at Cliff Pickover’s always excellent Reality Carnival. It took me a while to get it!

Previously on Neatorama: The Math Book: Milestones in the History of Math

Photographer Nikki Graziano takes pictures and then creates graphs of mathematical functions which map nicely to elements of the image. It’s a very neat and beautiful way of combining math, nature, and art together into a single image.

Most of us can’t tell our secant from our cotangent. But the forms are everywhere, and Nikki Graziano wants to help us see them. Graziano, a math and photography student at Rochester Institute of Technology, overlays graphs and their corresponding equations onto her carefully composed photos. “I wanted to create something that could communicate how awesome math is, to everyone,” she says.

Link

From the Upcoming ueue, submitted by thalin.

The Salary Theorem proves mathematically that those who know more make less money. Therefore, if you know nothing, you should be fabulously wealthy! Link -via Digg

Just another way I can prove to my kids that I am, in fact, a genius. -via reddit