Weierstrass functions Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere". Here is an example of one: It is not hard to show that this series converges for all x. In fact, it is absolutely convergent. It is also an example of a fourier series, a very important and fun type of series. It can be shown that the function is continuous everywhere, yet is differentiable at no values of x. Here's a graph of the function. You can see it's pretty bumpy. Below is an animation, zooming into the graph at x=1. Wikipedia and MathWorld both have informative entries on Weierstrass functions. back to Dr.
How To Slice A Bagel Along A Mobius Strip — And Why In the weeks before Doug Sohn closed down his legendary Chicago sausage joint Hot Doug’s, people were literally walking in the door and offering him a million dollars to stay open. This week on The Sporkful podcast, we’re featuring part one of our live show at the Taste of Chicago. I talk to Doug about why he walked away from all that money, and one of the top chefs in the world reveals his favorite candy bar. As part of our live show I also interviewed mathematician Eugenia Cheng, author of How To Bake Pi: An Edible Exploration of the Mathematics of Mathematics, who sliced a bagel along a Mobius strip live on stage. A Mobius strip, as you probably forgot, is a surface with only one side. If you were to start drawing a line down the middle of the strip and just keep going, you’d cover all the paper and end up right back where you started, without ever flipping it over. How did that make the bagel more delicious? “Well, it’s basically completely ridiculous,” Cheng explains.
What does 0^0 (zero raised to the zeroth power) equal? Why do mathematicians and high school teachers disagree Clever student: I know! Now we just plug in x=0, and we see that zero to the zero is one! Cleverer student: No, you’re wrong! which is true since anything times 0 is 0. Cleverest student : That doesn’t work either, because if then is so your third step also involves dividing by zero which isn’t allowed! and see what happens as x>0 gets small. So, since = 1, that means that High School Teacher: Showing that approaches 1 as the positive value x gets arbitrarily close to zero does not prove that . is undefined. does not have a value. Calculus Teacher: For all , we have Hence, That is, as x gets arbitrarily close to (but remains positive), stays at On the other hand, for real numbers y such that , we have that That is, as y gets arbitrarily close to Therefore, we see that the function has a discontinuity at the point . but when we approach (0,0) along the line segment with y=0 and x>0 we get Therefore, the value of is going to depend on the direction that we take the limit. that will make the function ! . as is whatever
12 Mind Blowing Number Systems From Other Languages Today is a big day for lovers of the number 12, and no one loves 12s more than the members of the Dozenal Society. The Dozenal Society advocates for ditching the base-10 system we use for counting in favor of a base-12 system. Because 12 is cleanly divisible by more factors than 10 is (1, 2, 3, 4, 6 and 12 vs. 1, 2, 5 and 10), such a system would neaten up our mathematical lives in various ways. But a dozenal system would require us to change our number words so that, for example, what we know as 20 would mean 24 (2x12), 30 would mean 36, and so on. Does that blow your mind a little too much? 1. Photo Courtesy of Austronesian Counting The Oksapmin people of New Guinea have a base-27 counting system. 2. Tzotzil, a Mayan language spoken in Mexico, has a vigesimal, or base-20, counting system. 3. 4. Though modern Welsh uses base-10 numbers, the traditional system was base-20, with the added twist of using 15 as a reference point. 5. 6. 7. 8. 9. 10. 11. 12.
K-MODDL > Tutorials > Reuleaux Triangle If an enormously heavy object has to be moved from one spot to another, it may not be practical to move it on wheels. Instead the object is placed on a flat platform that in turn rests on cylindrical rollers (Figure 1). As the platform is pushed forward, the rollers left behind are picked up and put down in front. An object moved this way over a flat horizontal surface does not bob up and down as it rolls along. The reason is that cylindrical rollers have a circular cross section, and a circle is closed curve "with constant width." What does that mean? Is a circle the only curve with constant width? How to construct a Reuleaux triangle To construct a Reuleaux triangle begin with an equilateral triangle of side s, and then replace each side by a circular arc with the other two original sides as radii (Figure 4). The corners of a Reuleaux triangle are the sharpest possible on a curve with constant width. Here is another really surprising method of constructing curves with constant width:
Famed number π found hidden in the hydrogen atom Three hundred and sixty years ago, British mathematician John Wallis ground out an unusual formula for π, the famed number that never ends. Now, oddly, a pair of physicists has found that the same formula emerges from a routine calculation in the physics of the hydrogen atom—the simplest atom there is. But before you go looking for a cosmic connection or buy any crystals, relax: There is probably no deep meaning to the slice of π from the quantum calculation. Defined as the ratio of the circumference of a circle to its diameter, π is one of the weirder numbers going. Its decimal representation, 3.14159265358979 …, never ends and never repeats. Deriving that formula didn't come easy for Wallis, says Tamar Friedmann, a mathematician and physicist at the University of Rochester (U of R) in New York.
Great Literature Is Surprisingly Arithmetic A good book evokes a variety of emotions as you read. Turns out, though, that almost all novels and plays provide one of only six “emotional experiences” from beginning to end—a rags-to-riches exuberance, say, or a rise and fall of hope (below, top). Researchers at the University of Vermont graphed the happiness and sadness of words that occurred across the pages of more than 1,300 fiction works to reveal the emotional arcs and discovered relatively few variations. A different study coordinated by Poland's Institute of Nuclear Physics found that sentence lengths in books frequently form a fractal pattern—a set of objects that repeat on a small and large scale, the way small, triangular leaflets make up larger, triangular leaves that make up a larger, triangular palm frond (below, bottom). Why analyze the mathematics of literature?
A quantum technique highlights math’s mysterious link to physics It has long been a mystery why pure math can reveal so much about the nature of the physical world. Antimatter was discovered in Paul Dirac’s equations before being detected in cosmic rays. Quarks appeared in symbols sketched out on a napkin by Murray Gell-Mann several years before they were confirmed experimentally. Einstein’s equations for gravity suggested the universe was expanding a decade before Edwin Hubble provided the proof. Nobel laureate physicist Eugene Wigner alluded to math’s mysterious power as the “unreasonable effectiveness of mathematics in the natural sciences.” But maybe there’s a new clue to what that explanation might be. Sign Up For the Latest from Science News Headlines and summaries of the latest Science News articles, delivered to your inbox At least that’s a conceivable implication of a new paper that has startled the interrelated worlds of math, computer science and quantum physics. Verifying a proof without actually seeing it is not that strange a concept.
Quanta Magazine After coming up with this architecture, the researchers used a bank of elementary functions to generate several training data sets totaling about 200 million (tree-shaped) equations and solutions. They then “fed” that data to the neural network, so it could learn what solutions to these problems look like. After the training, it was time to see what the net could do. The computer scientists gave it a test set of 5,000 equations, this time without the answers. For almost all the problems, the program took less than 1 second to generate correct solutions. Despite the results, the mathematician Roger Germundsson, who heads research and development at Wolfram, which makes Mathematica, took issue with the direct comparison. Germundsson also noted that despite the enormous size of the training data set, it only included equations with one variable, and only those based on elementary functions. Still, they agree that the new approach will prove useful.
Quanta Magazine We like to say that anything is possible. In Norton Juster’s novel The Phantom Tollbooth, the king refuses to tell Milo that his quest is impossible because “so many things are possible just as long as you don’t know they’re impossible.” In reality, however, some things are impossible, and we can use mathematics to prove it. People use the term “impossible” in a variety of ways. It can describe things that are merely improbable, like finding identical decks of shuffled cards. It can describe tasks that are practically impossible due to a lack of time, space or resources, such as copying all the books in the Library of Congress in longhand. Mathematical impossibility is different. The punishment for what was perhaps the first proof of impossibility was severe. More than a century later, Euclid elevated the line and the circle, considering them the fundamental curves in geometry. Although these problems are geometric in nature, the proofs of their impossibility are not.
The Subtle Art of the Mathematical Conjecture Mountain climbing is a beloved metaphor for mathematical research. The comparison is almost inevitable: The frozen world, the cold thin air and the implacable harshness of mountaineering reflect the unforgiving landscape of numbers, formulas and theorems. And just as a climber pits his abilities against an unyielding object — in his case, a sheer wall of stone — a mathematician often finds herself engaged in an individual battle of the human mind against rigid logic. In mathematics, the role of these highest peaks is played by the great conjectures — sharply formulated statements that are most likely true but for which no conclusive proof has yet been found. Remarkably, mathematics has elevated the formulation of a conjecture into high art. Like every art form, a great conjecture must meet a number of stringent criteria. The highest summits are not conquered in a single effort. A wonderful example of this phenomenon is the proof of Fermat’s Last Theorem by Andrew Wiles in 1994.