Let’s say, for example, you want to bet on one of the highlights of the British sporting calendar, the annual university boat race between old rivals Oxford and Cambridge. One bookie is offering 3 to 1 on Cambridge to win and 1 to 4 on Oxford. But a second bookie disagrees and has Cambridge evens (1 to 1) and Oxford at 1 to 2.
Each bookie has looked after his own back, ensuring that it is impossible for you to bet on both Oxford and Cambridge with him and make a profit regardless of the result. However, if you spread your bets between the two bookies, it is possible to guarantee success (see diagram, for details). Having done the calculations, you place £37.50 on Cambridge with bookie 1 and £100 on Oxford with bookie 2. Whatever the result you make a profit of £12.50.
I say relatively because there are literally millions of pages on the web just about blackjack statistics. For instance, it’s easy to see how you’ll lose money playing blackjack in the long run — card counting aside — by looking at this house edge calculator. The only real advantage to the player occurs with a one-deck shoe and a bunch of other pro-player rules, which I imagine are difficult to find at the casinos. (via big contrarian)
Now, here’s the part that really boggled me: the Consumption/Waste idea is a 1:1 correspondence (something in yields something out), what mathematicians call a linear function. The Parabola idea connects, pretty obviously, with parabolas — now we’re looking at x raised to the power of two. Annular Systems are modeled by circles which are given in analytic geometry by equations with both x^2 and y^2. Limits and Infinity, of course, become necessary in order to find the area of shapes under curves like parabolas and three-dimensional projections of circles.
Whoa. That is a tiny bit mind-blowing…do I really have time for a reread right now? (thx, nick)
Watch as David Attenborough signals his interest in mating with a male cicada. Scientists think that cicadas have 13- or 17-year mating cycles because, being prime numbers, those periods are not divisible by those periods of potential predators. From Stephen J. Gould:
Many potential predators have 2-5-year life cycles. Such cycles are not set by the availability of cicadas (for they peak too often in years of nonemergence), but cicadas might be eagerly harvested when the cycles coincide. Consider a predator with a life-cycle of five years: if cicadas emerged every 15 years, each bloom would be hit by the predator. By cycling at a large prime number, cicadas minimize the number of coincidences (every 5 x 17, or 85 years, in this case). Thirteen- and 17-year cycles cannot be tracked by any smaller number.
Newish episode of Radiolab about randomness: Stochasticity.
How big a role does randomness play in our lives? Do we live in a world of magic and meaning or … is it all just chance and happenstance? To tackle this question, we look at the role chance and randomness play in sports, lottery tickets, and even the cells in our own body. Along the way, we talk to a woman suddenly consumed by a frenzied gambling addiction, two friends whose meeting seems purely providential, and some very noisy bacteria.
Regarding the game of Who Can Name the Bigger Number?, Scott Aaronson shows that while 9^9^9^9 might cut the mustard in the first couple of rounds, the numbers and the notation used to express them get much more complicated.
Exponentials are familiar, relevant, intimately connected to the physical world and to human hopes and fears. Using the notational systems I’ll discuss next, we can concisely name numbers that make exponentials picayune by comparison, that subjectively speaking exceed 9^9^9^9 as much as the latter exceeds 9.
Geoffrey West of the Santa Fe Institute and his colleagues Jim Brown and Brian Enquist have argued that a 3/4-power law is exactly what you’d expect if natural selection has evolved a transport system for conveying energy and nutrients as efficiently and rapidly as possible to all points of a three-dimensional body, using a fractal network built from a series of branching tubes — precisely the architecture seen in the circulatory system and the airways of the lung, and not too different from the roads and cables and pipes that keep a city alive.
Joe liked the idea of measuring how long this number would be if it were set in type, which immediately called into question the choice of font. The number’s length would depend chiefly on the width of the font selected, and even listener-friendly choices like Times Roman and Helvetica would produce dramatically different outcomes. Small eccentricities in the design of a particular number, such as Times Roman’s inexplicably scrawny figure one, would have huge consequences when multiplied out to this length. But even this isn’t the hairy part. Where things get difficult, as always, is in the kerning.
In some cases, properly kerning the number resulted in a difference of more than 1000 feet for 12 pt. text.
In the complex formula L represents the number of lumps in the batter and C equals its consistency. The letter F stands for the flipping score, k is the ideal consistency and T is the temperature of the pan. Ideal temp of pan is represented by m, S is the length of time the batter stands before cooking and E is the length of time the cooked pancake sits before being eaten. The closer to 100 the result is — the better the pancake.
However, a commenter notes:
According to that formula, if you left the pancake batter standing for ten years, (s-e) would be large, and so the pancake would be near perfect. If you let it stand for the same time as you left the pancake to cool, (s-e) would be zero and the pancake would be infinitely bad.
In The Method, Archimedes was working out a way to compute the areas and volumes of objects with curved surfaces, which was also one of the problems that motivated Newton and Leibniz. Ancient mathematicians had long struggled to “square the circle” by calculating its exact area. That problem turned out to be impossible using only a straightedge and compass, the only tools the ancient Greeks allowed themselves. Nevertheless, Archimedes worked out ways of computing the areas of many other curved regions.
The same thing happened: something would look good at first and then turn out to be horrifying. For example, there was a book that started out with four pictures: first there was a windup toy; then there was an automobile; then there was a boy riding a bicycle; then there was something else. And underneath each picture it said, “What makes it go?”
I thought, “I know what it is: They’re going to talk about mechanics, how the springs work inside the toy; about chemistry, how the engine of the automobile works; and biology, about how the muscles work.”
It was the kind of thing my father would have talked about: “What makes it go? Everything goes because the sun is shining.” And then we would have fun discussing it:
“No, the toy goes because the spring is wound up,” I would say. “How did the spring get wound up?” he would ask.
“I wound it up.”
“And how did you get moving?”
“From eating.”
“And food grows only because the sun is shining. So it’s because the sun is shining that all these things are moving.” That would get the concept across that motion is simply the transformation of the sun’s power.
I know I’ve posted this one before but I’m probably gonna post it each time I run across it.
That’s chef Kin Jing Mark stretching and dividing dough into super-thin noodles. Seeing this when I was a kid made a great impression on me about the wonder of mathematics.
DARPA is soliciting research proposals for people wishing to solve one of twenty-three mathematical challenges, many of which deal with attempting to find a mathematical basis underlying biology.
What are the Fundamental Laws of Biology?: This question will remain front and center for the next 100 years. DARPA places this challenge last as finding these laws will undoubtedly require the mathematics developed in answering several of the questions listed above.
- The air in the Empire State Building weighs about 4 million pounds.
- The energy consumption of the world’s population will be greater than the energy coming from the sun in less than 500 years. (Peak photons?)
What’s surprising about such estimates is how often they are very close to the reality. This is especially true in a multi-step approximation, where over- and underestimates at various steps tend to cancel each other out, usually resulting in something not too far off from the truth.
Both Microsoft and Google use questions like these as part of their job interview process. We did a bunch of them in my freshman physics class; I loved them.
Banknote patterns fascinate me. I can get lost for hours in all the details, seeing how the patterns fit together, how the lettering works, the tiny security ‘flaws’ — they’re amazing. Central to banknote designs are Guilloche patterns, which can be created mechanically with a geometric lathe, or more likely these days, mathematically. The mathematical process attracted me immediately as I don’t have a geometric lathe and nor do I have anywhere to put one. I do, however, have a computer, and at the point I first started playing with the designs (mid-2004) Illustrator and Photoshop had gained the ability to be scripted.
In case you’re wondering, the most densely populated block group is one in New York County, New York — 3,240 people in 0.0097 square miles, for about 330,000 per square mile. The least dense is in the North Slope Borough of Alaska — 3 people in 3,246 square miles, or one per 1,082 square miles. The Manhattan block group I mention here is 360 million times more dense than the Alaska one; population densities vary over a huge range.
That’s approximately the same range from the height of an iPod to the diameter of the Earth. (via fakeisthenewreal)
Benoit Mandelbrot and Paola Antonelli talk about, among other things, fractals, self-similarity in architecture, algorithms that could specify the creation of entire cities, visual mathematics, and generalists.
This has been for me an extraordinary pleasure because it means a certain misuse of Euclid is dead. Now, of course, I think that Euclid is marvelous, he produced one of the masterpieces of the human mind. But it was not meant to be used as a textbook by millions of students century after century. It was meant for a very small community of mathematicians who were describing their works to one another. It’s a very complicated, very interesting book which I admire greatly. But to force beginners into a mathematics in this particular style was a decision taken by teachers and forced upon society. I don’t feel that Euclid is the way to start learning mathematics. Learning mathematics should begin by learning the geometry of mountains, of humans. In a certain sense, the geometry of…well, of Mother Nature, and also of buildings, of great architecture.
Speaking of the Yankees, Derek Jeter always seems to get a lot of credit for those four World Series victories in five years but a quick look at the OBP stats for those years shows that Bernie Williams was the engine driving that offense. Jeter’s a little overrated maybe?
Called “Hilbert” after the influential German mathematician, David Hilbert, the newly licensed software will be browser accessible and, utilizing AJAX technologies, will emulate the desktop version of the software with remarkable fidelity. “The magic of AJAX will allow OST to combine or ‘mash-up’ Mathematica with other web-based technologies to deliver and support high quality science and mathematics courses online such as the Calculus&Mathematica courses currently taught through NetMath at the University of Illinois and other universities,” explains Scott Gray, Director of the O’Reilly School of Technology.
Hilbert should be available before the end of the year.
Infinite Jest once again proved finite, although it’s taken me since August to get through it. This book was such a revelation the first time through that I was afraid of a reread letdown but I enjoyed it even more this time around…and got much more out of the experience too.
Right as I was finishing the book, I read a transcription of an interview with Wallace in which interviewer Michael Silverblatt asked him about the fractal-like structure of the novel:
MICHAEL SILVERBLATT: I don’t know how, exactly, to talk about this book, so I’m going to be reliant upon you to kind of guide me. But something came into my head that may be entirely imaginary, which seemed to be that the book was written in fractals.
DAVID FOSTER WALLACE: Expand on that.
MS: It occurred to me that the way in which the material is presented allows for a subject to be announced in a small form, then there seems to be a fan of subject matter, other subjects, and then it comes back in a second form containing the other subjects in small, and then comes back again as if what were being described were — and I don’t know this kind of science, but it just — I said to myself this must be fractals.
DFW: It’s — I’ve heard you were an acute reader. That’s one of the things, structurally, that’s going on. It’s actually structured like something called a Sierpinski Gasket, which is a very primitive kind of pyramidical fractal, although what was structured as a Sierpinski Gasket was the first- was the draft that I delivered to Michael in ‘94, and it went through some I think ‘mercy cuts’, so it’s probably kind of a lopsided Sierpinski Gasket now. But it’s interesting, that’s one of the structural ways that it’s supposed to kind of come together.
MS: “Michael” is Michael Pietsche, the editor at Little, Brown. What is a Sierpinski Gasket?
DFW: It would be almost im- … I would almost have to show you. It’s kind of a design that a man named Sierpinski I believe developed — it was quite a bit before the introduction of fractals and before any of the kind of technologies that fractals are a really useful metaphor for. But it looks basically like a pyramid on acid —
To answer Silverblatt’s question, a Sierpinski Gasket is constructed by taking a triangle, removing a triangle-shaped piece out of the middle, then doing the same for the remaining pieces, and so on and so forth, like so:
The result is an object of infinite boundary and zero area — almost literally everything and nothing at the same time. A Sierpinski Gasket is also self-similar…any smaller triangular portion is an exact replica of the whole gasket. You can see why Wallace would have wanted to structure his novel in this fashion.
What’s sort of great about it is that it will happen to everybody if you live long enough. If you were born in 2000, it happens instantaneously. The people who were born at the end of the century have to take care of themselves.
Basically, as the leaf grows it is constrained to a 2-d surface, but the cells of some leaves reproduce fast enough to require more surface area than a pi-r-squared plane surface can provide. Its only recourse is to buckle out-of-plane, giving the wrinkles. Since the exuberant growth continues as the leaf grows outward, the buckling process repeats and you get the multi-scale (ripples on ripples on ripples) shape that you see in kale and daffodils.
Cadaeic Cadenza is a 3834-word story by Mike Keith where each word in sequence has the same number of letters as the corresponding digit in pi. (thx, mark, who has more info on constrained writing) Related: The Feynman point is the sequence of six 9s which begins 762 digits into pi. “[Feynman] once stated during a lecture he would like to memorize the digits of pi until that point, so he could recite them and quip ‘nine nine nine nine nine nine and so on.’”
Stay Connected