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The Golden Ratio — that geometric expression of the Fibonacci sequence of numbers (1, 1, 2, 3, 5, etc.) — has influenced the way master painters created art and can be spotted occurring naturally in the seed arrangement on the face of a sunflower. But its serendipitous appearances aren't nearly as frequent as pop culture would have you believe, writes Samuel Arbesman at The Nautilus. In fact, one of the most common examples of mathematical perfection — the chambered nautilus shell — actually isn't. Even math can become part of the myths we tell ourselves as we try to create meaning in the universe.
Michael de Podesta has been doing the math on "Grime Dice" -- six sided cubes whose sides average out to 3.5, but whose face values are all radically different:
The interesting thing about these is that the odds of one die beating another are simple to calculate, but shift radically once you start rolling dice in pairs. It's a beautiful piece of counterintuitive probability math:
The amazing property of these dice is discernible when you use them competitively – i.e. you roll one dice against another. If you roll each of them against a normal dice then as you might expect, each dice will win as often as it will lose. But if you roll them against each other something amazing happens.
- Dice A will systematically beat Dice B
- Dice B will systematically beat Dice C
- Dice C will systematically beat Dice A
So the fact that Dice A beats Dice B, and Dice B beats Dice C does not ensure that Dice A will beat Dice C. Wow!
And how about this: If you ‘double up’ and roll 2 Dice A‘s against 2 Dice B‘s – the odds change around and now the B‘s will beat the A‘s ! Is that really possible? Well yes, and just to convince myself I wrote a Spreadsheet (.xlsx file) and generated the tables at the bottom of the article. If you download it you can change the numbers to try out other combinations.
A clip from the Discovery Channel's Dirty Jobs program on tanneries demonstrates the workings of a calculus-performing mechanical calculator that measures the surface-area of irregularly shaped hides with a fascinating and clever set of gears, calipers and ratchets.
Dirty Jobs - Tannery Mechanical Surface Integrator (Thanks, Dad!)
Jeremy Kun, a mathematics PhD student at the University of Illinois in Chicago, has posted a wonderful primer on probability theory for programmers on his blog. It's a subject vital to machine learning and data-mining, and it's at the heart of much of the stuff going on with Big Data. His primer is lucid and easy to follow, even for math ignoramuses like me.
For instance, suppose our probability space is and is defined by setting for all (here the “experiment” is rolling a single die). Then we are likely interested in more exquisite kinds of outcomes; instead of asking the probability that the outcome is 4, we might ask what is the probability that the outcome is even? This event would be the subset , and if any of these are the outcome of the experiment, the event is said to occur. In this case we would expect the probability of the die roll being even to be 1/2 (but we have not yet formalized why this is the case).
As a quick exercise, the reader should formulate a two-dice experiment in terms of sets. What would the probability space consist of as a set? What would the probability mass function look like? What are some interesting events one might consider (if playing a game of craps)?
Kevin Kelly told me about Phil Scale's new iOS application to teach kids arithmetic. It's called Math Fleet and it sounds great. (Phil created Kevin's Asia Grace photobook app, which is also wonderful.)
I've been an independent iOS app developer for four years, and my wife, Jennifer, and I work together from our home in Austin creating games and educational apps. Our newest app is called Math Fleet, an action game set in space where players must use quick math skills to save Earth from invasion, all while dodging asteroids and battling enemy star fighters.
The inspiration for the game came from our sons, Jack, Luke and Dylan (ages 7, 6 and 3) for whom I've downloaded and tried many educational apps and games. At the beginning of the summer we were all home together and I was brainstorming the next game, which I knew I wanted to set in space. I had my sons Jack and Luke playing a math game I had downloaded to earn playing time for the other games they really wanted to play. We all agreed though that the math game I had them playing, which cost me $5, wasn't a very good game and there really wasn't very much math in it. I knew I could write a better game and we started talking about what we would do differently, and in that moment, we decided our mission was to create not just a better game, but the most awesome action math game out there, and that Jack and Luke would be in the driver's seat guiding how the game would take shape.
It became the ultimate geek dad summer for me as I fully committed myself to making their ideas a reality, and some of their ideas were pretty challenging to implement. Such as Luke's idea that the user pilot multiple ships, customizable and upgradable -- the eventual foundation of the game; Or the Patrol Sector concept drawn up by my 7-year-old, Jack.
Throughout the eight month journey, as I coded, they tested prototypes of controls, menus, action sequences, effects, space weapons, and ships. They told me what they liked and what they didn't like, what they'd do differently, and in many cases they would contribute key design concepts delivered as crayon drawings or Lego models. I learned more about game design by watching them, listening, and discussing ideas than I learned from writing my previous two games.
Finally, after eight months and 300,000 lines of code, Math Fleet is officially released, and is hopefully everything we set out to create to create an exciting math game that kids actually want to play, while also challenging their minds by combining fast problem solving with the stress and distraction of piloting a space fleet.
Jeff Moser has a clear, fascinating enumeration of all the incredible math stuff that happens between a server and your browser when you click on an HTTPS link and open a secure connection to a remote end. It's one of the most important (and least understood) parts of the technical functioning of the Internet.
People sometimes wonder if math has any relevance to programming. Certificates give a very practical example of applied math. Amazon's certificate tells us that we should use the RSA algorithm to check the signature. RSA was created in the 1970's by MIT professors Ron *R*ivest, Adi *S*hamir, and Len *A*dleman who found a clever way to combine ideas spanning 2000 years of math development to come up with a beautifully simple algorithm:
You pick two huge prime numbers "p" and "q." Multiply them to get "n = p*q." Next, you pick a small public exponent "e" which is the "encryption exponent" and a specially crafted inverse of "e" called "d" as the "decryption exponent." You then make "n" and "e" public and keep "d" as secret as you possibly can and then throw away "p" and "q" (or keep them as secret as "d"). It's really important to remember that "e" and "d" are inverses of each other.
Now, if you have some message, you just need to interpret its bytes as a number "M." If you want to "encrypt" a message to create a "ciphertext", you'd calculate:
C ≡ Me (mod n)
This means that you multiply "M" by itself "e" times. The "mod n" means that we only take the remainder (e.g. "modulus") when dividing by "n." For example, 11 AM + 3 hours ≡ 2 (PM) (mod 12 hours). The recipient knows "d" which allows them to invert the message to recover the original message:
Cd ≡ (Me)d ≡ Me*d ≡ M1 ≡ M (mod n)