A Beautiful Math

John Tierney on fractal geometry:

It’s hard enough to make modern mathematics comprehensible in print, so I’m especially impressed to see anyone try to do it on television. Tonight, at 8 p.m. on PBS, Nova is presenting “Hunting the Hidden Dimension,” an hour-long documentary on what it calls a “compelling mathematical detective story,” the discovery of fractal geometry and its resulting applications. Of course, it doesn’t hurt that there are lots of beautiful examples of fractals the natural world — and the unnatural worlds of “Star Trek” and “Star Wars.”

The documentary, produced and directed by Michael Schwarz and Bill Jersey, tells how the mathematician Benoit Mandelbrot became obssessed with “roughness” because so much in nature was not explained by orderly classical shapes like cones and spheres. He developed equations to explain shapes ranging from clouds to broccoli, and the equations turned out to be useful in creating movies, building cell-phone antennas, developing stronger concrete and a myriad of other applications.

You can create your own version of the Mandelbrot set, the most famous fractal of all. And you don’t even have to solve the equations.

Read the whole thing and watch the documentary here.

Visualizing the Fourth Dimension

How to visualize the fourth (and higher) dimension:

A lot of mathematical art concerns itself with objects that exist in four or more dimensions. Even physicists have been telling us that we live in a ten dimensional universe. How are we, as three-dimensional beings, able to understand this?

Follow the link to find out.

On The Importance of Mathematics

(cartoon via xkcd)

Antonio Cangiano:

Mathematics is the queen of science and the language of nature. Its importance should be clear to any reasonable person. It is easy however to diminish the value of certain areas of research because they’re currently thought as having little practical use.

It is an understandable and even excusable fallacy that there are useful fields of math and useless ones, based on the perception of their applied or theoretical nature. But it’s still a misconception. Each theorem and discovery is a little piece of a larger puzzle that we conveniently categorize into aptly labeled macro-areas. Discoveries and mathematical ideas that are perceived as “useful” today because they’re applicable to engineering, for example, were at a certain point in time considered absolutely abstract and useless, or at least derived or intrinsically connected to some that were. Mathematics matters; all of it.

On the net I found an incredible lecture by the brilliant mathematician Timothy Gowers, entitled “The Importance of Mathematics”. In this keynote, Prof. Gowers makes a very strong case in favor of the value of math, of financing its relatively cheap research and its deep implications on human progress. You can watch the 8 parts that compose the whole video, in the following playlist:

Here is a a PDF of the transcript.

A Math Puzzle to Ponder

A very cool math trick from Steven Levitt:

White tells the following story about Quine: Upon hearing that White’s son Steve was not taught multiplication properly due to a switch in school districts, Quine sent him a letter with the following passage:

Since Stevie misses multiplication, he may enjoy gorging on this one. Well, so you have these two numbers, see, and you want to multiply one of them by the other. O. K., so you write them down more or less side by side, as potential headings of two potentially parallel potential columns, roughly thus:

19 27

Then you go to work on the left one, cutting it in half. Write the half underneath. No fractions, though. If it was odd, just forget the fraction. If it was 58 .. 537, just put down 29 .. 268 as its half. That’s near enough. Then, under that in turn, put its half (ignoring, again, the fraction if any); and so on, until you get down to 1. That completes your left-hand column.

19 27
9
4
2
1

Then go to work on the right-hand number, making a column under it by exactly the opposite method: doubling each time. This could go on forever, but don’t let it. Keep your right-hand column lined up with your left-hand column, entry by entry, and stop as soon as you are opposite the bottom of your left-hand column.

19 27
9 54
4 108
2 216
1 432

O. K., so now you have the two columns side by side. The next thing to do is to start in on the right-hand column and cross out a lot of it. Cross out all the entries which have even numbers opposite them in the left-hand column. Keep only those entries in the right-hand column which have odd numbers opposite them in the left-hand column. All right, now add up the right-hand column, what’s left of it after all the crossing out. The result, unless I have made a mistake somewhere, is the answer to your original multiplication problem.

19 27
9 54
4 108 [XXXed out]
2 216 [XXXed out]
1 432
—–
513 [19 * 27 = 513]

It may be that this information reaches Stevie a bit too late to be altogether useful. If I had tipped him off earlier, he would never had [sic] had to learn the multiplication table.

The algorithm actually works, which you will see if you play around with it. And indeed, although certain aspects of the storytelling are played up to make it seem like magic, after a little study it becomes clear (even to someone as mathematically handicapped as I am) why it works. If you like this sort of thing, it is fun to figure out.

Three Free Math Programs

Math-Blog:

In this short article I propose well known programs that will give you a lot of flexibility and math crunching fun. All of them have advantages and drawbacks and none of them can be considered perfect or infallible, but I consider them some of the best available today in their respective categories. They are rather general purose softwares, but there are plenty of other specialized open source programs if you have specific needs.

I’ve chosen one program for each of the 3 macro categories: symbolic, numeric and statistical computing, but you can expect quite a bit of overlapping and shared functionalities. Try the three of them, try the suggested alternatives and settle with the ones that you like and that meet your needs the best.

Follow the link to see his selections. I hope to try these out now that I have my field exam out of the way.

Quote: von Neumann on Mathematics

"In mathematics you don’t understand things. You just get used to them."

- John von Neumann, mathematician (1903 - 1957)

(HT Neatorama)

Mathematical Beauty

Euler's Relation:

One of the great wonders of mathematical world, Euler’s relation is like the Grand Canyon, Mount Everest and Niagara Falls rolled into one--what you see depends on how you look at it.

You’re probably familiar with the famous optical illusion of a painting of an old crone which suddenly changes into a beautiful young woman as your mental perspective changes. Both images are contained within the picture—they are different aspects of the same pattern of lines on a page. All you have to do is alter your internal viewpoint to see the difference.

Euler’s relation is a bit like this famous optical illusion but on a vastly grander scale. Imagine walking across a featureless landscape and stumbling across the raw natural beauty of Mount Everest. You’d have good reason to be pleased with your discovery but as you continue your journey you reach the breathtaking expanse of the Grand Canyon. And beyond that the thundering majesty of Niagara Falls.

The equivalent of Euler’s relation is a final vantage point that shows how all these entirely different wonders of the mathematical world are actually the same. The new perspective simply gives you the insight that connects them together.

Euler’s relation links five of the most fundamental concepts in mathematics in a simple and elegant formula. It says that when viewed in a particular way, the concepts of one and zero are the same as the concepts of the exponential power, e, the imaginary number, i, and the irrational number p.

And yet Euler’s relation is even more powerful. The equation in this work is actually a special case of a broader relation that links two entirely different branches of mathematics--geometry, the study of space, with algebra, the study of structure and quantity. Perhaps that's why the physicist and Nobel Laureate Richard Feynman called it the most remarkable formula in mathematics.

FURTHER READING
Mathworld and Planetmath.org offer derivations of Euler's relation

Read more on this remarkable equation here and here.

Tied Up In Knots

The mathematics of knots:

Call it Murphy's Law of knots: If something can get tangled up, it will. "Anything that's long and flexible seems to somehow end up knotted," says Andrew Belmonte, an applied mathematician at Pennsylvania State University in University Park. Belmonte has plenty of alarming anecdotal evidence. "It certainly happens in my house, with the cords of the venetian blind." But the knot scourge is a global one, as anyone who owns a desktop computer can confirm after peeking at the mess of connection cables and power cords behind the desk.

Now, scientists think they may have found out how and why things find their way into knotty arrangements. By tumbling a string of rope inside a box, biophysicists Dorian Raymer and Douglas Smith have discovered that knots—even complex knots—form surprisingly fast and often. The string first coils up, and then its free ends swivel around the other coils, tracing a random path among them. That essentially makes the coils into a braid, producing knots, the scientists say.

The results' relevance may go well beyond explaining the epidemic of tangled venetian blind cords. That's because spontaneous knots seem to be prevalent in nature, especially in biological molecules. For example, knottiness may be crucial to the workings of certain proteins (see "Knots in Proteins"). And knots can randomly form in DNA, hampering duplication or gene expression—so much so that living cells deploy special knot-chopping enzymes.

Read the whole thing.

(HT Instapundit)

Mathematical Challenges For the 21st Century

Benjamin Mann of DARPA has constructed a list of 23 challenges for mathematics for this century. Here are five of the top challenges:

1. The Mathematics of Quantum Computing, Algorithms, and Entanglement (DARPA 15) : “In the last century we learned how quantum phenomena shape our world. In the coming century we need to develop the mathematics required to control the quantum world.”

2. Settle the Riemann Hypothesis (DARPA 19) : “The Holy Grail of number theory.”

3. Geometric Langlands and Quantum Physics (DARPA 17) : “How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa?”

4. The Geometry of Genome Space (DARPA 15) : “What notion of distance is needed to incorporate biological utility?”

5. Algorithmic Origami and Biology (DARPA 10) : “Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.”

See the rest of the 23 here. [PDF]

Unwarping Smudged Fingerprints in Record Time

De-smudgifying fingerprints:

Gurus at the University of Warwick have developed a system that "identifies partial, distorted, scratched, smudged, or otherwise warped fingerprints in just a few seconds." The process is garnering attention thanks to its ability to spit out results in the blink of an eye after it "unwarps any fingerprint that has been distorted and creates a clear, digital representation that can then be mapped onto an image space of all other prints held on a database."

Reportedly, researchers have already established the Warwick Warp spinoff company to bring the technology to market, and they're looking in the commercial access control, financial transaction authorization and possibly even ID card / border control segments for opportunities.

Here's more:

Though it's not the company's focus, the Warp technology could have applications in forensics. After all, the system is designed to work with the kinds of non-ideal prints that unintentional situations tend to generate. The ability of the software to work with partial or smudged prints could increase the number of latent prints that investigators can use among those they find in the field.

Here's Charlie on NUMB3RS explaining how it works:

The Monty Hall Problem

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Did you say no? If so, your answer is wrong. You actually double your odds of winning if you switch doors.

Don't believe me? Try it out yourself. I did and verified after 45 tries that I would have doubled my chances of winning if I switched every time.

Watch this video explanation from NUMB3RS:

Still confused? Here's a visual proof of the problem:

I don't know about you, but I find this is absolutely fascinating! Like they said on the NUMB3RS video, this is completely counter-intuitive -- but it works.

Whoever said math isn't cool?

(HT Grey Matters)

NUMB3RS

A few days ago, I had a post about tit-for-tat strategy in game theory. I included a video clip from the TV show NUMB3RS. It is an intriguing TV show about a math genius college professor who helps his FBI agent brother solve complex cases using math. As a former math teacher (I taught math part-time for a couple years at a community college in Orlando), I love what I've seen of the show.

It looks like I'm not the only fan:

As some of you may have guessed from the video in my Monty Hall Dilemma post, I'm a fan of the TV show Numb3rs. If you're not familiar with the show, CBS has a description of the basics of the show, which is already in its 3rd season.

To get a feel for the show, CBS has posted several clips of Charlie Eppes (David Krumholtz) explaining various concepts over on YouTube:

Tracking A Sex Offender Killer
A Big Problem
Math and Shoeless Joe
Betting On The Horses
A Discussion of Randomness

There are many clips posted by fans. My favorites include the previously-mentioned Monty Hall Dilemma clip, as well as Game Theory. Shuzak's Numb3rs clip collection is also very impressive.

The math employed in the show is legitimate, with a team of technical advisors from Caltech providing the research for the show.

Naturally, several site have sprung up to help people better understand the math needed in the show. CBS itself took the first step, and arranged a partnership with Texas Instruments to develop the We All Use Math Everyday website, which includes activities for students that are related to each episode. The name of the site, of course, comes from a quote from the show opening.

Naturally, fans have sites delving into the math of the show, as well. At redhawke, their Numb3rs section is arranged by both episodes and mathematical concepts. Over at Northeastern University, their mathematics department has an entire blog devoted solely to Numb3rs. Those who want to learn about the various concepts can find themselves getting lost in these sites for hours.

I haven't seen many episodes yet, but hope to rent some of the previous seasons soon.

Game Theory: Tit for Tat

Here's a great explanation from the TV show NUMB3RS of the tit-for-tat strategy in an iterated prisoner's dilemma game.

Tit for tat is a highly effective strategy in game theory for the iterated prisoner's dilemma. It was first introduced by Anatol Rapoport in Robert Axelrod's two tournaments, held around 1980. Based on the English saying meaning "equivalent retaliation" ("tit for tat"), an agent using this strategy will initially cooperate, then respond in kind to an opponent's previous action. If the opponent previously was cooperative, the agent is cooperative. If not, the agent is not. This is similar to reciprocal altruism in biology.

Here is the basic set-up for the prisoner's dilemma game. Notice that regardless of what the other prisoner does, each prisoner is better off betraying the other rather than staying silent, even though their optimal cooperative strategy would be for both to stay silent. (Hence the dilemma.)

	Prisoner B Stays Silent	Prisoner B Betrays
Prisoner A Stays Silent	Each serves six months	Prisoner A serves ten years Prisoner B goes free
Prisoner A Betrays	Prisoner A goes free Prisoner B serves ten years	Each serves five years

The prisoner's dilemma has many applications in the analysis of economics, law, biology, politics, etc. It's ubiquity is what makes it such an interesting thing to study.

From what little I've seen of it, NUMB3RS is an excellent show. As a former math teacher, it is great seeing a show like this popularizing many of the cooler aspects and applications of various branches of mathematics.

Outside In

Ever wonder how to turn a sphere inside out? Watch this topology video to understand how.



See more great examples of mathematical visualizations here.

(HT Grey Matters)

The Mathematics of Procrastination

I'm sure I could have figured this out -- if I'd gotten around to it...

Here's another formula for procrastination and ten reasons why people procrastinate from Psychology Today. (HT Lifehacker)

Also, here is a video about how procrastination really works. (Unfortunately, this describes my life far better than I wish it did.)

Confession: Writing this post is helping me avoid that computer program I have due on Tuesday...

Mobius Transformations Demystified

Art of Problem Solving:

I found a nice animation explaining Mobius transformations. I feel like a lot of mathematics is like this: There's actually a very simple idea underlying what looks like a complicated concept. (I only learned recently that geometric inversion can be modeled by swapping the north and south hemispheres of the Riemann sphere.)

(HT Good Math, Bad Math)

'The Unreasonable Effectiveness of Mathematics in the Natural Sciences'

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. -- Eugene Wigner

Read Wigner's whole article, 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences'. (You can also download a PDF copy of the article and read its Wikipedia entry.) Wigner brings up the tremendous question of why mathematics, which was mostly developed independently of application, does such a marvelous job of describing the physical world? It is a deeply profound question if you stop to contemplate it, both from a scientific and theological perspective.

The short theological answer is that mathematics illuminates the underlying beauty of the structure of creation. While I strongly support this point of view, it still leaves the original question largely unanswered. Why is the universe able to be understood in a manner comprehensible by man? And why by tools that were not originally developed to do so?

Einstein himself was baffled by this:

The most incomprehensible thing about the universe is that it is comprehensible. -- Albert Einstein

How can it be that mathematics, being after all product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? -- Albert Einstein.

There is absolutely no question that math does a wonderful job of modeling the natural world. It is not without limitations, but the deeper we delve into the mathematical world, the deeper we are able to delve into the physical. Virtually all of physics, most of engineering, much of chemistry, and increasingly biology rests on mathematical application and understanding. It is quite amazing that this should be so.

Most of economic growth, whether it be from increases in agricultural production, computer technology, the continued advancement of transportation (air, land, and sea), advances in statistics and quality control, accounting and financial innovations, the evolving of venture capital and expansion of credit markets, etc., has math as a critical part of its foundation. Without the discovery of math, mankind would have never advanced to where it is today.

A much less settled question is how effective and appropriate mathematics is for modeling the social sciences. I'll save that discussion for another day...

Math Humor

What pi said to i:

The Value of Being "Numerate"

Numeracy:

the capacity for quantitative thought and expression

I was listening to an EconTalk podcast on my way to school today with Russ Roberts interviewing Dan Pink, author of A Whole New Mind. During the podcast, Pink said that he thinks it is critical for people to be both literate and numerate if they wanted to succeed in today's economy. This got me thinking and I couldn't agree more.

When I was living in Orlando, I used to teach math at Valencia Community College in Orlando. One of the questions I persistently got from my students was:

"Why do I need to know math?"

I think this chart succinctly answers this question. Compare the salaries of the fields that have significant quantitative orientation with those that do not.

While it is certainly true that it is possible to succeed in life without strong quantitative skills, they help stack the odds in your favor tremendously. What if you don't have strong math skills? My experience as both a teacher and a student is that nearly anyone is able to develop signifcant abilities in most areas through persistent practice. (See my previous posts on "Is A Star Born or Made?" , "The Invisibility Factor of Getting Good", and "The Myth of Talent" for more on this.)

When I was teaching, I had a 50+ year-old woman as a student who was going to college for her first time. She had grown up being told that women were no good at math and was delighted to discover later in life that she really could learn and excel at quantitative thinking. As much as any student I've ever had, her enthusiasm and excitement made teaching a true joy. In her own words: "you really can teach an old dog new tricks!" Anytime I get discouraged about struggling through learning something new, I just need to remember her example to get inspired to plow through.

I would strongly encourage anyone and everyone at any stage in life to build higher levels of numeracy. It makes life both more interesting and more fun and helps develop thinking skills that lead to higher paying jobs. For anyone interested in learning more about mathematical concepts from a layman's perspective, I highly recommend Ian Stewart's Concepts of Modern Mathematics. I am currently reading through it and it provides a concise, clearly communicated overview of many mathematical concepts in a manner accessible to nearly any dedicated reader who understands basic high-school-level algebra. I'd also recommend Mathematics: A Very Short Introduction by Timothy Gowers and Innumeracy by John Allen Paulos. I'd strongly encourage reading books like these to help build an intuitive grasp of what mathematics actually represents to help enhance your interest and understanding of mathematical concepts.

For anyone interested in studying math for economics, I've found nothing better than Alpha Chiang's Fundamental Methods of Mathematical Economics.

For those with highly developed quantitative skills, Dan Pink encouraged the necessity of developing strong communication skills and fostering creative thinking. For building these types of skills, I'd recommend Strunk and White's Elements of Style, Mortimer Adler's How to Read a Book, and Roger von Oech's A Whack on the Side of the Head. For economists, I'd also highly recommend Diedre McCloskey's Economical Writing. Dan Pink and Russ Roberts both praised Drawing on the Right Side of the Brain by Betty Edwards as a great place to start learning artistic skills. It is a book I hope to read soon.

The Ultimate Equation Sheet

For any of you engineering types out there, the National Council of Examiners for Engineering and Surveying has a fantastic 235-page reference handbook of engineering equations available as a free download. It covers a broad range of engineering topics and is incredibly comprehensive, covering many engineering fields (electrical, mechanical, chemical, civil, industrial, etc.). It also has significant material on mathematics and statistics. Looking at this really brings back memories of many late night cram sessions down at Virginia Tech.

If you're looking for a good engineering resource and reminders of what you might have forgotten, be sure to check it out! You might also like Engineering Formulas by Kurt and Reiner Gieck. I have the 6th edition of this pocket-sized reference book and highly recommend it.