The Truth about the Golden Ratio

Steven Strogatz, in a piece titled “Proportion Control” at The New York Times, dispels the myths behind the famous number known as the golden ratio (φequal to 1/2+√5/2):

Unfortunately, in the more than two millenniums since Euclid, the golden ratio has suffered from so much hype, numerology and wishful thinking that it’s become hard to separate the myth from the math. Many of its supposed occurrences in nature, anatomy, art and architecture don’t stand up to careful scrutiny. For example, you can find lots of books and Web sites claiming that the shell of the chambered nautilus obeys the golden ratio, but in reality, nautilus shells have average growth ratios between 1.24 and 1.43, quite far from 1.618.

So be skeptical the next time you see the golden ratio being used to sell blue jeans, stock tips or the perfect smile.

The upside is, if a nautilus can’t get its proportions golden, maybe I shouldn’t worry so much about mine.

Pass the nachos.

Love that conclusion.

Physicist Uses Math, Writes Paper, To Beat Traffic Ticket

What would you do to get out of a traffic ticket, if you were convinced you were innocent? Probably not as much as Dmitri Krioukov, a physicist based at the University of California San Diego, who was fined for (purportedly) running a stop sign. In a paper titled “Proof of Innocence,” Krioukov argues three physical phenomena combined at just the right time and misled the officer:

We show that if a car stops at a stop sign, an observer, e.g., a police ocer, located at a certain distance perpendicular to the car trajectory, must have an illusion that the car does not stop, if the following three conditions are satis ed: (1) the observer measures not the linear but angular speed of the car; (2) the car decelerates and subsequently accelerates relatively fast; and (3) there is a short-time obstruction of the observer’s view of the car by an external object, e.g., another car, at the moment when both cars are near the stop sign.

When Krioukov drove toward the stop sign the police officer was approximating Krioukov’s angular velocity instead of his linear velocity. This happens when we try to estimate the speed of a passing object, and the effect is more pronounced for faster objects. In Krioukov’s case, the police cruiser was situated about 100 feet away from a perpendicular intersection with a stop sign. Consequently, a car approaching the intersection with constant linear velocity will rapidly increase in angular velocity from the police officer’s perspective. A sneeze caused Krioukov to slam on the brakes hard as he approached the stop sign. With a potential car blocking the officer’s view for a split second, it appeared as though Krioukov never slowed down.

This mathematical description swayed the judge (or maybe he was simply impressed by Krioukov dedication in writing a paper on this personal incident), and the case was dismissed. What a way to get out of paying a traffic ticket!


(via Physics Central)

Finding Waldo with Mathematica

For those of you who are fans of Finding Waldo and have a bit of a nerdy side to you, you’ll appreciate that someone figured out how to find Waldo using Mathematica:

Finding Waldo with the help of Mathematica.

The author describes his technique and provides the relevant code:

First, I’m filtering out all colours that aren’t red

waldo = Import[""]; 
red = Fold[ImageSubtract, #[[1]], Rest[#]] &@ColorSeparate[waldo]; 

Next, I’m calculating the correlation of this image with a simple black and white pattern to find the red and white transitions in the shirt.

corr = ImageCorrelate[red,    Image@Join[ConstantArray[1, {2, 4}], 
ConstantArray[0, {2, 4}]],    NormalizedSquaredEuclideanDistance]; 

I use Binarize to pick out the pixels in the image with a sufficiently high correlation

and draw white circle around them to emphasize them using Dilation

pos = Dilation[ColorNegate[Binarize[corr, .12]], DiskMatrix[30]]; 

I had to play around a little with the level. If the level is too high, too many false positives are picked out.

Finally I’m combining this result with the original image to get the result above

found = ImageMultiply[waldo, ImageAdd[ColorConvert[pos, "GrayLevel"], .5]]



(via Kottke)

The Charlemagne Riddle and Pedigree Collapse

Robert Krulwich recounts a story/riddle about a guy who discovered he was related to Charlemagne the Great. But was he? He went to class one day, where the teacher had a lesson on genealogy:

The teacher says if you count your direct ancestors backward through time, the further back you go, obviously, the more ancestors you have. But when you do the numbers, something queer happens.

Go back to A.D. 800, he said, and the number of direct ancestors is, well, puzzling. You start with two grandparents, then four great-grandparents, then on to eight, 16, etc., and by the time you get to A.D. 800, the number averages to about 562,949,953,421,321. That’s a lot of people. In fact, that’s more people than have ever lived.

So if you go far back enough in history, everyone is related to Charlemagne. This answer is justified with the so-called pedigree collapse:

Without pedigree collapse, a person’s ancestor tree is a binary tree, formed by the person, the parents (2), the grandparents (4), great-grandparents (8), and so on. However, the number of individuals in such a tree grows exponentially and will eventually become impossibly high. For example, a single individual alive today would, over 30 generations going back to the High Middle Ages, have 230 or roughly a billion ancestors, more than the total world population at the time.

This apparent paradox is explained by shared ancestors. Instead of consisting of all unique individuals, a single individual may occupy multiple places in the tree. This typically happens when the parents of an ancestor are cousins (sometimes unbeknownst to themselves). For example, the offspring of two first cousins has at most only six great-grandparents instead of the normal eight. This reduction in the number of ancestors is pedigree collapse. It collapses the ancestor tree into a directed acyclic graph.

In some cultures, cousins were encouraged or required to marry to keep kin bonds, wealth and property within a family (endogamy). Among royalty, the frequent requirement to only marry other royals resulted in a reduced gene pool in which most individuals were the result of extensive pedigree collapse

So next time someone says they’re a direct descendant of someone, say, who was born before 1600, be highly, highly skeptical.

Inspirations: a Short Film Celebrating M.C. Escher

This is a beautiful short film celebrating M.C. Escher (1898-1972), the Dutch artist who explored a wide range of mathematical ideas with his woodcuts and lithographs. The filmmaker behind the film is Cristóbal Vila, who invites you to visit for more information about the film.

The film starts out with a view of a chessboard and what appear to be beans arranged on eleven of the board’s squares. This is a reference to the famous “Wheat and Chessboard Problem.” When the creator of the game of chess showed his invention to the ruler of the country, the ruler was so pleased that he gave the inventor the right to name his prize for the invention. The wise man asked the king: for the first square of the chess board, he would receive one grain of wheat (in some tellings, rice), two for the second one, four on the third one, and so forth, doubling the amount each time. The ruler, arithmetically unaware, quickly accepted the inventor’s offer, even getting offended by his perceived notion that the inventor was asking for such a low price. But when the treasurer started doing the calculations, it quickly surfaced that this was an impossible offer to fulfill. Given the request, the final tally would have been 18,446,744,073,709,551,616 2^64 – 1 = 18,446,744,073,709,551,615 grains!

Continuing along, we also see homages to such things as Fermat’s Last Theorem, Leonardo da Vinci’s The Vitruvian Man (Leonardo may have had some help in its creation), Hokusai’s The Great Wave, Jan van Eyck’s The Arnolfini Wedding, Gustav Klimt’s The Kiss (which I saw in person at the Belvedere Palace in Vienna, Austria), and much, much more. In essence, the short film contains a treasure-trove important cultural references. All of the artworks featured in the film may be seen here. All of the math references may be seen here.

If you’re interested in learning more about the creation of the film, take a look at the wireframes below:


(hat tip: Open Culture)

(Update 3/10/2012: Corrected the count of total grains from 2^64 to 2^64 – 1.)

The Intersection of Math and Pasta

The New York Times has a short piece on Sander Huisman, a graduate student in physics at the University of Twente in the Netherlands, who decided to plot pasta shapes on his favorite software, Mathematica (I prefer Matlab myself, though I’ve used Mathematica in college and grad school).

Mr. Huisman figured out the five lines or so of Mathematica computer code that would generate the shape of the pasta he had been eating — gemelli, a helixlike twist — and a dozen others. “Most shapes are very easy to create indeed,” he said.

Here is a rendering of one of the pasta shapes he posted to his blog:

Pasta Rendering in Mathematica

You can see the other Mathematica renderings in Sander’s blog post. Fun and tasty!


(via Gourmet Pigs)

Solving the Sudoku Minimum Number of Clues Problem

Three mathematicians — Gary McGuire, Bastian Tugemann, and Gilles Civario — spent a year working on a sudoku puzzle. Well, it’s a bit more complicated than that. The essential question they sought to answer: what is the minimum number of clues one must be provided to solve a sudoku puzzle? Turns out that one must see 17 clues (out of a total of 81 squares on a sudoku board) to solve the puzzle uniquely. From their paper, here is the abstract:

We apply our new hitting set enumeration algorithm to solve the sudoku minimum number of clues problem, which is the following question: What is the smallest number of clues (givens) that a sudoku puzzle may have? It was conjectured that the answer is 17. We have performed an exhaustive search for a 16-clue sudoku puzzle, and we did not find one, thereby proving that the answer is indeed 17. This article describes our method and the actual search

If you aren’t familiar with sudoku…the puzzle solver is presented with a 9×9 grid, some of whose cells already contain a digit between 1 and 9. The puzzle solver must complete the grid by filling in the remaining cells such that each row, each column, and each 3×3 box contains all digits between 1 and 9 exactly once. It is always understood that any proper (valid) sudoku puzzle must have only one completion. In other words, there is only one solution, only one correct answer.

And hence the interest in the minimum number of clues problem: What is the smallest number of clues that can possibly be given such that a sudoku puzzle still has only one solution?

There are exactly 6,670,903,752,021,072,936,960 possible solutions to Sudoku (about 6.7 * 10^21) . That’s far more than can be checked in a reasonable period of time. But due to various symmetry arguments (also known as equivalency transformations), many grids are identical, which reduces the numbers of grids to be checked to 5,472,730,538.

I am always on the lookout of mathematicians doing fun things, so if I find any papers on solving other types of games or puzzles, I will post the results here.


(via Technology Review)

On Understanding Advanced Mathematics

What’s it like to have an understanding of very advanced mathematics? A very detailed answer in a Quora post:

  • You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a small number of powerful general purpose “machines” (e.g. continuity arguments, combinatorial arguments, correspondence between geometric and algebraic objects, linear algebra, compactness arguments that reduce the infinite to the finite, dynamical systems, etc.). The number of fundamental ideas and techniques that people use to solve problems is pretty small — see for a partial list, maintained by Tim Gowers.
  • You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It’s not so much that you can “imagine” the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
  • Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being puzzled. For example, when answering a question about a high-dimensional space (e.g., whether a certain kind of rotation of a five-dimensional object has a “fixed point” which does not move during the rotation), you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don’t know that they shouldn’t be straining.) Instead…
  • When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about simple examples into much more impressive insights. For example, you might imagine two- and three- dimensional rotations that are analogous to the one you really care about, and think about whether they clearly do or don’t have the desired property. Then you think about what was important to those examples and try to distill those ideas into symbols. Often, you see that the key idea in those symbolic manipulations doesn’t depend on anything about two or three dimensions, and you know how to answer your hard question.
    As you get more mathematically advanced, the examples you consider easy are actually complex insights built up from many easier examples; the “simple case” you think about now took you two years to become comfortable with. But at any given stage, you do not strain to obtain a magical illumination about something intractable; you work to reduce it to the things that feel friendly.
  • You go up in abstraction, “higher and higher”. The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves — that is, you “zoom out” so that every function is just a point in a space, surrounded by many other “nearby” functions. Using this kind of “zooming out” technique, you can say very complex things in very short sentences — things that, if unpacked and said at the “zoomed in” level, would take up pages. Abstracting and compressing in this way allows you to consider very complicated issues while using your limited memory and processing power.
  • Understanding something abstract or proving that something is true becomes a task a lot like building something. You think: “First I will lay this foundation, then I will build this framework using these familiar pieces, but leave the walls to fill in later, then I will test the beams…” All these steps have mathematical analogues, and structuring things in a modular way allows you to spend several days thinking about something without feeling lost or frustrated. Andrew Wiles, who proved Fermat’s Last Theorem, used an “exploring” metaphor: “Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of—and couldn’t exist without—the many months of stumbling around in the dark that proceed them.”
  • You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any given mathematician can give a good answer. After two or three years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a tentative answer. This makes it more comfortable to be stumped by most problems; a sense that you know roughly what questions are tractable and which are currently far beyond our abilities is humbling, but also frees you from being intimidated, because you do know you are familiar with the most powerful apparatus we have for dealing with these kinds of problems.

(Hat tip: Chris Dixon)

On Shuffling and Randomness

From this fascinating piece in The Wall Street Journal, we learn about the randomness (or lack thereof) when shuffling cards:

The standard way to mix a deck of playing cards—the one used everywhere from casinos to rec rooms—is what is known as a riffle (or “dovetail”) shuffle. You begin by splitting the deck into two roughly equal stacks. Then you flick the cards with your thumbs off the bottoms of the piles in alternating fashion, interleaving the two stacks.

For games like blackjack or poker to be truly fair, the order of the cards must be completely random when the game begins. Otherwise a skilled cheat can exploit the lack of randomness to gain an advantage over other players.

How many riffle shuffles does it take to adequately mix a deck of 52 playing cards?

As it turns out, you have to shuffle seven times before a deck becomes truly scrambled. Not only that, the cards become mixed in a highly unusual way: The amount of randomness in the deck does not increase smoothly. The first few shuffles do little to disturb the original order, and even after six shuffles, you can still pick out distinctly non-random patches.

But right around the seventh shuffle something remarkable happens. Shuffling hits its tipping point, and the cards rapidly decay into chaos.

The seven-shuffles finding applies to messy, imperfect riffle shuffles. The deck might not be divided exactly in half, for instance, or the cards might be riffled together in a haphazard way. Far from undesirable, a little sloppiness is actually the key to a random shuffle.

A perfect (or “faro”) shuffle, meanwhile, wherein the deck is split precisely in half and the two halves are zippered together in perfect alternation, isn’t random at all. In fact, it’s completely predictable. Eight perfect shuffles will return a 52-card deck to its original order, with every card cycling back to its starting position.

And this doesn’t just work for 52 cards. A deck of any size will eventually return to its starting order after a finite sequence of faro shuffles, although the number of faros required isn’t always eight—and doesn’t increase linearly. If you have 104 cards, for instance, it takes 51 faros to restore the deck. For a thousand cards, it takes 36.

These findings are among the many fascinating results explored in Magical Mathematics, a dazzling tour of math-based magic tricks. The authors, Persi Diaconis and Ron Graham, are distinguished mathematicians with high-powered academic pedigrees. Both are also accomplished magicians who have taught courses on mathematical magic at Harvard and Stanford.

I’ve put the book on my to-read list.

Like Clockwork: Your Next Trivia Question at a Party

This post has very little to do with reading, but it’s a really awesome trivia/math question to ask at your next party (or wherever you happen to be).

A good lead-in to this question would be after you and the opposite party are looking at an analog clock..

Here’s the question: What is the first time after 9PM when the hour and minute hands of a clock are exactly on top of each other (that is, coincident)?

First, consider working out this problem on your own… If you’re curious, hit the jump link below to find out about the solution…

Continue reading