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The highly entertaining Vi Hart is back with another episode of Doodling in Math Class. Her videos are fun to watch, clear and easy to understand, and express the joy of scientific discovery:
This pattern is not just useful, not just beautiful, but inevitable. This is why science and mathematics are so much fun. You discover things that seem impossible to be true and then get to figure out why it’s impossible for them to not be.
-via Make
Related: a wonderful cartoon by Abstruse Goose that makes the same point.
Allegedly, this video shows a way to multiply large numbers by marking lines and counting their intersections. It seems to work — at least on these examples. It’s probably a good technique if you lack sufficient fingers and toes.
-via Nag on the Lake
Because they are awesome, Pat Ashforth, Steve Plummer and Ben Ashforth illustrate math through their crafting projects. Pictured above is their knitted version of Napier’s Bones, a calculation device invented by Scottish mathematician John Napier (1550-1617). George Hart explains how they work:
The image below shows how to arrange the bones if you wanted to multiply by 76495. For example, the bottom row, labeled 9 at left, allows you to read off 9 x 76495. The rightmost digit of the answer is the 5 seen in the triangle at right. Then read off the remaining digits by adding the two numbers in each parallelogram, carrying as necessary, e.g., 1+4 gives 5 as the next digit and 6+8=14 gives 4 as the following digit, with a carry of 1 into the digit after that. The result can be quickly read off as 688455.
Crafters’ Website -via Make
The correct answer is “An African or a European swallow?” Or, for partial credit, “blue”.
Pencils down.
Link -via Blame It on the Voices
Tim Chartier, a math professor at Davidson College, found a way to express a principle of calculus using the best of all possible source materials: chocolate. He created a series of enlarging charts featuring a growing number of chocolate chips:
If you count carefully, we use 83 milk chocolate chips of the 121 total. This gives us an estimate of 2.7438 for ?, which correlates to an error of about 0.378. [...]
What do you notice is happening to the error as we reduce the size of the squares? Indeed, our estimates are converging to the exact area. Here lies a fundamental concept of Calculus. If we were able to construct such chocolate chip mosaics with grids of ever increasing size, then we would converge to the exact area. Said another way, as the area of the squares approaches zero, the limit of our estimates will converge to ?. Keep in mind, we would need an infinite number of chocolate chips to estimate ? exactly, which is a very irrational thing to do!
Link -via That’s Nerdalicious!
Nick Sayers demonstrates his knowledge of geometry through a unique haircut:
The obtuse angles of each rhombus meet in groups of three, but the acute angles meet in groups of five, six, or seven, depending on the curvature. In the flatter areas, they meet in groups of six, like equilateral triangles, and in the areas of strong positive curvature they meet in groups of five, but in the negatively curved saddle at the back of the neck, there is a group of seven.
Link | Previously by Sayers: Geometric Sculpture Made from Coffee Stirrers
In a way, this actually makes it simpler. But I’ll probably end up using tax preparation software anyway.
-via Glenn Reynolds | Image: Daniel J. Velleman
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Did you hear the one about the state that tried to make pi equal to 3.2 -by law? It’s not a joke. It happened in Indiana in 1897.
The story of the “Indiana pi bill” starts with Edward J. Goodwin, a Solitude, Indiana, physician who spent his free time dabbling in mathematics. Goodwin’s pet obsession was an old problem known as squaring the circle. Since ancient times, mathematicians had theorized that there must be some way to calculate the area of circle using only a compass and a straightedge. Mathematicians thought that with the help of these tools, they could construct a square that had the exact same area as the circle. Then all one would need to do to find the area of the circle was calculate the area of the square, a simple task.
It can’t be done, but you don’t have to be a math whiz to be a state legislator. Besides, Goodwin had his reasons for pushing the bill to redefine pi. Read all about that strange episode at mental_floss. Link
The shapes of some of these plushies will be familiar to many Neatorama readers (standard normal distribution, chi-square, log normal…), but other, more uncommon distributions (Gumbel, Erlang, Cauchy) are also available.
From Etsy seller NausicaaDistribution, via the base of ln.
Comments Off Sapientia University has created a series of online videos that illustrate different sorting algorithms using folk dances:
What you have to do is just check that they are in fact implementing the algorithm correctly. The dancers have numbers stuck on their front and they do seem to look down and examine the value on another dancer before performing the dance routine dictated by the algorithm.
Embedded above is the bubble sort, as explained by a troupe performing the Hungarian “Csángó” dance.
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A fractal is a fragmented geometric shape that, when split into parts, each part is roughly a smaller copy of the whole, a property called self-similarity. And it makes some damn wild images when you start injecting color, layers or even candy.
Benoit Mandelbrot, the French mathematician who is known as the father of fractal geometry, died last week. Good has posted a slideshow in his honor.
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…on a lumpy leg. It is, rather, a guilloche pattern, typically seen on banknotes but also incorporated into other works of art and design. In the past such patterns might have been generated by a geometric lathe, but nowadays they can be created mathematically. The one above is generated by:
Found at The Ministry of Type, via Proof Math is Beautiful.
I haven’t seen it yet, but apparently last night’s episode of Futurama required a mathematical formula to explain a plot element. Producer David X. Cohen is noted for leaving real mathematical statements on screen during the show and had a staff mathematician compose an original theorem for that episode:
In an APS News exclusive, Cohen reveals for the first time that in the 10th episode of the upcoming season, tentatively entitled “The Prisoner of Benda,” a theorem based on group theory was specifically written (and proven!) by staffer/PhD mathematician Ken Keeler to explain a plot twist. Cohen can’t help but chuckle at the irony: his television-writing rule is that entertainment trumps science, but in this special case, a mathematical theorem was penned for the sake of entertainment.
Link via Geekosystem | Image: Fox
For many of us, the first time we appreciated the art of math was when we played with a Spirograph. However, it’s a long way from addition and subtraction to epicycloids, and very few of us actually study math that far. But those who do sometimes end up creating some very beautiful artworks based on mathematics and geometry.
Sculpture
Sculptor Bathsheba Grossman creates metal and crystal artworks of forms found in math, physics, biology, and astronomy. Grossmen shows us Borromean rings, hypercubes, gyroids, fractals, Calabi-Yau spaces, and interlaced sculptures based on the five Platonic solids. I particularly like this Voronoi network wrapped onto a Möbius toroid, sculpted in white glass.
Grossman created this beautiful lamp from one of her Ora series sculptures. Available in several lamp styles from Materialise.
Jewelry
The Julia set is a fractal equation that produces a series of rather pleasing spirals. Designer Marc Newson took that fractal shape and designed a necklace of 2,000 diamonds and sapphires that took jewelry craftsmen 1,500 hours to put together. Note that the necklace is not symmetrical, but still has a sense of balance. See how the jeweler, Boucheron, advertises the necklace.
Drawing
Probably the best known artist to use math concepts in his works is M.C. Escher. Many of his 2-dimensional drawings turned 3-dimensional geometry on its head. The lithograph titled Waterfall illustrates the concept of the Penrose triangle, also called the impossible triangle. Escher also explored tessellations in many of his drawings.
Computer Imaging
Paul Nylander was one of the developers of the Mandelbulb that we saw in a previous math post. He is a computer engineer and an artist who renders math and science concepts into colorful images including animated .gifs to help us visualize their 3- or 4-dimensional structures. Shown is a Dodeca-Spidroball, a variation on the spidron, which was invented by Daniel Erdely in 1979.
Belgian mechanical engineer Jos Leys renders and animates all kinds of math concepts into beautiful forms that boggle the mind. His artworks include fractals, Kleinian groups, inversive geometry, recursions, tessellations, knots, and tilings in both images and video renderings to show 3- and 4-dimensional effects. The image above is called Indra200, an example of “Kleinian jewelry“.
Other artists rendering math images worth checking out include Torolf Sauermann, Brian Johnston, Mehrdad Garousi, and the late Titia Van Beugen.
Video
Creating visual representations of math concepts became easier with computer rendering software and digital video capabilities. That doesn’t mean it is simple. Homporgo, the artist who created this video of a Mandelbox zoom said in a comment:
Believe me Bill, I wanted to go further too, but at the end part a single frame took 18 minutes to render, and the whole 1:27 minute video needed 12 days nonstop rendering. I felt thats more than enough at the time.
Twelve days! The result looks worth it to me. How about you? See more fractals on video in this post.
Previously at Neatorama: A Non-Math Look at Math Objects and A Non-Math Look at Math Shapes.
There are 43,252,003,274,489,856,000 possible varying positions for the Rubik’s Cube. A team of mathematicians and programmers determined that all of them can be solved within 20 moves:
With about 35 CPU-years of idle computer time donated by Google, a team of researchers has essentially solved every position of the Rubik’s Cube™, and shown that no position requires more than twenty moves.
Every solver of the Cube uses an algorithm, which is a sequence of steps for solving the Cube. One algorithm might use a sequence of moves to solve the top face, then another sequence of moves to position the middle edges, and so on. There are many different algorithms, varying in complexity and number of moves required, but those that can be memorized by a mortal typically require more than forty moves.
Link via Popular Science | Photo by Flickr user huangjiahui used under Creative Commons license
Quick! How many stars are on this flag? No, it’s not the American flag we use now, but a pattern with 51 stars, which will be needed if Puerto Rico becomes a state. Mathematician Skip Garibaldi worked out the most geometric layouts that will accommodate more stars if the flag has to be updated if and when states join the union. Slate posted an interactive flag calculator, in which you can enter a number of states, from one to a hundred stars, and see the best pattern according to a computer program Garibaldi created. Some numbers have more than one pattern, with up to six styles. Read all about it in this article from Chris Wilson at Slate. Link
Rene Descartes’ health may have been troubled, but his genius was completely intact. His survival led to an amazing perspective about why we are here, and the truth behind that notion.
Thus the whole of philosophy is like a tree. The roots are metaphysics, the trunk is physics, and the branches emerging from the trunk are all the other sciences, which may be reduced to three principal ones, namely medicine, mechanics and morals. By “morals” I understand the highest and most perfect moral system, which presupposes a complete knowledge of the other sciences and is the ultimate level of wisdom.
From the Upcoming 
ueue, submitted by lannaxe96.
The Monty Hall Problem is a common mathematical fallacy based on Monty Hall’s game show Let’s Make a Deal. It works like this:
Imagine that you’re in a game show and your host shows you three doors. Behind one of them is a shiny car and behind the others are far less lustrous goats. You pick one of the doors and get whatever lies within. After making your choice, your host opens one of the other two doors, which inevitably reveals a goat. He then asks you if you want to stick with your original pick, or swap to the other remaining door. What do you do?
It’s counterintuitive to many people, but switching doors will double your chances of winning:
The problem is that most people assume that with two doors left, the odds of a car lying behind each one are 50/50. But that’s not the case – the actions of the host beforehand have shifted the odds, and engineered it so that the chosen door is half as likely to hide the car.
At the very start, the contestant has a one in three chance of picking the right door. If that’s the case, they should stick. They also have a two in three chance of picking a goat door. In these situations, the host, not wanting to reveal the car, will always pick the other goat door. The final door hides the car, so the contestant should swap. This means that there are two trials when the contestant should swap for every one trial when they should stick. The best strategy is to always swap – that way they have a two in three chance of driving off, happy and goatless.
The bad news is that according to a scientific study, pigeons are better as this task than we are:
Each pigeon was faced with three lit keys, one of which could be pecked for food. At the first peck, all three keys switched off and after a second, two came back on including the bird’s first choice. The computer, playing the part of Monty Hall, had selected one of the unpecked keys to deactivate. If the pigeon pecked the right key of the remaining two, it earned some grain. On the first day of testing, the pigeons switched on just a third of the trials. But after a month, all six birds switched almost every time, earning virtually the maximum grainy reward.
Link via The Presurfer | Photo: Library of Congress
We’ve previously featured Cristóbal Vila’s animated depiction of Frank Lloyd Wright’s home Falling Water. Vila’s latest project, “Nature By Numbers”, illustrates how mathematical properties, such as the Fibonacci Sequence, pervade the natural world. The math of each part of the film is explained in detail at the link.
And now for something completely different. A math puzzle. Or conundrum, if you will.
In the figure to the left, the bar above the number 9 indicates that it is to be repeated forever. For the remainder of this post, we will represent that concept by several nines with an ellipsis (.999…).
Now, here is the conundrum. .9 repeating is EQUAL TO ONE. Not CLOSE to one, mind you, but EQUAL to one.
Nonsense, you reply. It is obviously less than one. Not by much – by an infinitely small amount, in fact. But the simple fact (?) that it is not one is enough to demonstrate that it can’t be equal to one. It’s as close as you can get to one without being one.
Wrong. It is in fact equal to one, and that fact can be demonstrated mathematically in several ways.
The most easily understood is to revert to other familiar repeating digits. Everyone knows that 1/3 is 0.333… and that 2/3 is 0.666… If you add them together, you get 3/3, which is one.
But now note that the sum of the decimals on the right side of the equation is 0.999…
Therefore, one is equal to (not close to) .999…
You don’t agree? Then try this. Subtract .999… from one. What you have is 0.000… An infinitely long string of zeroes, which can only be equal to zero. And if the subtraction of .999… from one leaves zero, then the .999… must be one. But, you say, there’s a one at the end that string of zeroes. No, there isn’t, because the string of 9s doesn’t end.
There are other proofs at the Polymathematics blog, along with a long series of comments, an update refuting the counterarguments, and a final refutation of the most stubborn skeptics.
Link.
That’s 123 billion digits more than the previous number. Computer scientist Fabrice Bellard ran his calculations on a desktop computer, taking 131 days to run the program and then check the results:
Previous records were established using supercomputers, but Mr Bellard claims his method is 20 times more efficient.
The prior record of about 2.6 trillion digits, set in August 2009 by Daisuke Takahashi at the University of Tsukuba in Japan, took just 29 hours.
However, that work employed a supercomputer 2,000 times faster and thousands of times more expensive than the desktop Mr Bellard employed.
I blogged about that record at the time.
Link via Geekologie | Image: flickr user Paul Adam Smith, used under Creative Commons license
The Salary Theorem proves mathematically that those who know more make less money. Therefore, if you know nothing, you should be fabulously wealthy! Link -via Digg
The Fibonacci sequence, named after a 13th Century Italian mathematician, is a sequence of numbers in which every third number is the sum of the previous two numbers. This ring and others like it by Etsy seller Holmes Craft is an homage to that mathematical sequence in that the beads are organized according to the first four Fibonacci numbers.
Link via Technabob | Math Explanation | Photo: Holmes Craft
Professor Matthew Weathers went the extra mile for his math lecture Wednesday at Biola University. Who says math isn’t fun? -via Cynical-C
Michael Tennesen writes in Scientific American that biologists suspect that robins, baby chicks, rhesus monkeys, and parrots may have the ability to count. Although they may not have fixed numerals, they have have concepts of relative quantities:
Elizabeth Brannon of Duke University has conducted similar experiments with rhesus monkeys, getting them to match the number of sounds they hear to the number of shapes they see, proving they can do math across different senses. She also tested the monkeys’ ability to do subtraction by covering a number of objects and then removing some of them. In all cases, the monkeys picked the correct remainder at a rate greater than chance. And although they might not grasp the deeper concept of zero as a number, the monkeys knew it was less than two or one, conclude Brannon and her colleagues in the May Journal of Experimental Psychology: General.
Although Brannon feels that animals do not have a linguistic sense of numbers—they aren’t counting “one, two, three” in their heads—they can do a rough sort of math by summing sets of objects without actually using numbers, and she believes that ability is innate. Brannon thinks that it might have evolved from the need for territorial animals “to access the different sizes of competing groups and for foraging animals to determine whether it is good to stay in one area given the amount of food retrieved versus the amount of time invested.”
Image: U.S. Department of the Interior
Chris Matyszczyk explains that the laws of probability indicate when you should settle for one prospective mate, and when you should keep on looking. There’s a point of diminishing returns in a succession of relationships when you should marry before your prospects start to get worse:
So for a long time, mathematicians believed that, given 100 choices (each of which has to be chosen or discarded after the interview) you should discard the first 50 and then choose the next best one. (The assumption also is that if you don’t choose the first 99, you have to choose number 100, which, again, seems rather realistic to me. I know so many people who have chosen the last resort out of perceived necessity rather than, say, happiness.)
The “Discard 50 then Choose the Next Best” method apparently gives you a 25 percent chance of choosing the best candidate.
However, then along came John Gilbert and Frederick Mosteller of Harvard University. I do not believe they were married. However, they came upon the idea that the magic number is, in fact, 37. Yes, you should stop after 37 candidates and choose the next best one. This number was apparently derived by taking the number 100 and dividing by e, the base of the natural logarithms (around 2.72). And it apparently increases your chances of the best choice to 37 percent.
Link via The Corner
Image: U.S. Department of Energy
