Category Archive 'Mathematics'
12 Feb 2019

Penrose Tiles

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In 1974, Roger Penrose, a British mathematician, created a revolutionary set of tiles that could be used to cover an infinite plane in a pattern that never repeats. In 1982, Daniel Shechtman, an Israeli crystallographer, discovered a metallic alloy whose atoms were organized unlike anything ever observed in materials science. Penrose garnered public renown on a scale rarely seen in mathematics. Shechtman won the Nobel Prize. Both scientists defied human intuition and changed our basic understanding of nature’s design, revealing how infinite variation could emerge within a highly ordered environment.

At the heart of their breakthroughs is “forbidden symmetry,” so-called because it flies in the face of a deeply ingrained association between symmetry and repetition. Symmetry is based on axes of reflection—whatever appears on one side of a line is duplicated on the other. In math, that relationship is reflected in tiling patterns. Symmetrical shapes such as rectangles and triangles can cover a plane with neither gap nor overlap, and in an ever-repeating pattern. Repeated patterns are called “periodic” and are said to have “translational symmetry.” If you move a pattern from place to place, it looks the same.

Penrose, a bold, ambitious scientist, was interested less in identical patterns and repetition, and more in infinite variation. To be precise, he was interested in “aperiodic” tiling, or sets of tiles that can cover an infinite plane with neither gap nor overlap, without the tiling pattern ever repeating itself. That was a challenge because he couldn’t use tiles with two, three, four, or six axes of symmetry—rectangles, triangles, squares, and hexagons—because on an infinite plane they would result in periodic or repeated patterns. That meant he had to rely on shapes believed to leave gaps in the tiling of a plane—those with forbidden symmetries.

Penrose turned to five-axis symmetry, the pentagon, to create his plane of non-repeating patterns, in part, he has said, because pentagons “are just nice to look at.” What was remarkable about Penrose tiles was that even though he derived his tiles from the lines and angles of pentagons, his shapes left no awkward gaps. They snugged together perfectly, twisting and turning across the plane, always coming close to repetition, but never quite getting there.

Penrose tiling captured public attention for two major reasons. First, he found a way to generate infinitely changing patterns using just two types of tiles. Second, and even more spectacular, his tiles were simple, symmetrical shapes that on their own betrayed no sign of their unusual properties.

Penrose made several versions of his aperiodic tile sets. One of his most famous is known as the “kite” and the “dart.” The kite looks like the kids’ toy of the same name, and the dart looks like a simplified outline of a stealth bomber. Both divide cleanly along axes of symmetry and each has two simple, symmetrical arcs on their surface. Penrose established one placing rule: for a “legal” tile placement these arcs must match up, creating contiguous curves. Without this rule, kites and darts can be placed together in repeating patterns. With this rule, repetition never comes. The kite and the dart tile forever, dancing around their five axes, creating starbursts and decagons, winding curves, butterflies and flowers. Shapes recur but new variations keep creeping in.

Edmund Harriss, an assistant clinical professor in mathematical studies at the University of Arkansas, who wrote his Ph.D. thesis on Penrose tiles, offers a comparison. “Imagine you’re on a world that is just made up of squares,” Harriss says. “You start walking, and when you get to the edge of the square, and the next square is exactly the same, you know what you’re going to see if you walk forever.” Penrose tiling has the exact opposite nature. “No matter how much information you have, how much you’ve seen of the tiling, you’ll never be able to predict what happens next. It will be something that you’ve never seen before.”


10 Jul 2018

Alternative Math

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22 Jun 2018

Exeter Prof: “Mathematics to Blame for Global Disparities in Wealth”

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Campus Reform shares the latest breakthrough in thought from today’s Academy.

In a chapter for a new textbook, University of Exeter professor Paul Ernest warns that mathematics education can cause “collateral damage” to society by training students in “ethics-free thought.”

He even argues that since money involves mathematics, math is “implicated in the global disparities of wealth” because math students are taught to value “detached” and “calculative” reasoning.


10 Jul 2017

Calculating the Odds at Gettysburg

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Military historians have long debated whether Lee’s decision to attack the Union center on the third day of the Battle of Gettysburg with Pickett’s Division ever had any chance of succeeding.

Up at Norwich University in Vermont, Michael J. Armstrong (with Steve Sondergen) was the most recent to have a try at settling the dispute.

We used computer software to build a mathematical model of the charge. The model estimated the casualties and survivors on each side, given their starting strengths.

We used data from the actual conflict to calibrate the model’s equations. This ensured they initially recreated the historical results. We then adjusted the equations to represent changes in the charge, to see how those affected the outcome. This allowed us to experiment mathematically with several different alternatives.

The first factor we examined was the Confederate retreat. About half the charging infantry had become casualties before the rest pulled back. Should they have kept fighting instead? If they had, our model calculated that they all would have become casualties too. By contrast, the defending Union soldiers would have suffered only slightly higher losses. The charge simply didn’t include enough Confederate soldiers to win. They were wise to retreat when they did.

We next evaluated how many soldiers the Confederate charge would have needed to succeed. Lee put nine infantry brigades, more than 10,000 men, in the charge. He kept five more brigades back in reserve. If he had put most of those reserves into the charge, our model estimated it would have captured the Union position. But then Lee would have had insufficient fresh troops left to take advantage of that success.
Ammunition ran out

We also looked at the Confederate artillery barrage. Contrary to plans, their cannons ran short of ammunition due to a mix-up with their supply wagons. If their generals had better coordinated those supplies, the cannons could have fired twice as much. Our model calculated that this improved barrage would have been like adding one more infantry brigade to the charge. That is, the supply mix-up hurt the Confederate attack, but was not decisive by itself.

Finally, we considered the Union Army. After the battle, critics complained that Meade had focused too much on preparing his defences. This made it harder to launch a counter-attack later. However, our model estimated that if he had put even one less infantry brigade in his defensive line, the Confederate charge probably would have succeeded. This suggests Meade was correct to emphasize his defense.

Pickett’s Charge was not the only controversial part of Gettysburg. Two days earlier, Confederate Gen. Richard Ewell decided against attacking Union soldiers on Culp’s Hill. He instead waited for his infantry and artillery reinforcements. By the time they arrived, however, it was too late to attack the hill.
Was Ewell’s Gettysburg decision actually wise?

Ewell was on the receiving end of a lot of criticism for missing that opportunity. Capturing the hill would have given the Confederates a much stronger position on the battlefield. However, a failed attack could have crippled Ewell’s units. Either result could have altered the rest of the battle.

A study at the U.S. Military Academy used a more complex computer simulation to estimate the outcome if Ewell had attacked. The simulation indicated that an assault using only his existing infantry would have failed with heavy casualties. By contrast, an assault that also included his later-arriving artillery would have succeeded. Thus, Ewell made a wise decision for his situation.


I’m afraid I do not buy the analysis on Ewell’s decision one bit. The Union 1st and XIth Corps were retiring in disorder late in afternoon having been beaten in a hard fight west of Gettysburg. Ewell’s Corps was arriving from the North, on the right flank of the collapsing Union line. What do you suppose would have happened to the Union Army if Stonewall Jackson had survived the Battle of Chancellorsville, six weeks earlier, and been commanding that corps instead of Ewell? Those computer simulations up at Norwich are clearly not accurately calculating for momentum and morale.


For every Southern boy fourteen years old, not once but whenever he wants it, there is the instant when it’s still not yet two o’clock on that July afternoon in 1863, the brigades are in position behind the rail fence, the guns are laid and ready in the woods and the furled flags are already loosened to break out and Pickett himself with his long oiled ringlets and his hat in one hand probably and his sword in the other looking up the hill waiting for Longstreet to give the word and it’s all in the balance, it hasn’t happened yet, it hasn’t even begun yet, it not only hasn’t begun yet but there is still time for it not to begin against that position and those circumstance which made more men than Garnett and Kemper and Armistead and Wilcox look grave yet it’s going to begin, we all know that, we have come too far with too much at stake and that moment doesn’t need even a fourteen-year-old boy to think This time. Maybe this time with all this much to lose than all this much to gain: Pennsylvania, Maryland, the world, the golden dome of Washington itself to crown with desperate and unbelievable victory the desperate gamble, the cast made two years ago.

—William Faulkner, Intruder in the Dust, 1948.

29 Jun 2017


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Hat tip to Karen L. Myers.

14 Mar 2017

Happy Pi Day!


26 Feb 2017

Math for the Social Justice Major

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Yale Classmate Seattle Sam writes:

I created a course that I think will be in next year’s Yale course catalog.

Math for the Social Justice Major

Mathematics was devised by old white men who sought to oppress the uneducated masses. In this course we will explore a more empathetic approach to the subject.

The course will explore questions such as:

How does the number 6 make you feel?

If John has 6 marbles and Sue has 2, isn’t that unfair?

How can there be any “incorrect” answers?

Isn’t identifying a number as “positive” or negative” stereotyping?

If you identify with 5 more than 4, why shouldn’t that be a solution to 2+2=?

What did Euclid know and when did he know it?

Isn’t a null set non-inclusive?

What should you do if the solution to an equation make you feel unsafe?

Shouldn’t we just deem the Parallel Postulate proved?

What’s the point of carrying pi out to more than two decimals?

Aren’t < and > judgmental symbols?

Who are you to determine that a fraction is improper?

Why do you think prime numbers have only a token even member?

Why shouldn’t an inverse tangent have the same value as a cosine?

Aren’t right angles reactionary?

Are there really any absolute values?

Why should binomials and polynomials be considered deviants?

Isn’t a Real Number just your perception?

Just because a number can’t be expressed as a ratio of integers, why should it be called irrational?

16 Jan 2016

Music and Math

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How is it that Beethoven, who is celebrated as one of the most significant composers of all time, wrote many of his most beloved songs while going deaf? The answer lies in the math behind his music. Using the “Moonlight Sonata”, we can begin to understand the way Beethoven was able to convey emotion and creativity using the certainty of mathematics.


The standard piano octave consists of 13 keys, each separated by a half step. A standard major or minor scale uses 8 of these keys with 5 whole step intervals and 2 half step ones.


The first half of measure 50 of “Moonlight Sonata” consists of three notes in D major, separated by intervals called thirds that skip over the next note in the scale. By stacking the notes first, third, and fifth notes – D, F sharp, and A – we get a harmonic pattern known as a triad.


But, these aren’t just arbitrary magic numbers. Rather, they represent the mathematical relationship between the pitch frequencies of different notes, which form a geometric series. The stacking of these three frequencies creates ‘consonance’, which sounds naturally pleasant to our ears. Examining Beethoven’s use of both consonance and dissonance can help us begin to understand how he added the unquantifiable elements of emotion and creativity to the certainty of mathematics.

For a deeper dive into the mathematics of the “Moonlight Sonata”, watch the TED-Ed Lesson Music and math: The genius of Beethoven – Natalya St. Clair

Animation by Qa’ed Mai

Via Ratak Monodosico.

31 Oct 2015

The Mathematics of Vampire-Human Relations



Ella Morton reviews the academic literature addressing the possibility of vampire-humanity peaceful coexistence and the alternative hypothesis which argues that hungry vamps will simply eat themselves out of human prey.

A surprisingly large number of academic studies—as in, more than one—have applied mathematical modeling to the concept of human-vampire co-existence. … these papers look at whether Earth’s vampire population would inevitably annihilate humanity, and, if so, how long it would take.

11 Oct 2015

Intellectual Challenge: Did He or Didn’t He?

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Shinichi Mochizuki

Scientific American reports that a Japanese math genius, three years ago, quietly posted the proof to an important number theory conjecture. The problem is that his proof is so abstruse that nobody else can understand it.

Sometime on the morning of August 30 2012, Shinichi Mochizuki quietly posted four papers on his website.

The papers were huge—more than 500 pages in all—packed densely with symbols, and the culmination of more than a decade of solitary work. They also had the potential to be an academic bombshell. In them, Mochizuki claimed to have solved the abc conjecture, a 27-year-old problem in number theory that no other mathematician had even come close to solving. If his proof was correct, it would be one of the most astounding achievements of mathematics this century and would completely revolutionize the study of equations with whole numbers.

Mochizuki, however, did not make a fuss about his proof. The respected mathematician, who works at Kyoto University’s Research Institute for Mathematical Sciences (RIMS) in Japan, did not even announce his work to peers around the world. He simply posted the papers, and waited for the world to find out.

Probably the first person to notice the papers was Akio Tamagawa, a colleague of Mochizuki’s at RIMS. He, like other researchers, knew that Mochizuki had been working on the conjecture for years and had been finalizing his work. That same day, Tamagawa e-mailed the news to one of his collaborators, number theorist Ivan Fesenko of the University of Nottingham, UK. Fesenko immediately downloaded the papers and started to read. But he soon became “bewildered”, he says. “It was impossible to understand them.”

Fesenko e-mailed some top experts in Mochizuki’s field of arithmetic geometry, and word of the proof quickly spread. Within days, intense chatter began on mathematical blogs and online forums (see Nature). But for many researchers, early elation about the proof quickly turned to scepticism. Everyone—even those whose area of expertise was closest to Mochizuki’s—was just as flummoxed by the papers as Fesenko had been. To complete the proof, Mochizuki had invented a new branch of his discipline, one that is astonishingly abstract even by the standards of pure maths. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” number theorist Jordan Ellenberg, of the University of Wisconsin–Madison, wrote on his blog a few days after the paper appeared.

Three years on, Mochizuki’s proof remains in mathematical limbo—neither debunked nor accepted by the wider community. Mochizuki has estimated that it would take an expert in arithmetic geometry some 500 hours to understand his work, and a maths graduate student about ten years. So far, only four mathematicians say that they have been able to read the entire proof.

Read the whole thing.

His papers (You want the Inter-universal Teichmuller Theory papers.)

09 Aug 2015

Fourier Series Using Circles

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14 Mar 2015

Happy Pi Day!

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