Category Archive 'Mathematics'
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.

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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.

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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

xkcd

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Hat tip to Karen L. Myers.

14 Mar 2017

Happy Pi Day!

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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.

Beethoven1

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.

Beethoven2

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.

Beethoven3

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

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Vampire

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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Mochizuki
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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Pi

01 Dec 2014

Sir Isaac Newton

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IsaacNewton

Nautilus profiles the great man, thusly:

Describing his life, shortly before his death, Newton put his contributions this way: “I don’t know what I may seem to the world, but, as to myself, I seem to have been only like a boy playing on the sea shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay undiscovered before me.”

One thing Newton never did do, actually, was play at the seashore. In fact, though he profited greatly from occasional interaction with scientists elsewhere in Britain and on the Continent—often by mail—he never left the vicinity of the small triangle connecting his birthplace, Woolsthorpe, his university, Cambridge, and his capital city, London. Nor did he seem to “play” in any sense of the word that most of us use. Newton’s life did not include many friends, or family he felt close to, or even a single lover, for, at least until his later years, getting Newton to socialize was something like convincing cats to gather for a game of Scrabble. Perhaps most telling was a remark by a distant relative, Humphrey Newton, who served as his assistant for five years: he saw Newton laugh only once—when someone asked him why anyone would want to study Euclid.

Newton had a purely disinterested passion for understanding the world, not a drive to improve it to benefit humankind. He achieved much fame in his lifetime, but had no one to share it with. He achieved intellectual triumph, but never love. He received the highest of accolades and honors, but spent much of his time in intellectual quarrel. It would be nice to be able to say that this giant of intellect was an empathetic, agreeable man, but if he had any such tendencies he did a good job suppressing them and coming off as an arrogant misanthrope. He was the kind of man who, if you said it was a gray day, would say, “no, actually the sky is blue.” Even more annoying, he was the kind who could prove it. Physicist Richard Feynman voiced the feelings of many a self-absorbed scientist when he wrote a book titled, What Do You Care What Other People Think? Newton never wrote a memoir, but if he had, he probably would have called it I Hope I Really Pissed You Off, or maybe, Don’t Bother Me, You Ass.

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Lock, Stock, & History is similiarly irreverent.

Today we consider the great scientist Isaac Newton to be one of the greatest geniuses of history. After all he developed many laws and theories in the fields of physics, optics, mathematics, and astronomy which are still very relevant today. However if you actually met Sir Isaac Newton today, I guarantee you would think him to be a nutjob.

While Newton is celebrated today for his many scientific breakthroughs, his works in other, less scientific fields are largely forgotten. A dedicated alchemist and occultist, Newton spent much of his time working on experiments that are today mostly considered outright bizarre. A devoted follower of many interesting occult sects, Newton spent years trying to determine the “sacred geometry” of Solomon’s Temple, with hopes of mathematically divining the secrets of God. He also spent much time and energy trying to find and de-crypt the “Bible Code”. In a detailed study of the Bible, Newton made a prediction for the end of world using the chronology of the Holy Book. According to Newton, the world should come to an end in 2060 AD. Newton calculated the end of the world specifically “to put a stop to the rash conjectures of fanciful men who are frequently predicting the time of the end, and by doing so bring the sacred prophesies into discredit as often as their predictions fail.” Eat your hearts out Mayans!

Of all of Newton’s discoveries, from gravity to refraction of light, from divining the location of Atlantis to discovering how to communicate with angels, Newton believed his most important work was in creating the Philosopher’s Stone. Newton believed that with the Philosophers Stone he could have everlasting life and be able to turn lead into gold. He spent years, if not decades studying the work of the noted alchemist Nicholas Flammel and other alchemists, with the believe that he was about to make a breakthrough at any moment. In fact, to Newton the discovery of the Philosopher’s Stone was so important that all his other discoveries were trivial when compared to his work in alchemy. His obsession with the stone caused him to have a weird set of priorities. After developing calculus, he kept his results to himself for over 30 years because he didn’t think it was important and “disliked intellectual matters”.

Finally some of Newton’s experiments were just downright kooky and creepy. According to writings in his notebook, one experiment involved him sticking a needle into his eyeball and twirling it around to analyze how light traveled through his optic nerve,

    I tooke a bodkine (needle) & put it betwixt my eye & [the] bone as neare to [the] backside of my eye as I could: & pressing my eye [with the] end of it (soe as to make [the] curvature a, bcdef in my eye) there appeared severall white darke & coloured circles r, s, t, &c. Which circles were plainest when I continued to rub my eye [with the] point of [the] bodkine, but if I held my eye & [the] bodkin still, though I continued to presse my eye [with] it yet [the] circles would grow faint & often disappeare untill I removed [them] by moving my eye or [the] bodkin.

In another strange experiment, Newton stared directly at the sun for as long as he could bare with the same objective of his “needle experiment”.

While dedicated to the discovery of the Philosopher’s Stone, his work would all be in vain as he died in 1727. He never did figure out how to turn lead into gold.

19 Oct 2014

The Golden Ratio

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GoldenRatio


Golden Ratio φ = (1+sqrt(5))/2 = 1.6180339887498948482…

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0. Two quantities a and b are said to be in the golden ratio φ if

(a+b)/a = a/b = φ

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ:

(a+b)/a = 1+ b/a = 1+1/φ

Therefore: 1+1/φ = φ
Multiplying by φ gives: φ^2 – φ – 1 = 0

Using the quadratic formula, two solutions are obtained::

φ = (1- sqrt(5))/2 or φ = (1+sqrt(5))/2

Because φ is the ratio between positive quantities φ is necessarily positive:

φ = (1+sqrt(5))/2 = 1.6180339887498948482…

See more at Golden Ratio.

Image: Phi (golden number) by Steve Lewis.

From Ratak Monodosico.

17 Jul 2014

Applying Math to Myth

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SagaConnections
Network combining the five major Icelandic sagas. White nodes represent characters who appear in more than one saga. There is a large overlap of characters from Laxdæla Saga (green) and Njáls Saga (red). The other sagas are Egil (blue), Vatnsdaela (yellow), and Gisla (light blue).

Veronique Greenwood, at the Verge, describes a fascinating application of the techniques of statistical physics to identify patterns and relationships in medieval literature.

An unusual article recently appeared in the magazine of the Royal Statistical Society and American Statistical Association.

It featured web-like diagrams of lines connecting nodes, a hallmark of research that analyzes networks. But each node, rather than being a plain dot, was the head of a burly, red-bearded Viking sporting a horned hat, his tresses blowing in the wind.

This whimsical-seeming piece of scholarship went on to describe the social network of more than 1,500 characters in the Icelandic Sagas, epic tales about the colonization of Iceland around a thousand years ago that were first written down a few hundred years after that. It was the work of a pair of statistical physicists, Ralph Kenna of University of Coventry in the UK and his graduate student Pádraig Mac Carron, now at Oxford, who are applying the tools of their trade to works of epic literature, legend, and myth.

For this particular analysis, they painstakingly recorded the relationship of every settler in 18 sagas. The resulting web of interactions helped shed light on theories humanities scholars have been discussing for years, and even picked up on some previously unnoticed patterns. Their work is part of a movement that promises a new way to approach old questions in literature, history, and archaeology, with fanciful diagrams as just the appetizer.

Demonstration of social network analysis, with red lines representing unfriendly connections and green lines representing friendly ones.

The story of how Kenna and Mac Carron got here begins with the Irish tale of the cattle-raid of Cooley, or the Táin Bó Cúailnge. That yarn tells how the warrior-queen Medb of Connacht rallies an army to steal a fine bull from Ulster, and how youthful Cúchulainn, an Ulster folk hero, stands against her. Complete with a maiden prophet with three pupils in each eye, wild chariot rides, and an enormous cast of characters, it’s a story to grip anyone’s imagination.

It’s a story that Kenna and Mac Carron, who are both Irish, have known since childhood. Several years ago, Kenna, who has a successful career as a physicist, found his thoughts returning to mythology. It wasn’t as big a departure as it might seem at first. “In statistical physics, you’re dealing with objects such as gasses that are comprised of molecules and atoms,” he says. “The system consists of many small entities, and so many of them you cannot deal with them individually, you have to deal with them statistically.” Some physicists have started to use similar methods to look at how large numbers of people interact to produce aspects of human society, and Kenna wondered whether they could be applied to myths and stories. The Táin, which comes to us in pieces from many different manuscripts, the oldest nearly 1,000 years old, is considered literature rather than historical account. But it might still encode, in a way statistics can reveal, information about the society that produced it. Math might also help classify tales in a new way, quantitatively, in addition to the usual qualitative classifications.

Hat tip to Karen L. Myers.

14 Oct 2013

XKCD Meets Ayn Rand

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Hat tip to Karen L. Myers.

07 May 2013

“Jame Austen, Game Theorist”

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The latest wrinkle in the contemporary Jane Austen boom is described at Science Blog:

Austen’s novels are game theory textbooks,” Michael Suk-Young Chwe writes in “Jane Austen, Game Theorist,” which Princeton University Press published April 21. “She’s trying to get readers to use their higher thinking skills and to think strategically.”

At its most basic level, game theory assesses all the choices available to two (or more) people in a given situation and assigns a numerical value to the benefit each person reaps from each choice. Often, the choice that is most valuable to one player comes at the expense of the other; hence, game theory’s best-known phrase — “zero-sum game.” But just as frequently, there is a choice with unexpected benefits for both players.

“In game theory, you make choices by anticipating the payoffs for others,” Chwe explains.

Chwe argues that Austen explores this concept in all six of her novels, albeit with a different vocabulary than the one used by Nash, von Neumann and other game theory greats some 150 years later. In Austen’s romantic fiction, this type of strategic thinking is described as “penetration,” “foresight” or “a good scheme.”

In “Pride and Prejudice,” for instance, Mrs. Bennet, a mother eager to marry off her five daughters, sends her oldest, Jane, on horseback to a neighboring estate, even though she’s aware a storm is on the way. “Mrs. Bennet knows full well that because of the rain, Jane’s hosts will invite her to spend the night, thus maximizing face time with the eligible bachelor there, Charles Bingley, whom Jane eventually marries,” Chwe said.

In “Persuasion,” the unmarried heroine, Anne Elliot, is approached by Sophia Croft, the sister of a man whose marriage proposal Anne spurned eight years earlier — a decision she still bitterly regrets. Mrs. Croft casually asks Anne whether she’s heard that her brother has married. Anne flinches, thinking the reference is to her former beau, Captain Frederick Wentworth, but relaxes upon learning that Mrs. Croft is actually referring to their younger brother, Edward.

“It’s hard to imagine a better way for Mrs. Croft to gauge Anne’s visceral interest in her unmarried brother,” said Chwe, a UCLA associate professor of political science (whose last name is pronounced like “chess” without the “ss”). The rest of the novel involves schemes to give Captain Wentworth so many signals of Anne’s enduring love that he finds the courage to propose to her again.

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