Category Archive 'Mathematics'
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.

14 Mar 2013

Applied Math

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click on image

22 Sep 2012

Voting by the Dumb and Ill-Informed

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Beyond a certain point of negative understanding, of course, it is better for someone not to vote.

Joseph C. McMurray discusses the Marquis de Condorcet’s mathematical analysis favoring decision-making by larger numbers of people.

An interesting, if somewhat uncommon, lens through which to view politics is that of mathematics. One of the strongest arguments ever made in favor of democracy, for example, was in 1785 by the political philosopher-mathematician, Nicolas de Condorcet. Because different people possess different pieces of information about an issue, he reasoned, they predict different outcomes from the same policy proposals, and will thus favor different policies, even when they actually share a common goal. Ultimately, however, if the future were perfectly known, some of these predictions would prove more accurate than others. From a present vantage point, then, each voter has some probability of actually favoring an inferior policy. Individually, this probability may be rather high, but collective decisions draw information from large numbers of sources, mistaking mistakes less likely.

To clarify Condorcet’s argument, note that an individual who knows nothing can identify the more effective of two policies with 50% probability; if she knows a lot about an issue, her odds are higher. For the sake of argument, suppose that a citizen correctly identifies the better alternative 51% of the time. On any given issue, then, many will erroneously support the inferior policy, but (assuming that voters form opinions independently, in a statistical sense) a 51% majority will favor whichever policy is actually superior. More formally, the probability of a collective mistake approaches zero as the number of voters grows large.

Condorcet’s mathematical analysis assumes that voters’ opinions are equally reliable, but in reality, expertise varies widely on any issue, which raises the question of who should be voting? One conventional view is that everyone should participate; in fact, this has a mathematical justification, since in Condorcet’s model, collective errors become less likely as the number of voters increases. On the other hand, another common view is that citizens with only limited information should abstain, leaving a decision to those who know the most about the issue. Ultimately, the question must be settled mathematically: assuming that different citizens have different probabilities of correctly identifying good policies, what configuration of voter participation maximizes the probability of making the right collective decision?

It turns out that, when voters differ in expertise, it is not optimal for all to vote, even when each citizen’s private accuracy exceeds 50%. In other words, a citizen with only limited expertise on an issue can best serve the electorate by ignoring her own opinion and abstaining, in deference to those who know more. …

This raises a new question, however, which is who should continue voting: if the least informed citizens all abstain, then a moderately informed citizen now becomes the least informed voter; should she abstain, as well?

Mathematically, it turns out that for any distribution of expertise, there is a threshold above which citizens should continue voting, no matter how large the electorate grows. A citizen right at this threshold is less knowledgeable than other voters, but nevertheless improves the collective electoral decision by bolstering the number of votes. The formula that derives this threshold is of limited practical use, since voter accuracies cannot readily be measured, but simple example distributions demonstrate that voting may well be optimal for a sizeable majority of the electorate.

The dual message that poorly informed votes reduce the quality of electoral decisions, but that moderately informed votes can improve even the decisions made even by more expert peers, may leave an individual feeling conflicted as to whether she should express her tentative opinions, or abstain in deference to those with better expertise. Assuming that her peers vote and abstain optimally, it may be useful to first predict voter turnout, and then participate (or not) accordingly: when half the electorate votes, it should be the better-informed half; when voter turnout is 75%, all but the least-informed quartile should participate. …

If Condorcet’s basic premise is right, an uninformed citizen’s highest contribution may actually be to abstain from voting, trusting her peers to make decisions on her behalf. At the same time, voters with only limited expertise can rest assured that a single, moderately-informed vote can improve upon the decision made by a large number of experts. One might say that this is the true essence of democracy.

His conclusion seems to accord with observed results. Ordinary people are surprisingly well able to correct the follies and delusions which too commonly afflict the experts and elites, but there are also people so clueless that they are always going to vote wrong.

02 Jul 2012

Good Tip

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Hat tip to Jose Guardia.

23 Jun 2012

Gandalf’s Function

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Hat tip to The Meta Picture via Kathleen Wagner.

30 May 2012

The Archimedes Palimpsest

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“How do you read a two-thousand-year-old manuscript that has been erased, cut up, written on and painted over [i.e., a palimpsest]? With a powerful particle accelerator, of course! Ancient books curator William Noel tells the fascinating story behind the Archimedes palimpsest, a Byzantine prayer book containing previously-unknown original writings from ancient Greek mathematician Archimedes and others.”

Archimedes Palimpsest website

Hat tip to Dot Porter.

28 Sep 2010

Trigonometry Illustrated

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From reddit.

22 Jan 2010

Yoshimoto Cube

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Two stellated rhombic dodecahedrons can be folded into a cube. “A very impressive piece of engineering.”

1:15 video

Hat tip to Forgetomori, forwarded by Robert Breedlove.

16 Jan 2010

The Predator Prey Ecology of Vampires and Humans in (Pre-Apocalypse) Sunnyvale, California

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A typical Sunnyvale vampire (Spike) preying on a typical female citizen (Willow).

Brian Dalen Thomas addresses the vexed question of human vampire ecology in the Sunnyvale, California of Joss Wheedon’s Buffy the Vampire Slayer.

Excerpts:

(W)e know from the sign in “Lover’s Walk” that the human population of Sunnydale is 38,500. …

Sunnydale’s human population growth rate is 10% annually, which is certainly at the high end for a budding California community.

A vampire feeds every three days, and encounters about one hundred potential victims in the course of a day, meaning that 1 out of every 300 encounters involves a little refreshment.

An individual vampire sires a victim every other year, or once per 240 feedings.

Buffy and her Slayerettes, busy little beavers that they are, annually stake about 1/3 of the vampires plaguing Sunnydale.

Vampires are flocking to Sunnydale, since the Hellmouth is the underwordly equivalent of Silicon Valley, and the demon labor market is just too good to be true. Thus, we’ll assume a yearly migration rate of about 10%, or the same as for the humans.

A Model

What follows is based on some of the simpler theoretical understandings of predator-prey population dynamics. I’m assuming that human populations are not controlled solely by vampire predation (i.e.- in the absence of vampires, the human population would still eventually be limited by some other factor, like food supply, disease, or access to a well written weekly news magazine. I like The Economist myself, but that’s clearly a digression).

If we let H stand for the size of the human population and V stand for the size of the vampire population, then we can represent the changes in each population over time with a pair of differential equations:

dH/dt = rH (K-H)/K -aHV

dv/dT = baHV + mV – sV

where r is the intrinsic growth rate of the human population, incorporating natural rates of both birth and death as well as immigration

K is the human carrying capacity of the habitat in question

a is a coefficient that relates the number of human-vampire encounters to the number of actual feedings

b is the proportion of feedings in which the vampire sires the victim (i.e.- this is the vampire birth rate)

m is the net rate of vampire migration into Sunnydale

s is the rate at which the Scoobies stake vampires (assumed to be the only important source of vampire deaths).


The following graph shows human population sizes on the horizontal axis and vampire population sizes on the vertical axis. Each line represents a trajectory through time (the tail of each line, scattered around the outer edge of the figure, shows the “initial population size” where we started the model in motion). Any point on a line represents a combination of human and vampire population sizes – a step, if you will, in that beautiful dance between Buffy and the Minions of Evil. Notice that wherever we “start” the trajectories, they all spiral in towards our equilibrium state, indicated in the center by an
asterisk.

Hat tip to Robert M. Breedlove.

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