Social networks in fact and fiction

SIAM News arrived this afternoon and had an interesting story on the front page: Applying math to myth helps separate fact from fiction.

In a nutshell, the authors hope to get some insight into whether a myth is based on fact by seeing whether the social network of characters in the myth looks more like a real social network or like the social network in a work of deliberate fiction. For instance, the social networks of the Iliad and Beowulf look more like actual social networks than does the social network of Harry Potter. Real social networks follow a power law distribution more closely than do social networks in works of fiction.

This could be interesting. For example, the article points out that some scholars believe Beowulf has a basis in historical events, though they don’t believe that Beowulf the character corresponds to a historical person. The network approach lends support to this position: the Beowulf social network looks more realistic when Beowulf himself is removed.

It seems however that an accurate historical account might have a suspicious social network, not because the events in it were made up but because they were filtered according to what the historian thought was important.

Baroque computers

From an interview with Neal Stephenson, giving some background for his Baroque Cycle:

Leibniz [1646-1716] actually thought about symbolic logic and why it was powerful and how it could be put to use. He went from that to building a machine that could carry out logical operations on bits. He knew about binary arithmetic. I found that quite startling. Up till then I hadn’t been that well informed about the history of logic and computing. I hadn’t been aware that anyone was thinking about those things so far in the past. I thought it all started with [Alan] Turing. So, I had computers in the 17th century.

Fourier series before Fourier

I always thought that Fourier was the first to come up with the idea of expressing general functions as infinite sums of sines and cosines. Apparently this isn’t true.

The idea that various functions can be described in terms of Fourier series … was for the first time proposed by Daniel Bernoulli (1700–1782) to solve the one-dimensional wave equation (the equation of motion of a string) about 50 years before Fourier. … However, no one contemporaneous to D. Bernoulli accepted the idea as a general method, and soon the study was forgotten.

Source: The Nonlinear World

Perhaps Fourier’s name stuck to the idea because he developed it further than Bernoulli did.

Related posts

History of weather prediction

I’ve just started reading Invisible in the Storm: The Role of Mathematics in Understanding Weather, ISBN 0691152721.

The subtitle may be a little misleading. There is a fair amount of math in the book, but the ratio of history to math is pretty high. You might say the book is more about the role of mathematicians than the role of mathematics. As Roger Penrose says on the back cover, the book has “illuminating descriptions and minimal technicality.”

Someone interested in weather prediction but without a strong math background would enjoy reading the book, though someone who knows more math will recognize some familiar names and theorems and will better appreciate how they fit into the narrative.

Related posts

Size of ancient and modern bureaucracies

According to The History of Rome, episode 126, Diocletian increased the size of the Roman imperial bureaucracy from around 15,000 people to around 30,000.

I wanted to compare the size of the bureaucracy that ran the Roman Empire to the size of the bureaucracy that runs Houston, TX. This page suggests that the city of Houston has about 68,000 employees. But far more people work for government in other capacities than work for the city. According to Table 1 of this page, the latest estimate is that 361,800 in the Houston MSA work in the government sector. And about 22 million people work in the government sector nation wide.

Please don’t leave comments saying the Roman Empire and Houston are not directly comparable. Of course they’re not. But still, a very rough comparison is interesting.

Related post: Pax Romana

Oldest series for pi

Here’s an interesting bit of history from Julian Havil’s new book The Irrationals. In 1593 François Viète discovered the following infinite product for pi:

\frac{2}{\pi} = \frac{\sqrt{2}}{2}\frac{\sqrt{2+\sqrt{2}}}{2}\frac{\sqrt{2 + \sqrt{2+\sqrt{2}}}}{2} \cdots

Havil says this is “the earliest known.” I don’t know whether this is specifically the oldest product representation for pi, or more generally the oldest formula for an infinite sequence of approximations that converge to pi. Vièta’s series is based on the double angle formula for cosine.

The first series for pi I remember seeing comes from evaluating the Taylor series for arc tangent at 1:

\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots

I saw this long before I knew what a Taylor series was. I imagine others have had the same experience because the series is fairly common in popular math books. However, this series is completely impractical for computing pi because it converges at a glacial pace. Vièta’s formula, on the other hand, converges fairly quickly. You could see for yourself by running the following Python code:

    from math import sqrt

    prod = 1.0
    radic = 0.0

    for i in range(10):
        radic = sqrt(2.0 + radic)
        prod *= 0.5*radic
        print 2.0/prod

After 10 terms, Vièta’s formula is correct to five decimal places.

Posts on more sophisticated and efficient series for computing pi:

Ancient understanding of tides

In his essay On Providence, Seneca (4 BC – 65 AD) says the following about tides:

In point of fact, their growth is strictly allotted; at the appropriate day and hour they approach in greater volume or less according as they are attracted by the lunar orb, at whose sway the ocean wells up.

Seneca doesn’t just mention an association between lunar and tidal cycles, but he says tides are attracted by the moon. That sounds awfully Newtonian for someone writing 16 centuries before Newton. The ancients may have understood that gravity wasn’t limited to the pull of the earth, that at least the moon also had a gravitational pull. That’s news to me.

The 1970s

Here’s a perspective on the 1970s I found interesting: The decade was so embarrassing that climbing out of the ’70s was a proud achievement.

The 1970s were America’s low tide. Not since the Depression had the country been so wracked with woe. Never — not even during the Depression — had American pride and self-confidence plunged deeper. But the decade was also, paradoxically, in some ways America’s finest hour. America was afflicted in the 1970s by a systemic crisis analogous to the one that struck Imperial Rome in the middle of the third century A.D. … But unlike the Romans, Americans staggered only briefly before the crisis. They took the blow. For a short time they behaved foolishly, and on one or two occasions, even disgracefully. Then they recouped. They rethought. They reinvented.

Source: How We Got Here: The 70’s: The Decade That Brought You Modern Life—For Better or Worse

An algebra problem from 1798

The Lady’s Diary was a popular magazine published in England from 1704 to 1841. It contained mathematical puzzles such as the following, published in 1798.

What two numbers are those whose product, difference of their squares, and the ratio or quotient of their cubes, are all equal to each other?

From Benjamin Wardhaugh’s new book A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing.

See also my brief review of How to Read Historical Mathematics by the same author.