Babylonian Trigonometry

Last weekend all the science news sites and many regular news outlets carried stories about this 3,700-year-old Babylonian tablet, known as Plimpton 322. They all reported that it had either been shown to be (for the bold majority) or claimed to be (for the cautious few) a trigonometric table based on ratios rather than angles. But these stories were obviously written by reporters whose math is about as good as their cuneiform, so I was unable to tell from them what had actually been claimed, how plausible it was, or how this table of ratios was supposed to work. (I saw in several places the claim that this table is more accurate than modern trig tables, which is absurd since with a computer we can run the numbers out to a million decimal places if we feel like it.) I hate writing about things I don't understand, so I held off.

But now the Times has found a mathematically literate reporter to do a follow-up story, and his explanation is fascinating. He asks, how would a Babylonian scribe have gone about solving this problem:

Suppose that a ramp leading to the top of a ziggurat wall is 56 cubits long, and the vertical height of the ziggurat is 45 cubits. What is the distance x from the outside base of the ramp to the point directly below the top?

Perhaps using a table like the one on Plimpton 322. Plimpton 322 is a list of 15 numerical triads, all of which are Pythagorean triples – numbers for which a² + b² = c², like 3² + 4² = 5². But these are not randomly chosen triads. They are placed in the ascending order of the ratio of the hypotenuse and the short side.

A Babylonian faced with the ziggurat word problem may have found it easy to set up: a right triangle with the long side, or hypotenuse, 56 cubits long, and one of the shorter sides 45 cubits. Next, the problem solver could have calculated the ratio 56/45, or about 1.244 and then looked up the closest entry on the table, which is line 11, which lists the ratio 1.25.

From that line, it is then a straightforward calculation to produce an answer of 33.6 cubits. In their paper, Dr. Mansfield and Dr. Wildberger show that this is better than what would be calculated using a trigonometric table from the Indian mathematician Madhava 3,000 years later.

The modern answer is 33.3317, so obviously the Plimpton 322 method is not perfect. But it would be useful in many circumstances. Which one reason why I think this is a perfectly plausible explanation of this mysterious tablet; some Babylonian scribes were obviously fascinated by the relationship between mathematical arcana and real-world problems, and this is just the sort of thing they might have been into.

The second reason is that it conforms to our understanding of how math has evolved. It seems that many general theories, like the fully-developed Greek theory of angles, are proceeded by numerous glimpses of the underlying principles, and by lists of examples of the theory at work. The generalization of a principle is usually the last stage of a long process. This was especially true in the ancient world, when mathematical advances were usually driven by specific problems in surveying or astronomy. As this example shows, Babylonian scribes often made use of right triangles in their calculations, but so far as we know they never wrote down what we call the Pythagorean Theorem in its abstract form. Some moderns insist that they must have understood it even if they did not write it down, but that is probably a mistake. The drive to create perfectly expressed universal theorems is not that common even among modern people who use complex math; modern physicists and statisticians use mathematical tricks all the time that have never been formalized or proved, because they seem to work, and Babylonian scribes probably used math in the same way.

Because of the great interest, Historia Mathematica has made the original paper by Mansfield and Wildberger available online; but I still recommend  Kenneth Chang's Times piece as the better explanation.

The Mathematician Plays Chess with the Devil

From an article on math prodigy turned Fields Medal winner Terry Tao:

The true work of the mathematician is not experienced until the later parts of graduate school, when the student is challenged to create knowledge in the form of a novel proof. It is common to fill page after page with an attempt, the seasons turning, only to arrive precisely where you began, empty-handed — or to realize that a subtle flaw of logic doomed the whole enterprise from its outset. The steady state of mathematical research is to be completely stuck. It is a process that Charles Fefferman of Princeton, himself a onetime math prodigy turned Fields medalist, likens to ‘‘playing chess with the devil.’’ The rules of the devil’s game are special, though: The devil is vastly superior at chess, but, Fefferman explained, you may take back as many moves as you like, and the devil may not. You play a first game, and, of course, ‘‘he crushes you.’’ So you take back moves and try something different, and he crushes you again, ‘‘in much the same way.’’ If you are sufficiently wily, you will eventually discover a move that forces the devil to shift strategy; you still lose, but — aha! — you have your first clue.

RIP Benoit Mandelbrot

Benoit Mandelbrot was a British mathematician best known for his work on fractal geometry; he died yesterday at the age of 85. A fractal is a shape that becomes more and more complex the closer you look at it. The most famous example is the Mandelbrot Set, a shape of infinite complexity defined by a simple equation: , where C is the set of complex numbers; if Z in this relation does not tend toward infinity, the number is in the set. The more you magnify this fabulous shape, the more detail you see.

The most interesting real-world consequence of fractal geometry is that the length of a line depends on the scale at which you measure it. If you measure the coastline of Britain on a world map, you get one number, but as you look at the coast on maps of higher resolution you see more and more small features -- inlets, peninsulas, rocks -- that you missed in the global view, and the length of the coastline keeps getting longer. Mandelbrot pointed this out in a famous article titled, "How long is the coastline of Britain?" In my experience it is impossible to persuade even most archaeologists that this is true; people want something like the length of the Mississippi River or the surface area of a person's skin to be a number that is independent of the scale of your map or the length of your ruler. But the world is not like that. As Mandelbrot told the NY Times a few years ago,

Here is a question, a staple of grade-school geometry that, if you think about it, is impossible. The length of the coastline, in a sense, is infinite.

Mandelbrot had a knack for getting the attention of people who had only a little interest in math, and it is typical that the equation to which he put his name is not only mathematically interesting but gorgeous.

Probability Made Easy

Another great Steven Strogartz math column in the NY Times, this one about probability:

The probability that one of these women has breast cancer is 0.8 percent. If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 7 percent that she will still have a positive mammogram. Imagine a woman who has a positive mammogram. What is the probability that she actually has breast cancer?. . .

The right answer is 9 percent.

How can it be so low? . . . the analysis becomes almost transparent if we translate the original information from percentages and probabilities into natural frequencies:

Eight out of every 1,000 women have breast cancer. Of these 8 women with breast cancer, 7 will have a positive mammogram. Of the remaining 992 women who don’t have breast cancer, some 70 will still have a positive mammogram. Imagine a sample of women who have positive mammograms in screening. How many of these women actually have breast cancer?

Since a total of 7 + 70 = 77 women have positive mammograms, and only 7 of them truly have breast cancer, the probability of having breast cancer given a positive mammogram is 7 out of 77, which is 1 in 11, or about 9 percent.

As this example shows, many probability problems become much simpler when you start by imagining a complete set of possible outcomes, in this case, for 1000 women who have had mammograms. To figure out the probability of rolling an 8 on two dice -- regular six-sided dice -- you can start by listing all the possible outcomes (there are 36, 6x6) and just counting how many add up to 8.

Limits

Stephen Strogatz has been writing a long series of columns at the NY Times in which he teaches the basics of math, from addition on up. The latest explains how finding the area of a circle led Archimedes very close to the discovery of calculus. It's very clear, and anyone who never took calculus but is curious what it is all about might find it interesting.