Night at the Mathemuseum

For Memorial Day, we decided to treat the Ladybug to a movie at the local movieplex, and being there reminded me of a bunch of movie-related posts I’ve been meaning to make, which I’ll do throughout the week.

To kick it off, let’s begin with the movie we actually saw.   We took the Ladybug to go see Night at the Museum 2, which was surprisingly entertaining, in that I was able to sit through all 105 minutes of its run-time despite the presence of Ben Stiller onscreen for almost all of them.

In fact, my only major gripe with this movie is mathematical.   The plot (such as it is) involves an evil pharaoh who needs to discover a secret combination to push into an enchanted tablet to unleash unspeakable power.   Or something; it doesn’t matter.     What ends up happening is the following:

  1. The numeric combination is “the secret at the base of the pyramids,” which is immediately identified by Amelia Earhart as pi
  2. Albert Einstein then states that pi is “3.14159265, to be exact.”
  3. The evil pharaoh then pushes this into the tablet, which consists of a 3×3 grid of squares, as if it were a giant telephone pad.

What?

Now granted, a movie built around sentient statues, undead dinosaurs, bird-headed Egyptian gods, and a surprisingly flight-worthy Wright flier is clearly not striving for any kind of scientific or quantitative realism, but that’s not the bit that bugged me.

No, these things offended my sense of mathematical aesthetics and history, and in the spirit of mathematical edification, I’ll tell you why, point by point.

Point 1: pi is not encoded in the Great Pyramid.   This idea is based on the new-agey notion that the Egyptians encoded their knowledge of transcendental numbers such as pi or the golden ratio in the dimensions of the Great Pyramid of Cheops.   Specifically, pi is supposedly approximated by twice the ratio of the base of the pyramid to its height.

However, there’s no bit of mathematical history to support this as being anything other than coincidental. In fact, most archeological evidence indicates that the Egyptians were relatively unconcerned with the value we nowadays know as pi.   For example, the process by which Egyptians computed the area of a circle, for example, is not to take the radius squared and multiply it by some constant; rather, Egyptian surveyors were instructed to instead first subtract from the diameter of the circle 1/9th the length of the diameter, and then to multiply that value by itself.     In fact, this process is equivalent to approximating the area of a circle by the area of a square whose side length is 8/9 the diameter of the circle, which would mean that ancient Egyptians were more or less approximating pi by the value 256/81, which is roughly 3.16, not 3.14.

The Great Pyramid mumbo-jumbo itself became popular only in the early 1920s (which would at least explain why Amelia Earhart was so quick to know it), and it’s largely due to the predictions of bone fide psychic/crank Edgar Cayce, who connected the “divine proportions” of the ancient Egyptians with, among other things, the re-emergence of the lost city of Atlantis.   In fact, several of his followers (who quickly became known to Egyptologists as pyramidiots) were so taken with proving Cayce correct that they were actually caught gently filing down corners of the Great pyramid to make the approximations that much more accurate.

Point 2: pi is not exactly 3.14159265.   That’s a really accurate approximation, but that’s all it is.   The number pi is irrational (transcendental, even), so it cannot be written exactly as any finite decimal number, no matter how long.

What’s worse — Albert Einstein would have known better.   For shame.

Point 3: No Egyptian would understand “3.14159265.” Even assuming that ancient Egyptians did treat the number pi with any special reverence — and given that their standard approximation of it is only accurate to one decimal place, this is highly unlikely! —   they still wouldn’t work with something like 3.14159265.   The reason why is two fold.

First, even though the ancient Egyptians used a base-10 number system like we do now, they didn’t use a positional number system.   To explain: expressing a number like one-hundred twenty-five as “125” assumes you’re using a positional number system: the order in which the digits appear is just as important has the digits themselves.   Just think about it: even through 125 and 251 and 152 and 521 all have the same digits (namely 1, 2, and 5), they all represent very different numbers.   However, standard Egyptian hieroglyphic numbering was not positional, it was additive: it consisted of a special symbol for each of the numbers 1, 10, 100, 100, etc., and then used multiple copies of each symbol to construct a number.   For example, to write 125 in hieroglyphics, an Egyptian scribe could write

which consists of five | symbols (each meaning 1), two arches (each representing 10) and a single spiral (representing 100).   The order is immaterial — each of the following values still represent 125:

Mathematical history buffs might point out that high priests and other top-tier ancient Egyptians did not use hieroglyphics for their day-to-day recording and calculations involving numbers.   They used an abbreviated shorthand called hieratic script, but even this shorthand was an additive ciphered system — it had separate symbols for the numbers 1, 2, …,   9, then another set for 10, 20, …, 90, and then another set for 100, 200, …, 900, and so on.   For example, to write 125, a scribe would only need to write the symbol for 100, 20, and 5, such as

But again, the order didn’t matter.   The symbol

still means the same thing.   By way of another comparison, the number 152 and 125 have the same digits but refer to different numbers — their relative position is what determines this.   In contrast, in hieratic the number 152 uses a completely different set of symbols that 125:

An expression like 3.14159265 assumes a positional numbering system, and that’s just something the Egyptians didn’t work with.

That brings me to the second reason why Egyptians wouldn’t work with something like 3.14159265.   Even assuming they were to understand that this meant the value found by dividing 314159265 by 100000000, they never would have expressed the answer as the single fraction 314159265/100000000.

The single most defining aspect of the Egyptian number system is its reliance on unit fractions.   To describe any quantity less than 1, Egyptians always and invariably expressed it as a sum of fractions of the form 1/n, so-called “unit” fractions.   For example, an ancient Egyptian would never talk about the fraction 2/5, they would instead say that 5 into 2 yields 1/3 + 1/15; they’d never talk about 2/89, but rather 1/60 + 1/356 + 1/534 + 1/890.   Hell, even though the Egyptians treated pi as 256/81, they never would have thought to express it that way; instead, it would have been written as 3 + 1/7 + 1/57 + 1/10773. I have no idea how the Egyptians would have expressed this more accurate approximation of 3.14159265, but it would not be as a decimal; it’d be something closer to 3 + 1/8 + 1/61 + 1/5020.

So go and enjoy the show… and be better informed, too!

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