Monday, May 22, 2017

Lesson 15-9: The Isoperimetric Theorems in Space (Day 159)

This is what Theoni Pappas writes on page 142 of her Magic of Mathematics:


This is another introduction page. Chapter 6 of the Pappas book is "Mathematical Magic From the Past," and the actual title page for this chapter is on page 143.

But as usual, Pappas doesn't simply leave page 142 blank. Instead, she fills this page by counting from zero to eleven in several different numeration systems:

-- Hindu-Arabic
-- Babylonian
-- Greek
-- Egyptian Hieroglyphic
-- Chinese Script
-- Hebrew
-- Chinese Rod Numerals
-- Roman
-- Egyptian Hieratic
-- Mayan
-- Binary Numerals

Not all of these numbers can be rendered easily in ASCII though. The Hindu-Arabic numerals from zero to eleven are very simply rendered in ASCII:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11...

That's right -- the standard numerals we use today are actually Hindu-Arabic. The Roman numerals are obviously easy to type in ASCII as well:

I, II, III, IV, V, VI, VII, VIII, IX, X, XI...

Notice that the Roman, as well as some of the others, begin with one rather than zero, as these systems lack a zero. The final set of numbers easy to type in ASCII are the binary numerals:

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011...

With the Greek numerals, the number 1 is A and 2 is B, but we can't do 3 in ASCII. That's because the A is actually "alpha" and the B is "beta," but the 3 is "gamma," which doesn't look like any letter that we can type in ASCII. Number 9 is "theta," a letter which we associate with angles.

As it happens, both Greek and Hebrew simply use the first ten letters of the alphabet for 1-10. The symbol for 1 in Hebrew is "aleph," which Cantor used to denote infinity. The fact that every letter in Hebrew and Greek is a number is used in the "Bible Code," where people try to find hidden messages in the scriptures by converting the words in the original language to numbers.

Lesson 15-9 of the U of Chicago text is on "The Isoperimetric Theorems in Space." These are the 3D analogs of the theorems we discussed on Friday.

Isoperimetric Theorem (space version):
Of all solids with the same surface area, the sphere has the most volume.
Of all solids with the same volume, the sphere has the least surface area.

We don't even bother trying to prove these theorems. As we've seen, the 2D proofs are very difficult, so imagine how much more so the 3D proofs would be.

This is the final lesson in the U of Chicago text. Here is how the U of Chicago closes the text:

"The Isoperimetric Theorems involve square and cube roots, pi, polygons, circles, polyhedra, and spheres. They explain properties of fences, soap bubbles, and sponges. They demonstrate the broad applicability of geometry and the unity of mathematics. Many people enjoy mathematics due to the way it connects diverse topics. Others like mathematics for its uses. Still others like the logical way mathematics fits together and grows. We have tried to provide all these kinds of experiences in this book and hope that you have enjoyed it."

Well I for one have certainly enjoyed this text, and I hope you, the readers of this blog, have as well.

One of the bonus Exploration questions mentions the ancient Carthaginian queen Dido. I wrote about her last year as well:

"According to one of the legends of history, Dido, of the Phoenician city of Tyre, ran away from her family to settle on the Mediterranean coast of North Africa. There she bargained for some land and agreed to pay a fixed sum for as much land as could be encompassed by a bull's hide."

"Her second bright idea was to use this length to bound an area along the sea. Because no hide would be needed along the seashore she could thereby enclose more area."

We know that the solution to the Isometric Problem is the circle -- the curve that encloses the most area for its length. We've also seen questions in which we are to maximize area by building a fence along a river to enclose a rectangular area -- the answer is a rectangle whose width is exactly half of its length. Combining these two ideas, we can solve the Dido problem:

"According to the legend, Dido thought about the problem and discovered that the length of hide should form a semicircle."

So we see that without water, the largest area is a circle -- with water, it's a semicircle. If we restrict to rectangles, without water the largest area is a square -- with water, it's a semi-square (that is, half of a square, or a rectangle whose width is half of its length).

"[Dido's new lover] Aeneas was a man on a mission, and he soon departed to found a new civilization in Rome. Dejected and distraught, Dido could do no more for Aeneas than to throw herself on a blazing pyre so as to help light his way to Italy...Rome made no contributions to mathematics whereas Dido might have."

By the way, let's tie this back to Pappas and ask, what numerals would Dido have used? Carthage is actually derived from the Phoenician culture. Notice that the Square One TV video "The Mathematics of Love" seems to imply that the Phoenicians used our (current) numerals (in contrast with the Roman), even though Phoenician has nothing to do with Hindu-Arabic. So in the end, I don't know what numerals Carthaginians might have used.

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