The Important Theorems Are the Beautiful Ones

Dr. Gene Chase guest blog author here again.

What makes a math theorem important?

The usual answer is that it is either beautiful or useful. If like me you think that being useful is a beautiful thing, then important theorems are the beautiful ones.

But what makes a theorem beautiful? For example, why is the Theorem of Pythagoras widely regarded as beautiful: and a, b, and c are not 0 if and only if a, b, and c are the sides of a right triangle? (OK, break into small groups and discuss this among yourselves! An answer appears at the bottom of this post.)

But the theorem 1223334444 = 1223334443 + 1 is not beautiful, won’t you agree?

If the theorem is geometric, we can appeal to visual beauty. For example, three circles pairwise tangent have a beautiful property that is animated here.

But beautiful theorems do not have to be geometric. Numbers are beautiful. For example, Euclid’s theorem that there are an infinity of primes is beautiful. No one has been able to draw a beautiful picture about that, although people have tried from astronomer and mathematician Eratosthenes in 200 BC to science fiction writer and mathematician Stanislaw Ulam in 1963.

For $15 you can have a mathematical theorem named after you. But I can guarantee that it won’t be beautiful. So if you want a theorem named after you, give Mr. Chase the $15 instead and he’ll find one for you. Don’t use 1223334444 = 1223334443 + 1. I claim that as “Dr. Gene Chase’s theorem.”

Answer to discussion question above: Most folks say that a beautiful theorem has to be “deep,” which is just a metaphor for “having many connections to many other things.” For example, the Theorem of Pythagoras has to do with areas, not squares specifically. The semicircle on the hypotenuse of a right triangle has an area equal to the sum of the areas of the semicircles on the adjacent sides. And so for any three similar figures.

Do you remember the joy that you feel when you first learned that two of your friends are also friends of each other? That’s the joy that a mathematician feels when she discovers that the Theorem of Pythagoras and the Theorem of Euclid are intimate with each other. But I’ll leave that connection to another post.

Math is about surprising connections. Which is to say, it’s about beauty.

200 Countries over 200 Years: Visual Data Analysis

Dr. Gene Chase guest author again.

The video at Joy of Stats shows 200 countries’ health and wealth over 200 years. The augmented reality (AR) presentation makes it interesting even if you’re not a geek like me.

My favorite part of Statistics isn’t numerical. It’s graphical. Visual data analysis is powerful because our eyes coupled with our minds are able to see patterns that no amount of means, modes, medians, and standard deviations can show.

Math vocabulary sometimes makes sense

This is the first guest post from John Chase’s dad, also a math teacher.  Thanks, son, for letting me post to your blog.

Gene Chase:  I was taking a shower today when I figured out why I always confused the words “sequence” and “series.”  2, 3, 4, 5, … is a sequence; 2+3+4+5 is a series.  Until today, I thought that my confusion was because “series” and “sequence” both begin with “s.”  Now I see the real problem!  Teachers would say “sum the following series.”  They should have said “evaluate the following series,” since the series is already a sum.

Comment from John Chase:   In non-mathematical contexts we don’t differentiate between the two. We think of “television series” and a “series” of cars in a line at an intersection. How mathematically sloppy!

Gene Chase:  Yes, usually mathematical language is general language made more precise, not less precise.  For example, if you tell a story elliptically, you leave things out of it; if you tell the story parabolically, you give an analog of the story; if you tell the story hyperbolically, you embellish it.  The corresponding geometric figures have eccentricities which are either between 0 and 1 (ellipse), precisely equal to 1 (parabola), or greater than 1 (hyperbola).

This makes sense when you remember that “elliptic” is Greek for “defective,” “para” is Greek for “along side,” and “hyper” is Greek for “beyond.”