March Mathness

Time for math!

I have a special fondness for the month of March. For one thing, it contains the Vernal Equinox — one of my favorite days, because it heralds six months of light. (As a Minnesotan, Spring has much more impact than it did when I lived in Los Angeles.)

March is when the weather elves begin preparing for the April Showers that create May Flowers. It’s when baseball Spring Training is in full swing with the regular season looming (lately, even at the end of the month; this year on the 28th).

It also contains some important birthdays: Albert Einstein (3/14) and Emmy Noether (3/23), to name two, and in their honor I have myriad math posts planned!

I assume everyone knows basically who Albert Einstein was (although many may not be familiar with his other major contributions, such as Brownian motion and the photoelectric effect). Obviously, he’s the guy behind (Special and General) Relativity.

Emmy Noether is not well-known outside the mathier sciences, but she is actually one of the most significant mathematicians ever. What’s more, her work with symmetry underpins much of basic physics. Conservation laws, and Quantum Field theory, are both based on mathematical symmetry.

Those who read my Special Relativity series (March of 2015) will recall I used Al (Einstein) and Em (Noether) as my “Alice and Bob” (or as I prefer in these gender-neutral times: “Alex and Blair”).

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I know many people don’t like math, or feel they can’t get it.

In many (if not all) cases, I think this is due to an extremely bad math education. Few math teachers seem to tap into the fascination and beauty of math. Most of us learned math by rote with no insight into the need or power of the tools we learned.

“Imaginary” (complex) numbers are a good example. They’re based on an apparent absurdity, the square root of negative one.

But negative numbers can’t have square roots, so how is it even possible?

Isn’t that a contradiction? Why would we even do that?

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It turns out we need them.

It also turns out they sort of pop out naturally if you try to find the roots to certain simple equations. For example:

Obviously, x equals +2 and -2. But what about:

The only way this works is if, somehow, x² is -4, and that requires the concept of the square root of negative one, which we call i.

The answers to the second equation, then, are +2i and -2i.

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If you didn’t follow that, don’t worry about it, it’s not important.

What’s important is realizing that many of the seemingly bizarre things that pop out of math turn out to have the same necessity and obviousness as do complex numbers.

One of the things that’s so fascinating and beautiful about math is how one thing leads to another. It seems that, once we invent (or discover) counting, everything else follows.

Further reading: See Inevitable Math and Beautiful Math.

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I also have some very technical posts coming that pick up where I left off with Quaternions. (They’ll all be Sideband posts to signal that they’re down in the technical weeds.)

I’ve been self-studying rotation in different dimensions (one through four) with the eventual goal of fully understanding this:

Which is a common visualization of the “rotation” of a tesseract.

The model was easy enough to make based on ones I’ve seen, but I wanted to fully understand what’s going on. (For that matter, I just wanted to fully understand tesseracts, let alone their rotation.)

Crucially, I wanted the math behind that “rotation” — I wanted the mathematical model.

Which, no doubt, I could find online, if I wanted, because this is all very well-plowed ground. I’m not on the leading edge of anything here; I’m just exercising my aging (failing!) brain. I want to figure it out myself as much as possible.

It’s that this low-hanging fruit of simple rotations is my speed. It’s a level of math I can actually make sense of and explore on my own.

There is also that discovering these simple old things for myself, I have that well-known territory as reference to validate (or more often correct) my humble efforts.

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I’ve also benefited greatly from YouTube videos (and Wikipedia).

In particular, I’m hugely indebted to the 3Blue1Brown channel, although I’ve also learned a lot from Mathologer (which is also really fun). There is also the Numberphile channel, which explores the nooks and crannies of math.

But 3Blue1Brown is possibly the best YouTube math channel of all because of the way it brings out the beauty and purpose of math. If you like math at all, it’s a channel more than worth exploring.

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So,… I guess,… “Look out! Here comes the math!”

But it’s gonna be fun.

Really!

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As an example, check this out:

[click for big]

Here’s a video describing that technique:

Not that I (or likely most of us) have much application to find square roots, and when we do we probably prefer calculated accuracy.

What interested me was the error between the estimate and the correct value. I wondered what it looked like.

As you can see, the error is very high with smaller numbers, but the larger the number, the smaller the potential error.

(On the flip side, who remembers the really large perfect squares? Most of us likely cut out after 12² = 144.)

This is one of the great benefits of computers: the easy ability to visualize complex data.

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Down the road I’ll explore the importance of math with regard to pizza, so stay tuned!

On the other hand, I’m also going to get into:

So be warned?

(It’s the math behind (part of) that tesseract rotation!)