Folded into the mixed baklava of my 2018, was a special mathematical bit of honey. With the help of some excellent YouTube videos, the light bulb finally went on for me, and I could see quaternions. Judging by online comments I’ve read, I wasn’t alone in the dark.
There does seem a conceptual stumbling block (I tripped, anyway), but once that’s cleared up, quaternions turn out to be pretty easy to use. Which is cool, because they are very useful if you want to rotate some points in 3D space (a need I’m sure many of have experienced over the years).
The stumbling block has to do with quaternions having not one, not two, but three distinct “imaginary” numbers.
Last week I did a little jazz riff on the idea of “story space” — where all the stories live — and how the interesting stories we want to hear are all improbable to the point of having zero chance of actually happening (unless, gasp, statistics can lie).
I thought I’d return to that basic story space idea and, in the process, finally deal with a note that’s been on my idea board for years. My problem has been that, while the idea the note expresses seemed interesting enough, I’ve never quite seen how to turn it into a post. I’m not even sure the idea makes any real sense, let alone is worth trying to write about.
However that’s never stopped me before, and it’s (almost) Chillaxmus, so cue the music, it’s riff time again…
When I was a high school kid, my dad and I sometimes played a game where one of us would make up a secret code, write a message in that code, and the other would try to decipher the message. We generally used simple substitution ciphers, so it was an exercise in letter frequency analysis and word guessing.
There’s a cute secret code I found in a book back then that really stuck with me because of the neat way it looks. It also stuck with me because it’s so simple that once you learn it, you really can’t forget it.
So for some Saturday fun, I thought I’d share it with you.
But my brain is full!
You may have noticed that, in a number of recent posts, the topic has been math. The good-bad news is that there’s more to come (sorry, but I love this stuff). The good-good news is that I’m done with math foundations. For now.
To wrap up the discussion of math’s universality and inevitability — and also of its fascination and beauty — today I just have some YouTube videos you can watch this Sunday afternoon. (Assuming you’re a geek like me.)
So get a coffee and get comfortable!
In the recent post Inevitable Math I explored the idea that mathematics was both universal and inevitable. The argument is that the foundations of mathematics are so woven into the fabric of reality (if not actually being the fabric of reality) that any intelligence must discover them.
Which is not to say they would think about or express their mathematics in ways immediately recognizable to us. There could be fundamental differences, not just in their notation, but in how they conceive of numbers.
To explore that a little, here are a couple of twists on numbers:
Oh, no! Not math again!
Among those who try to imagine alien first contact, many believe that mathematics will be the basis of initial communication. This is based on the perceived universality and inevitability of mathematics. They see math as so fundamental any intelligence must not only discover it, but must discover the same things we’ve discovered.
There is even a belief that math is more real than the physical universe, that it may be the actual basis of reality. The other end of that spectrum is a belief that mathematics is an invented game of symbol manipulation with no deep meaning.
So today: the idea that math is universal and inevitable.
Put on your arithmetic caps, dear readers. Also your math mittens, geometry galoshes and cosine coats. Today we’re venturing after numeric prey that lurks down among the lines and angles.
There’s no danger, at least not to life or limb, but I can’t promise some ideas won’t take root in your brain. There’s a very real danger of learning something when you venture into dark territory such as this. Even the strongest sometimes succumb, so hang on to your hats (and galoshes and mittens and coats and brains).
Today we’re going after vectors and scalars (and some other game)!
I’ve written here before about chaos theory and how it prevents us from calculating certain physical models effectively. It’s not that these models don’t accurately reflect the physics involved; it’s that any attempt to use actual numbers introduces tiny errors into the process. These cause the result to drift more and more as the calculation extends into the future.
This is why tomorrow’s weather prediction is fairly accurate but a prediction for a year from now is entirely guesswork. (We could make a rough guess based on past seasons.) Yet the Earth itself is a computer — an analog computer — that tells us exactly what the weather is a year from now.
The thing is: it runs in real-time and takes a year to give us an answer!
No doubt those who regard quantum physics or Einstein’s relativity or even just trigonometry as an impenetrable thicket of unknowable terms and ideas have a hard time believing science could be easy. The lingo alone seems to create an exclusive “members only” club.
The trick is: easy (or difficult) compared to what? Many scientists now disdain philosophy (apparently forgetting what we now call science was once called natural philosophy). They point to the advances of science in the last 500 (or whatever) years and then say that philosophy hasn’t been nearly as successful in 2000 years.
But that’s because science is easy. It’s philosophy that’s hard!