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.
You’ve been waiting for the other shoe to drop, right? The tick to finally tock? (My clock is — as usual — running a bit behind; this should be #62, but that’s another story.) Today’s tale involves electro-mechanical logic! Computing with relays rather than solid-state gates.
Rather than the tick-tock of a mechanical clock, the tock-tick of lots and lots of relays! Aisle after aisle of racks of relays, many thousands of them all clicking away like chattering insects. That’s what is (or was) inside some of those windowless buildings found in every neighborhood with local phone service.
However, today the focus is quite a bit smaller…
A different, more personal, anniversary… Looking back at the five years of this blog (or, for that matter, the many more years of this blogger), I can’t help but reflect on how man- and mice-plans go astray.
Case in point: Sidebands. Originally, as the name implies, meant as out-of-the-norm posts. True of the first few, but by #41 they’d become too inclusive, not much different from regular posts. The idea needed retooling; it was over a year before I posted #42. (And I added Brain Bubbles for minor-topic or very short posts.)
Since then, I’ve tried to restrict Sidebands to extremely technical topics. (For example, #59. I should have made that tesseract post #60, and today doubly wish I had.)
As a diversion for the weekend: Have you ever wondered why computers run so hot? No? Okay, I’ll tell you. It’s actually kind of a hoot. (We’ll get back to the more serious topic of algorithms and AI, and wrap up that series, next week.)
You kind of have to wonder. Humankind has gone from oil and gas lamps, to incandescent copper filaments, to fluorescent lights, and now to LEDs. The trend here seems towards cooler more efficient light sources. But computers seem to need bigger and bigger fans!
The short answer: It’s all those short circuits!
evaluate(2B || !2B)
Hamlet’s famous question, “To be or not to be?” is just one example of a question with a yes/no answer. It’s different from a question such as, “What’s your favorite color?” or, “How was your day?” What it boils down to is that the young Prince’s question requires only one bit to answer, and that bit is either yea or nay.
Computers can be very good at answering yes/no questions. We can write a computer program to compare two numbers and tell us — yea or nay — if the first one is bigger than the second one. Computers are also very good at calculations (they’re just big calculators, after all). For example, we can write a computer program that divides one number by another.
But there are questions computers can’t answer, and calculations they can’t make.
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:
We’re still motoring through numeric waters, but hang in there; the shore is just ahead. This is the last math theory post… for now. I do have one more up my sleeve, but that one is more of an overly long (and very technical) comment in reply to a post I read years ago. If I do write that one, it’ll be mainly to record the effort of trying to figure out the right answer.
This post picks up where I left off last time and talks more about the difference between numeric values and how we represent those values. Some of the groundwork for this discussion I’ve already written about in the L26 post and its followup L27 Details post. I’ll skip fairly lightly over that ground here.
Essentially, this post is about how we “spell” numbers.
In this post I’ll show how Set Theory allows us to define the natural numbers using sets. It’s admittedly a very abstract topic, but it’s about something very common in our experience: counting things. Seeing how numbers are defined also demonstrates (contrary to some false notions) that there is a huge difference between a number and how that number is “spelled” or represented.
Note: I am not a mathematician! This topic is right on the edge of my mathematical frontier. I wanted this addendum to the previous post, but be aware I may misstep. I welcome any feedback from Real Mathematicians!
But go on anyway… keep reading… I dare ya!
Be warned: these next Sideband posts are about Mathematics! Worse, they’re about the Theory of Mathematics!! But consider sticking around, at least for this one. It fulfills a promise I made in the Infinity is Funny post about how Georg Cantor proved there are (at least) two kinds of infinity: countable and uncountable. It also connects with the Smooth or Bumpy post, which considered differences between the discrete and the continuous.
This first one is pretty easy. The actual math involved is trivial, and I think it’s fascinating how the Yin/Yang of separate units versus a smooth continuum seems a fundamental aspect of reality. We can look around to see many places characterized by “bumpy” or “smooth” (including Star Trek). (The division lies at the heart of the conflict between Einstein’s Relativity and quantum physics.)
So let’s consider Cantor.
Today I want to tie up last time’s post about animation before moving on to other things. I’m sure I’ll return to the topic of making movies with POV-Ray and FFmpeg; it’s just too much fun, and I have tons of ideas. (I can finally do a really decent animation for the Special Relativity article I’m planning for Albert’s birthday.)
Firstly, I’ll discuss the animation initialization file, the ANI.INI file, and show you how the multiple segments are managed. Secondly, I’ll talk about the output files — all those frames we generate — and what to do with them.
Plus, I have a couple of important announcements!