In recent posts I’ve presented the complex numbers and the complex plane. Those were just stepping stones to this post, which involves a basic fact about the Mandelbrot set. It’s something that I stumbled over recently (after tip-toeing around it many times, because math).
This is one of those places where something that seems complicated turns out to have a fairly simple (and kinda cool) way to see it when approached the right way. In this case, it’s the way multiplication rotates points on the complex plane. This allow us to actually visualize certain equations.
With that, we’re ready to move on to the “heart” of the matter…
In the first post I explained why the mathematical “imaginary” number i is “real” (in more than one sense of the word). That weird number is just a stepping stone to the complex numbers, which are themselves stepping stones to the complex plane.
Which, in turn, is a big stepping stone to a fun fact about the Mandelbrot I want to write about. (But we all have to get there, first.) I think it’s a worthwhile journey — understanding the complex plane opens the door to more than just the Mandelbrot. (For instance, Euler’s beautiful “sonnet” also lives on the complex plane.)
As it turns out, the complex numbers cause this plane to “fly” a little bit differently than the regular X-Y plane does.
Graph of ax2 for diff a values.
(green < 1; blue = 1; red > 1)
This is a little detour before the main event. The first post of this series, which explained why the imaginary unit, i, is important to math, was long enough; I didn’t want to make it longer. However there is a simple visual way of illustrating exactly why it seems, at least initially, that the original premise isn’t right.
There is also a visual way to illustrate the solution, but it requires four dimensions to display. Three dimensions can get us there if we use some creative color shading, but we’re still stuck displaying it on a two-dimensional screen, so it’ll take a little imagination on our part.
And while the solution might not be super obvious, the problem sure is.
Yes, this is a math post, but don’t run off too quickly. I’ll keep it as simple as possible (but no simpler), and I’ll do all the actual math so you can just ride along and watch. What I’m about here is laying the groundwork to explain a fun fact about the Mandelbrot.
This post is kind of an origin story. It seeks to explain why something rather mind-bending — the so-called “imaginary numbers” — are actually vital members of the mathematical family despite being based on what seems an impossibility.
The truth is, math would be a bit stuck without them.
Here’s yet another unplanned post, mostly because there was something important I forgot to mention yesterday, but also because I started watching three different Netflix shows (or maybe call it two-and-a-half), and all three are fit for a Sci-Fi Saturday post, so here I am again.
I dither about three because one of them was wasn’t new, it was season two I started of Siempre Bruja. But I hadn’t yet seen any of Lost in Space or the new Chilling Adventures of Sabrina. I’ve been suspicious of the former, and the latter isn’t quite my cup of tea on several counts.
But first you should know about (Your) CloudLibrary!
Wow, for the third time this month (third time in a week) I’ve realized the day calls for a post I hadn’t planned. The first time was when the MLB delayed the baseball season. The second time — the very next day — was Pi Day and Albert Einstein’s birthday.
This time it’s the equinox (and a friend’s birthday; shout out!). For those of us in the northern hemisphere it’s the spring (vernal) equinox, and that’s my favorite of the four annual solar node points (two equinoxes; two solstices). It means we have a whole half a year of light ahead.
So I just had to post something.
When possible, I try to find a theme for the Wednesday Wow posts. Last time, for instance, the theme was aviation and fireworks (two things you wouldn’t normally think went together, but in one case they delightfully did).
The problem is that I’m jaded and have seen a lot, so I can be hard to impress. Not lots of things raise to my highest rating, Wow! Fortunately, I’m not so far gone I can’t still see a world filled with wonder, some of which drops my jaw.
The theme, such as it is, concerns measurements, especially tiny and precise ones. Like, for instance, Planck Length tiny.
I hadn’t really planned to, but it’s both Pi Day and Albert Einstein. As a fan of both the number and the man, it seems like I should post something.
But I’ve written a lot about pi and Einstein, so — especially not having planned anything — I don’t have anything to say about either right now. In any event, I’m more inclined to celebrate Tau Day when we double the pi(e). I do have something that’s maybe kind vaguely of pi-ish. It’s something I was going to mention when I wrote about Well World.
It’s just a little thing about hexagons.
These days, during the regular baseball season, pretty much every game is broadcast on TV by someone, so it doesn’t take a lot of effort to see every game your team plays (it generally just takes money). But during Spring Training it’s rare that a game your team plays is televised.
So I was looking forward to watching a Twins game yesterday — one of those rare Spring Training games that was being televised. After several months of winter (and no baseball), I’m ready to start seeing games again.
But I turned on the TV to discover the MLB cancelled Spring Training!