#
Category Archives: Sideband

This is a Sideband to the previous post, **The 4th Dimension**. It’s for those who want to know more about the rotation discussed in that post, specifically with regard to axes *involved with* rotation versus axes *about which* rotation occurs.

The latter, rotation about (or around) an axis, is what we usually mean when we refer to a *rotation axis*. A key characteristic of such an axis is that coordinate values on that axis *don’t change* during rotation. Rotating about (or on or around) the **Y** axis means that the **Y** coordinate values never change.

In contrast, an axis *involved with* rotation changes its associated coordinate values according to the angle of rotation. The difference is starkly apparent when we look at rotation matrices.

Continue reading

3 Comments | tags: 2D, 3D, 4D, column vector, matrix math, matrix transform, rotation, rotation matrix, unit vector, vectors | posted in Math, Sideband

In the last installment I introduced the idea of a *transformation matrix* — a square matrix that we view as a set of (vertically written) vectors describing a new *basis* for a transformed space. Points in the original space have the same relationship to the original basis as points in the transformed space have to the transformed basis.

When we left off, I had just introduced the idea of a *rotation matrix*. Two immediate questions were: How do we create a rotation matrix, and how do we use it. (By extension, how do we create and use *any* matrix?)

This is where our story resumes…

Continue reading

1 Comment | tags: 3Blue1Brown, column vector, linear algebra, matrix math, matrix multiplication, rotation, unit vector, vectors | posted in Math, Sideband

For me, the star attraction of March Mathness is **matrix rotation**. It’s a new toy (um, *tool*) for me that’s exciting on two levels: Firstly, it answers key questions I’ve had about rotation, especially with regard to **4D** (let alone **3D** or easy peasy **2D**). Secondly, I’ve never had a handle on matrix math, and thanks to an extraordinary YouTube channel, now I see it in a whole new light.

Literally (and I do mean “literally” literally), I will never look at a matrix the same way again. Knowing how to look at them changes everything. That they turned out to be exactly what I needed to understand rotation makes the whole thing kinda wondrous.

I’m going to try to provide an overview of what I learned and then point to a great set of YouTube videos if you want to learn, too. Continue reading

16 Comments | tags: 3Blue1Brown, column vector, complex numbers, linear algebra, matrix math, matrix multiplication, rotation, trigonometry, unit vector, vectors | posted in Math, Sideband

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.

Continue reading

16 Comments | tags: complex numbers, math, math theory, mathematics, natural numbers, number systems, numbers, octonion, quaternion, real numbers | posted in Math, Sideband

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…

Continue reading

Comments Off on Sideband #61: Tock | tags: binary logic, boolean logic, electro-mechanical, flip-flop, full-adder, half-adder, inverter, logic, logic gate, PBX, relay logic, telephones | posted in Sideband

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.)

Continue reading

Comments Off on Sideband #60: Tick | tags: 60-61, Anniversary | posted in Life, Sideband

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!

Continue reading

15 Comments | tags: CMOS, computer chip, computer heat, computers, CPU, CPU heat, digital, electronics, FET, hardware, heat, MOSFET, transitor | posted in Computers, Sideband

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.

Continue reading

2 Comments | tags: Alan Turing, algorithm, calculation, Cantor's Diagonal, computers, discrete mathematics, Halting Problem, Turing, Turing Halting Problem | posted in Computers, Sideband

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:

Continue reading

7 Comments | tags: alien math, Frederik Pohl, Heechee, math, math origins, math theory, more math, prime numbers, real numbers, surreal number | posted in Math, Opinion, Sideband

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.

Continue reading

1 Comment | tags: base 10, base 2, base 8, Frederik Pohl, Heechee, irrational numbers, Leopold Kronecker, natural numbers, number bases, number names, numbers, pi, prime numbers, rational numbers | posted in Math, Sideband