BB #90: The Growth Paradigm

My final post in 2023 was about growth curves. It focused on the difference between geometric growth versus exponential growth — which turns out to be not much — and compared them to polynomial growth (see that post for the math-y details; this post isn’t a math post, so relax and read on).

A key characteristic of all these growth curves is that they grow without limit. If we treat the horizontal axis as time, then the longer the growth continues along the curve, the greater whatever growing grows.

The problem is that nothing in the real world can grow infinitely without limit. At some point, something has to give.

Continue reading


BB #89: The Irrational Square

Consider the lowly square, a four-sided shape with sides of equal length meeting at right angles. The embodiment of what we’re referring to when we refer to square miles, square kilometers, square inches, or square whatevers. The two-dimensional version of any one-dimensional length.

A trivially easy shape to draw, all you need is a straight edge and a compass — the latter for ensuring your corners are right angles (see Plato’s Divided Line for more on using a straight edge and compass). The only simpler shape is the circle.

Yet the simple square threw early mathematicians into a serious tizzy!

Continue reading


Winter Finally Showed Up!

February 15, and winter finally decided show up:

And it’s awfully pretty, I gotta give it that!

Continue reading


BB #88: Boltsmann Brains

An article in a recent issue of New Scientist caught my attention on two counts: firstly, in what it said about my old friend wavefunction collapse and the measurement problem; and secondly, in mentioning Boltzmann Brains. Both set off my “Yeah, but!” reaction.

I’ll touch (as briefly as possible) on the first point, but this little Bubble is mainly about the second one.

Boltzmann Brains bug me.

Continue reading


BB #87: Two = Zero!

You may have, at some point, seen one of those bits where a series of seemingly simple math operations somehow end up proving that 1=0 or something equally clearly wrong. Most of them accomplish their joke by sneaking in a hidden division by zero. From that point on, all bets are off (see Divide by Zero).

Recently, on a YouTube channel I follow, I saw a clever example that uses a much sneakier trick. It’s harder to spot because the operation it uses is legit in two of the three possible cases.

The gag, of course, uses the third one.

Continue reading


Sideband #80: Divide by Zero

You may remember learning way back in grade school that you can’t divide by zero. You may remember being told that division by zero is undefined. But have you ever wondered why we can’t divide by zero? Couldn’t the answer just be zero? We get zero when we multiply by zero, so why not when we divide?

But dividing is the opposite (or inverse) of multiplying, so if multiplying by zero gives zero, then maybe dividing by zero gives us… infinity? But infinity isn’t a number (it’s an idea), so that doesn’t work, either.

In this post I’ll dig into why division by zero is undefined.

Continue reading


Friday Notes (Jan 19, 2024)

The first Friday Notes of 2024. (The title nicely caps the 2024, 2023, 2024, 2023 series from the titles of the previous four posts.) We’re well past the glut of holidays, and it’s back to business. There is also the fresh-start sense of the new year. I’m already thinking about all the stuff I want to throw away this coming Spring Cleaning.

My various piles of notes grow smaller and smaller! In some cases, because I deleted notes that seemed to have aged beyond their lifetime. But these Notes posts have been instrumental. In some areas, I’m actually scraping the bottom of the barrel.

But I still have notes, so here we go again…

Continue reading


2023: All the Charts

This post contains the post statistics charts I didn’t have room for in the 2023 wrap-up post. There are a lot of them, and I don’t imagine they’re of much interest to anyone but me. I post them mainly to document the blog. A set of milestones, if you will.

I confess to being fascinated — and utterly befuddled — by which posts receive attention and which little piggies get none. It’s particularly interesting to me when an old post — especially one that never got many views — suddenly gets noticed and sometimes even slightly popular.

Mostly the stats are similar to previous years, but 2023 did bring a few surprises.

Continue reading


2024: Looking Ahead

The other face of Janus looks forward to the Christmases yet to come. Or words to that effect. The last post looked back at some highlights from 2023; this one looks forward to the hopes, plans, and guesses for 2024.

Hopes and plans are easy enough, but when it comes to guesses, I’ve never thought much of futurists. I don’t have much truck with “could” and “might” when it comes to the actual world. Science fiction is my favorite fiction, but futurists claim to be prescient, and I very much doubt it.

Regardless (and heedless), some hopes, plans, and wild-ass guesses, for 2024.

Continue reading


2023: Looking Back

I thought this year’s Janus posts — one looking back, one looking forward — would be unusually late, but it turns out they’re only one day later than last year’s pair (assuming I post what I’ve just started writing today). Hosting my furry nephew Bentley over New Year’s accounts for the (thoroughly delightful!) delay this year and last.

As usual, this post looks back at the previous year, 2023, and tomorrow’s post (assuming I write it and post it) looks forward to what 2024 might hold in store (guesses and hopes).

And, of course, there are stats and charts.

Continue reading