August 12, 2022

Friday Notes (Aug 12, 2022)

It’s been a few minutes since my last post. Lately, the effort of writing hasn’t seemed worth the almost non-existent return. I find I’ve lost faith in humanity, and the phrase that seems most resonant is: “Really, when you come right down to it, what’s the point of it all?” I think, at least in our case, the Fermi Paradox seems resolved.

Perhaps more crucially, this damned dark cloud over me seems all I can write about. Everything else seems ephemeral. If we can’t solve our most basic human problems (education, race, gender, poverty, pollution) then the rest of it really is fiddling while Rome burns.

It makes me angry. Humanity can do better than this. I think.

I’ve been angry about this since I was in high school (where one friend gave me the moniker The Angry Young Man). In the 50 years since I’ve seen those concerns and fears justified. My argument now has a literal trump card. (Never was there a debate I more wanted to lose. The unfunny thing is that all my life people have told me how smart I am, yet no one ever believes a word I say.)

One reason I haven’t posted lately is that I’m as tired of hearing me rant as anyone. It’s a lot easier to just ignore it. I’ve stopped following the news at all. Blame Apple, at least in part, for that. Speaking of which, the general awfulness of technology companies is yet another dark cloud I can’t seem to escape. It’s tempting to completely unplug it all.

That I no longer feel welcome, let alone embraced, by WordPress is another weight on the “Why Post?” side of the scale. The title “Happiness Engineer” has become insultingly ironic. That said, the era of long-form blogging seems passed (therefore past), so I’m sure they don’t see much reason to provide serious support.

It seems every tech company’s public-facing support invariably refuses to take ownership of an issue and invariably fobs me off to someone else. Two recent examples:

Reported an obvious server issue to Overdrive, the company behind the Libby library app I use (been a great app… until recently). The response was that I should be sure the app is up-to-date (duh) and to reboot my computer. Their servers began working again shortly thereafter with no action on my part.
Reported to Amazon an issue about being unable to remove from my Library list, library books transmitted from Libby to Kindle app. Once the loan expires, they can no longer be read (or even opened), but there is no option to remove them from the list. The response was a huge email with zillions of links and a long-winded explanation about why they couldn’t make any effort to help me until I sent them a ton of information about the problem, the operating system, my devices, etc.

That’s the other sneaky trick support organizations use to not do any work. They bounce the ball back into your court and hope you’ll be as discouraged as I was and just drop it.

Is it any wonder I’m angry?

I’ve written plenty about our love of shit-covered raisins. Because we love the raisins so much. But if we were more willing to say “No!” to the shit, maybe we might eventually end up with just the raisins. Or at least a thinner coating of shit.

I’ve focused a lot about how we do it with movies and TV shows, but we excuse crap in too many aspects of life. We accept buggy technology and half-assed support for it because we’re so in love with the technological raisins. (And are we sure they aren’t actually mouse droppings?)

I’ve related to Howard Beale ever since I met him way back in 1976. Somehow his call to urgency and outrage devolved into our narcissistic preoccupied love of being offended and tweeting about it.

Somewhere recently I bumped into a meme featuring a gal saying to a guy something along the lines of, “you think it’s cool to hate things, but it isn’t, it’s boring.”

I think it highlights an important Yin-Yang difference in how people see the world. As in many (most? all?) such cases, I’m not sure the two groups are capable of fully understanding each other, of appreciating what the other side sees. As best as I (as someone sensitive to the smell of shit) can tell, some folks not only don’t mind the shit, they don’t see it as shit in the first place.

Which, I admit, I cannot fathom. To me, a lot of shit is objectively shit, not a matter of taste. I don’t understand the willingness to hand-wave away the especially shitty aspects of something in favor of exclaiming about the raisin. Nothing can be perfect, life is flawed, but I wish we had better values and standards.

An episode of House, M.D. really stayed with me. It featured a guy with a rare disease that removed his social filters. He speaks his every passing thought. With disastrous results. It nearly ends his marriage, it seriously alienates both wife and daughter, and pisses off friends, co-workers, and the doctors.

It’s a fascinating treatise on the necessity of social lies. Or a question about that necessity. Imagine a society based on total honesty (I’ve seen such imagined in science fiction stories). My big takeaway from the story was the expressed idea that some people wouldn’t say those things even with no social filters. Some people are as nice inside as outside and don’t need them.

Which I cannot fathom even more. Surely everyone has dark thoughts, at least sometimes. Surely everyone thinks things in passing they would never share because they don’t actually think that — not with their full mind.

As a trivial example, wouldn’t you like to just slap someone that you’re finding particularly annoying? Alternately, wouldn’t you like to fuck someone you find particularly sexy? But aren’t those just passing thoughts some small part of your mind introduces? Aren’t these things your full mind repudiates?

Or do some people really never feel irritation or inappropriate lust? Are some people really that nice?

§ §

Well, that wandered away from anything I meant to talk about when I sat down, but it’s a mode I may try to go with more often. Just start rambling and see what comes out. The original mode of web logging.

I do have some Friday Notes, though…

§ §

The Unit Circle

In a way, pi, the ratio between a circle’s radius and circumference, is a bridge from the countable to the uncountable. The usual form, a unit circle centered on the origin, intersects the real number line at -1 and +1, the absolute values of which are, by construction, the circle’s radius.

If we multiply that -1 to +1 interval by pi, we get the circumference of the circle.

It’s a trivial observation, really, but I wondered if it might account for why pi shows up in so many seemingly unrelated places in physics math (and even math in general).

Pi connects the linear to the nonlinear, lines to circles.

The thought occurred while thinking about Euler’s famous bit of mathematical beauty:

$\displaystyle{e}^{{i}\pi}+1=0$

It’s been called a “sonnet” and it really is pretty awesome. [See Beautiful Math for details.] Part of what makes it so cool is the contrast between the transcendental numbers e and pi versus the integers one and zero (not to mention the imaginary unit, i). Seeing the way pi connects these just adds another jewel to the crown.

Math Pop Quiz: You know that:

$\displaystyle\sqrt{1}={1}\;\;\textsf{and}\;\;\sqrt{-1}={i}$

But can you figure out:

$\displaystyle\sqrt{i}=\textsf{?}$

If you know, or can figure out, the answer, you have a decent grasp of complex numbers. (In particular, the complex number plane. Hint, hint.)

Speaking of countable infinity, once I met a math teacher who didn’t realize the rational numbers (which have the form P/Q) are countable. It’s understandable. Given any two rational numbers, no matter how close, it’s always possible to construct a rational number between them:

$\displaystyle\textsf{given:}\;\;\frac{p_a}{q_a}<\frac{p_b}{q_b},\;\;\frac{p_c}{q_c}=\frac{2{p_a}{q_b}+({p_a}{q_b}-{q_a}{p_b})}{2{q_a}{q_b}}$

Which could certainly give one the idea they have the same uncountable infinity as the real numbers.

The key here is a successor function. Given some number, N, is there a function that gives the next number in order? With integers, the successor function is obviously +1; we just add one to N to get the next number in order.

But with real numbers, there is no successor function. For example, given the real number pi, there is no “next” number in order. The concept is invalid in the real numbers. Which is why they’re uncountable. If you can’t list them, you can’t count them.

Not sure why I got to pondering this recently. Something passing triggered it. I realized I’d never tried to write down the successor function for the rational numbers. Given an arbitrary rational number P/Q, what’s the “next” number in some order?

It does depend on an ordering. If we can always find new numbers between existing numbers, the normal sort order won’t do. But we can order the rational numbers in a grid by numerator and denominator:

Enumerating the Rational Numbers. Numerators are columns, denominators are rows. Numbers in yellow are equal to one. Those above are larger; those below are smaller.
[click for full-size version]

This obviously extends infinitely to the left and infinitely down as numerators and denominators get bigger and bigger. (That there are two infinite series can also suggest an uncountable infinity despite that both these series are countable. That both are countable makes their union countable.)

We read (enumerate) the P/Q pairs as a linear list along the diagonals that slant to the upper right. The first handful, then, are: 1/1, 1/2, 2/1, 1/3, 2/2, 3/2, 1/4, 2/3…

The series is infinite the same way 1, 2, 3,… is infinite. The successor function is simply:

$\displaystyle\frac{P_2}{Q_2}=\frac{P_1+1}{Q_1-1}\;\;\mathsf{if}\;({1}<{Q_1})\;\mathsf{else}\;\;\frac{1}{P_1+Q_1}$

Which gives either the rational number to the upper-right on the chart or the one that starts the next diagonal if we’re at the top of the chart. It was interesting to see what the successor function (or at least a successor function) for the rationals looks like. (But I’m easily amused mathematically speaking.)

I had the idea of creating a maze based on a tree of Collatz sequences (see Math Games #1). As it turns out, that’s a much harder proposition than I expected.

Due to the nature of a tree of Collatz sequences, there doesn’t seem to be an easy algorithm that packs the tree into the maze. After a lot of messing around, here’s the closest I came to success:

Maze based on Collatz sequences. [click for full-size]

The red cell indicates a place where the algorithm needed to branch to a new path but found it physically impossible. The structure of the data requires the ability to re-arrange the maze around a new path, and that’s not a capability I included in the foundation code. (Given how the maze is represented as a data structure, it would be a complex proposition, and I never saw the need. Until now.)

I’ve shelved the project for now, but that result is intriguing enough that I’d like to come back to it someday. It did result in a long-scheduled cleaning, revising, and thorough documenting, of the original maze generation code, so it was quite productive.

It also allowed me to try an idea I’ve had in the back of my mind: An animation of the maze generation code creating a maze as well as the maze solver code finding the correct path through that maze. The cleanup and revision made that easy to accomplish:

But the result isn’t as interesting as I’d hoped. Mostly, it’s just long. I added some information balloons, but I don’t know how much they helped. (And I really need to add music or sound to these.)

§ §

After considerable thought I’ve decided to provide the world with a definitive definition of two important terms:

Life: Something that wants. (Mostly to not die.)

Consciousness: Something with an opinion. (Probably a wrong one.)

Now we can all stop debating what these things are. You’re welcome.

§ §

I’ll post more about this soon, but Minnesota artist Todd Ronning finished my lake carving:

[click for 1920×1080 version]

And I love it! It’s Thompson Lake in Canada where I camped and fished nearly every summer for over 20 years. [See Canadian Camping 1996]

Stay a-mazed, my friends! Go forth and spread beauty and light.

∇

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

This entry was posted on Friday, August 12th, 2022 at 7:11 am and tagged with maze generator, mazes, pi, rational numbers and posted in Friday Notes. You can follow any responses to this entry through the RSS 2.0 feed.

16 responses to “Friday Notes (Aug 12, 2022)”

Wyrd Smythe
August 12th, 2022 at 9:12 am

One thing that might lead one down the wrong path (or not) regarding √+1 versus √-1 versus √i is that:

$\displaystyle{+1}^{2}\!=\!{+1},\;\;\;{-1}^{2}\!=\!{+1},\;\;\;{i}^{2}\!=\!{-1}$

It may (or may not) be helpful to consider cases that don’t involve unity (1.0):

$\displaystyle{(+2)}^{2}\!=\!{+4},\;\;\;{(-2)}^{2}\!=\!{+4},\;\;\;{(+2i)}^{2}\!=\!{(-2i)}^{2}\!=\!{-4}$

One way through this is to remember that complex numbers have multiple forms:

$\displaystyle{z}=(a+bi)=\eta(\cos\theta+{i}\sin\theta)=\eta{e}^{i\theta}$

And we can use the last one productively:

$\displaystyle{2i}^{2}=\left({2}{e}^{i\frac{\pi}{2}}\right)^{2}={2}^{2}\!\left({e}^{i\pi\frac{1}{2}}\right)^{2}={4}{e}^{i\pi}={4}(-1)=-4$

And also:

$\displaystyle(-2i)^{2}=\left({-2}{e}^{i\frac{\pi}{2}}\right)^{2}=(-2)^{2}\!\left({e}^{i\pi\frac{1}{2}}\right)^{2}={4}{e}^{i\pi}=-4$

Which might offer some insight to the nature of √i.

Or not.

Reply
- Wyrd Smythe
  August 13th, 2022 at 1:22 am
  
  These equalities might help:
  
  $\displaystyle\eta{i}=\eta\sqrt{-1}=\eta\sqrt{{e}^{i\pi}}=\eta{e}^{i\pi\frac{1}{2}}=\eta{e}^{i\frac{\pi}{2}}$
  
  Because we know from Euler’s Identity that:
  
  $\displaystyle{e}^{{i}\pi}=-1$
  
  Reply
- Wyrd Smythe
  August 14th, 2022 at 12:58 pm
  
  Ready for the answer? First some background. We know from a basic rule of exponents that:
  
  $\displaystyle{e}^{i0}={e}^{0}=1$
  
  Because any value to the zeroth power is one. From Euler’s Identity we also know that:
  
  $\displaystyle{e}^{i\pi}=-1$
  
  And Euler’s Function gives us the usual form, (a+bi), for any complex number given its exponential form:
  
  $\displaystyle{e}^{i\alpha}=\cos(\alpha)+{i}\sin(\alpha)$
  
  The presence of i in the exponent puts us on the complex number plane, which gives us two complex values for +1 and -1:
  
  $\displaystyle{e}^{i0}=\big(+\!\!1+0i\big),\;\;\;{e}^{i\pi}=\big(-\!\!1+0i\big)$
  
  Now, recall that multiplication is rotation on the complex plane. And that any number can be viewed as the multiplicative identity, (+1+0i), rotated and expanded (i.e. multiplied) to some value. Multiplying the identity by minus one, (-1+0i), is a 180° rotation. That’s why multiplying -1×-1 gives +1 — the -1 is rotated 180° back to +1.
  
  Which makes it clear how i works. If -1 is a 180° rotation, and i×i=-1, then multiplying by i is a 90° rotation. This is why the imaginary axis is orthogonal to the real axis.
  
  To find √i we can first note that:
  
  $\displaystyle\sqrt{i}=\sqrt{\sqrt{-1}}=\sqrt[4]{-1}={-1}^{\frac{1}{4}}$
  
  We can find the exponential form for i by using Euler’s Identity:
  
  $\displaystyle{i}=\sqrt{-1}=\sqrt{{e}^{i\pi}}=\left({e}^{i\pi}\right)^{\frac{1}{2}}={e}^{i\frac{\pi}{2}}$
  
  And from there to find the square root of i:
  
  $\displaystyle\sqrt{i}=\sqrt{\sqrt{{e}^{i\pi}}}=\left(\left({e}^{i\pi}\right)^{\frac{1}{2}}\right)^{\frac{1}{2}}=\left({e}^{i\pi}\right)^{\frac{1}{4}}={e}^{i\frac{\pi}{4}}$
  
  And, of course:
  
  $\displaystyle\sqrt{i}=\left(-1\right)^{\frac{1}{4}}=\left({e}^{i\pi}\right)^{\frac{1}{4}}$
  
  Euler’s Function gives us the usual form for i:
  
  $\displaystyle{e}^{i\frac{\pi}{4}}=\cos(\frac{\pi}{4})+{i}\sin(\frac{\pi}{4})=\left(\frac{1}{\sqrt{2}}+{i}\frac{1}{\sqrt{2}}\right)$
  
  Which, of course, is the vector angled at 45°, halfway between +1 and i. Squaring that vector rotates it another 45° for the 90° vector for i, (0+1i).
  
  Reply
Mark Edward Jabbour
August 12th, 2022 at 9:52 am

I was wondering about you. I’ve been having almost the same experience. I’ve finished a piece called “My People” (7,300 words), but haven’t posted it. Written at the suggestion of my psych-girl. So hang in there brother. FYI – I like your insights into the human condition. (I’m a psych-guy.) It’s nice to know I’m not totally alone. 🙂 cheers.

Reply
- Wyrd Smythe
  August 12th, 2022 at 4:19 pm
  
  I think a lot of older adults are experiencing a form of PTSD these days. The constant bombardment of sensibility for decades culminating in Mr. Toad’s Wild Political Ride in 2016-2020 plus a virulent disease that killed more Americans than WWII (and in far shorter time). Future shock meets shell shock, and we’re all wondering WTF happened to anything we once considered normal.
  
  I’ve found it does help writing about it even if one doesn’t make it public. I let it all flow and then go back and delete lots of paragraphs. Or entire posts sometimes. That one part of my mind versus my whole mind thing. Pain leads to anger and that leads to hate if one doesn’t watch oneself. (Fear has a more direct path to hate and is harder to correct.)
  
  Anyway, no, not alone. It’s a big lifeboat. We’re bailing as fast as we can!
  
  Reply
  - Mark Edward Jabbour
    August 14th, 2022 at 3:00 pm
    
    “Pain leads to anger and that leads to hate if one doesn’t watch oneself. (Fear has a more direct path to hate and is harder to correct.)
    
    Yep. And a lot of people are in a lot of pain (even if they’re not aware of it – You’re 85%)
    
    Anyways, I read your posts and enjoy them, even if I don’t always agree. And the technical stuff ? Not my domain. (And psych-girl says, “Old dog stuff?” with a wry smile.)
  - Wyrd Smythe
    August 14th, 2022 at 7:35 pm
    
    Speaking as an old dog trying to learn new tricks, it’s #$%@* hard! I just hope it keeps my mind from turning into mush.
SelfAwarePatterns
August 12th, 2022 at 1:54 pm

Good to see a post from you Wyrd. I was starting to wonder if you’d sworn off blogging for good. Although sorry you’re feeling so down and frustrated. Wish I had some pearls of wisdom, but truth be told I’m struggling myself right now. All I can do is second Mark: hang in there! Often it’s all we can do.

Those definitions of life and consciousness are as good as anyone else’s I’ve seen. The devil’s in the details, but of course that’s always true.

Reply
- Wyrd Smythe
  August 12th, 2022 at 4:27 pm
  
  I’ve been wondering the same thing. At the very least I needed some time off. And I’ve realized retirement life has a sameness that’s starting to feel old. I think I need a vacation from retirement.
  
  As I said to Mark, a lot of us older adults seem to be struggling. Whatever personal issues, plus aging, are bad enough, but factor in the last decade or so, and the noise factor alone is wearying (let alone whatever assault on one’s values one perceives).
  
  It’s always amused me how both the devil and heaven are found in the details. 😈👼🏼
  
  Reply
  - SelfAwarePatterns
    August 12th, 2022 at 7:27 pm
    
    I can see taking time off from blogging. I’ve actually been struggling myself lately with motivation to post. In my case, some of it’s related to work pressure again. I wasn’t supposed to be having to deal with this stuff by now.
    
    On vacationing from retirement, I don’t know if it’s what you’re looking for, but now is a great time for a retired programmer to find work that’s part time and remote. Although you might be looking for something with more people interaction, or just something different from what you did before.
  - Wyrd Smythe
    August 12th, 2022 at 7:53 pm
    
    I can well recall the effect work issues had on me. Sometimes it made me post more — venting, really — and sometimes it shut me down. One thing about work (and about life in general): It never fucking ends. By “it” I mean the bullshit.
    
    And I can see it getting harder after years of blogging on a focused set of topics. I do have the “advantage” of a diverse set of topics. A disadvantage in terms of readership, but it does give me more things to write about. My little coding and research projects provide a lot of content, too. I suspect a lot of my posts from here on will be either in the venting or social observation mode or related to those projects. I’m increasingly leaning away from media reviews. On some level I feel silly posting yet another review of whatever TV show or movie. (I do like writing about books, though.)
    
    Ha! The headhunters nagged me when I retired, and I’d guess my background would still have some value to them, but the last thing I want is to go back to work. I spent a lifetime fulfilling other people’s expectations, and now that seems anathema to me. By vacation I mean something like a road trip to another city to watch a ballgame in a baseball park I’ve only ever seen on TV. And I’ve always wanted to see a rocket launch. Seems like everyone is doing it these days, so it shouldn’t be hard to find one.
  - SelfAwarePatterns
    August 13th, 2022 at 10:45 am
    
    I’ve always steered clear of any work topics on the blog except for the occasional oblique reference. Some of it is because I’ve seen people’s careers affected by social media posts. But mostly because I’m usually not interested in having conversations about it.
    
    That said, I’ve never felt hemmed in by topics. There are bloggers who stick to single subjects, which has never appealed to me. I remember one who only blogged about consciousness. When he did an out of band historical post, some of us urged him to branch out more since it was interesting, but he never did. He writes fiction now. I’d read his posts about it if he did any.
    
    In my case, I generally blog about anything I’d like to have a conversation on. But it’s an excellent point that motivational issues may be due to simply not posting on what we’re currently interested in. Something I need to remember.
    
    Seeing a launch sounds like an excellent thing to do in retirement. With all the delays, it’s tough to fit into a scheduled vacation.
Anonymole
August 12th, 2022 at 2:56 pm

Go forth and spread beauty & light, unless you’re a curmudgeonly skeptic where you can spread misery and darkness.

Does that maze algo /always/ result in a solvable maze?

Reply
- Wyrd Smythe
  August 12th, 2022 at 4:35 pm
  
  Yes. The generator creates a tree structure containing all cells by cutting random paths branching from random existing paths until no cells are left. The exit cell is guaranteed to be in some branch of the tree, the root of which is the entrance.
  
  Speaking as a dedicated irascible curmudgeon, what one feels is one thing, but what one spreads is another. Every action is a pebble in a pond with ripples that spread outwards. It’s hard but I try to spread good ripples (with ridges).
  
  Reply
- Wyrd Smythe
  August 12th, 2022 at 7:32 pm
  
  And that tree structure, BTW, is why the left-hand rule always works for solving the maze. It’s just a matter of traveling the tree until you find the desired node. (You can defeat the left-hand rule by building a maze with paths that loop within the maze. Then the rule just takes you in circles.)
  
  Reply
BB #85: More Fraction Fun | Logos con carne
April 22nd, 2023 at 10:10 am

[…] So, the rational numbers — good old p/q fractions — though still infinite are countably infinite (see this post for details). […]

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