Hey, John — Matt Parker mentioned in one of his ellipse videos the fact that every elliptical ratio has its own pi-like constant. He just quickly rattles the fact off, but never delves into it. Do you know of any research into trying to characterize the family of pi? I mean, beyond its evil cousins.
For a circle, pi is the ratio of the circumference to its diameter. Every ellipse also has a circumference-to-diameter ratio. Well, two ratios, since ellipses have both major and minor diameters. You might think that there would be some kind of clever formula that let you calculate this ratio, but there isn’t! Instead, these pi-like numbers for ellipses are expressed as integrals:
Scroll down to “Complete elliptic integral of the second kind”. That is your search term for looking it up. It is kind of a surprise that there isn’t some neat formula for calculating the circumference of an ellipse. The formula given is:
C = 4 a E(e)
The function E(e) here can be calculated in a few different ways, but it is really just defined as an integral that measures the length of a single ellipse arc.
Here, e is eccentricity. E(0) therefore gives π/4 since a circle has eccentricity 0. E(1) also therefore gives 1. So the E(e) function goes from π/4 to 1 as e goes from 0 to 1.
Yes, definitely. Pi is just the perimeter of the circle, and varpi is the perimeter of the lemniscate. If you use three points, you get three tear-drops, and you can compute the perimeter of that.
Let’s call it a trilemniscate. ;)
Here’s a 3d plot of it. If you rotate to view it from +Z downward, then you’ll see the trilemniscate, which is where the volume intersects with the XY plane. Note I subtracted 1 from the product in order to visualize the plane intersection. (And you can turn off the 3 points version and turn on the 2 points version to compare.)
One interesting note about 2 points vs 3 points. The area inside the lemniscate and trilemniscate is the same! (True for more points, as long as they’re evenly space on a circle). The perimeter, of course, goes to infinity as you add more points.
I mean the concept of distance from 3 points introduces a mess of metrics or even measure theory.
2 points always have a shortest path between each other, so the constant is about this fact. For 3 points you have the whole universe of possible triangle shapes to contend with.
Shortest path between two points still depends on your metric.
For instance, if you're constrained to travel along the surface of Earth, your shortest path is going to travel along a great circle, rather than pass through the interior of the sphere.
That said, you could, for instance, pick the three vertices of an equilateral triangle (using the Euclidean distance as your metric of choice, as we do in order to derive the lemniscate and the circle), and again deal with the product of the distances from each vertex.
You again start with small circles around each vertex, which eventually expand to a single looping curve, and then into ovals encircling the entire triangle.
> I'm not enough of a cultural relativist to believe there's a civilization that cares more about the shape ∞ than the shape ◯.
Maybe these are "logarithmic" beings, as opposed to us "linear" beings? The lemniscate is based on geometric mean, which is basically multiplicative mean and/or mean in log-space -- as opposed to the additive mean in linear space.
If we assume we are linear beings good at intuitive addition but somewhat bad at intuitive multiplication, there could exist beings which live in log-space and whose minds are based on multiplication. Their circle would be the lemniscate.
Humans are actually intuitively log scale thinkers. That is, humans without the kind of early arithmetic training that Westerners get will think more in terms of ratios than differences. There are theories it is more evolutionarily adaptive.
Isn't it also related to our physical perception? Both hearing and vision at least have somewhat logarithmic properties (e.g. response to point-source brightness, and hearing frequency response)
aside: As the Professor points out, the ratio of pi to its evil twin is ~1.198, the arithmetic-geometric mean of sqrt(2) and 1. The geometric part involves a square root, and square roots are expensive. So I was like, well, if the AM converges to GM, then due to AM-GM-HM inequality, it must converge to the harmonic mean as well. And the HM does not need an expensive square root!
Its quite wild that the AM GM convergence is almost immediate - in just 2 steps, whereas to get a decent convergence for the Gauss's constant via HM, you need like 15 steps.You can dispense with expensive operators like square root but you end up paying for it with numerous iterations.
The c value you compute depends on computing the b value, though. It's not a recursion carried out in a way which avoids square roots. It's just carrying out the same AM-GM sequence computation, and then taking a certain weighted harmonic mean over that sequence, which converges just because that original sequence converges anyway.
Hmm. Why only 2? Why not 3 points? Can you find an interesting curve produced by a constant product of distances from N points? Maybe even in higher dimensions, for 1 point, you have a sphere. What is the shape for 2 points? Is it more like an hourglass-like double droplet?
On the 3 points bit: One and two points are special. In each of these cases, there is, up to translations and uniform scaling, only one configuration. When you have three points, though, there are as many configurations as there are similar triangles. You could probably get a number for each similarity class of triangle, but you shouldn't expect to get a constant across all classes.
> Back before Twitter became a Nazi bar, I issued a challenge there: find a whole series of numbers like pi, each with its own bunch of formulas. @duetosymmetry took me up on this and invented the numbers ϖₙ: (...)
Having that shape become more important to a civilisation than the circle because it has something to do with the geometry of hyperspace seems like it could be an interesting conceit for a sci-fi setting.
This somehow reminds me of Egyptian mathematics where they refused to admit to the existence of any fraction with a numerator other than 1 (except for 2/3).
Learning how to expand e.g. 3/7 into 1/n + 1/m + ... using their methods was a fascinating experience.
I wouldn't want to suffer under such constraints day to day but it was one of the most memorable parts of the History of Mathematics course I took alongside what was other a mostly pure maths degree.
Egan would probably be my first thought of somebody who could take a concept like that and make something well worth reading out of it.
Second thought would probably be Derek Künsken. (no claim he's necessarily the second best option but he's definitely the second author I've read recently enough to have the name of in brain cache to come to mind as "could almost certainly pull it off")
Bob Shaw's Night Walk has something like that as a major plot point.
It's not aliens but humans, and it's not an 8-loop geometry, but without spoiling it too much it's safe to say that discovering how hyperspace works is the central concept guiding the story.
Shakespeare often spelt the same word differently at different times. If it was good enough for Billy Shakespeare, it should be good enough for modern-day mathematicians, forsooth.
The first of Shakespeare plays predate the first published English documentary. It was uncommon for spellings to be inconsistent or change between writings to be easier for a particular audience (in this case, actors) to be able to read.
I’m still making my way through it, but reading a history of shakespearean/elizabethan england, the first written publications of shakespeare’s plays that were accessible to the general public weren’t written by the man himself (if indeed he was singular).
There were entire efforts put towards pirating the plays by writing them, mostly from memory. It’s believed that someone in the crowd creating a stenographic copy would’ve been noticed so this is a less likely explanation. The memorial effort likely involved both audience and actors. “Official” versions meant to direct the stage productions might have been smuggled out or lost and found.
I haven’t gotten to the part yet that connects to the standard versions we have today. Some official versions were released to correct the record on bad pirated versions. Sometimes theaters would sell official versions to shore up funds.
Maybe this would explain the multiple shakespeare theory as well as writing inconsistencies?
Yeah; frankly, in almost all languages, some early works of literature tend to be THE thing that establishes canonical spelling. A lot of this is simply that they act as an argument-settler when two people can't agree how something "ought to be" spelled. In fact, sometimes they go so far as to warp pronunciation, cementing little verbal quirks that only some speakers had.
Why stop at greek or arabic when you can go all the way to sanskrit?
The words for sine and cosine derive from the sanskrit jiva (meaning bowstring, i.e., the chord of a circle)[1]. Sine and cosine were respectively jya and koti-jya, which got transcribed into arabic without the vowel (where it meant nothing). They then pronounced the vowel in the wrong place, calling it jeb (which meant pocket or fold in arabic)[2]. Then this wrong word got translated into latin as sinus (fold), and hence we have sine and cosine!
A healthy mixture was always preferred in maths and science. This is occasionally taken to extremes; the name reverse transcriptase, an enzyme used by retroviruses, is a combo of English, Latin and Greek!
Interesting. I'm not sure we can really call these arabic-derived, though. They do seem to ultimately trace back to fairly unrelated arabic words, but their first use in mathematics (much later) seems to have come in the form of a mixture of words from European languages. The two examples I gave seem to be more legitimately Arabic in origin.
Nadir always seemed very obviously Arabic to me. Weirdly, I first encountered it in a book on category theory, and only after that did I start to hear it used in everyday English to mean the opposite of 'apex'.
Dolphin, music (from muse), logic, ethics, physics, mathematics, pharmacy, angel, comedy, drama. The list of Greek loan words that are shared by many European languages goes on and on
Edit: I think almost every word with "ph" in it is from Greek and "th" in languages other than English.
If I saw ϖ in the wild I would have assumed it was an omega (ω) with a macron over it. Makes me wonder how many more varient Greek letters are out there.
Ancient, Ancient Greek had three additional letters: an F like character, a double lambda character, and P sounding character that looked like a lollipop. In case you need some additional symbols
The post mentions that ϖ is called “varpi”; I just wanted to add that this is actually short for “variant of pi”, also known as an “archaic form of pi” from old Greek writing.
There are infinite integers [1]. So even if we just look at basic polygons — shapes formed by connecting some (integer) number of points with straight lines — we easily get infinite shapes.
Math is crazy. The universe is crazy. Happy holidays!
Can’t have harmonics (i.e. harmonic oscillations, or any oscillations really) without e, though. sine and cosine are both sums of e, and if you look at the beauty of analytical sinusoid signals (which only have one component in the entire spectrum, lacking their negative frequency one) it’s just one exponential and nothing else.
However, the lemniscate of Bernoulli may be visually more pleasing; it has a parametrization very similar to the lemniscate of Gerono, except that both axes are scaled by a factor of 1/(sin(t)^2 + 1) = 2/(3 - cos(2t)):
scale = 2 / (3 - cos(2t));
x = scale cos(t);
y = scale * sin(2*t) / 2;
It looks like this:
There are the Lucky Numbers https://en.wikipedia.org/wiki/Lucky_number. Generated by a variant of the Sieve of Eratosthenes, they're believed to have a similar distribution to the primes while not having similar multiplicative properties.
Just like advertising--if they have earned my attention by saying things I want to read, then they have the right to dilute its quality with whatever else they want, up until it net doesn't interest me anymore. In this case, the jab is tiny and the quality content is bountiful.
For some reason, I imagined a number where every digit of pi was transformed into a [9-digit] and that it has special properties. This one is more magical, though.
Pizarro = Pi + Bizarro. Also there was an evil person that beared this name, Francisco Pizarro, the conquistador that kickstarted the genocide against the Incas. See https://en.m.wikipedia.org/wiki/Francisco_Pizarro
Curiously that made the thread better for me and the author's opinion about Twitter is exactly as true as the opposite opinion, that it is now the unfiltered source of objective truth. Or do you believe your opinions on the threads value or twitters reputation is special?
What's more likely: (i) famous mathematician expressing his frustration regarding how his previous internet community is now full of Nazis, or (ii) famous mathematician casually saying Nazis suck so to be perceived as morally superior by some random readers?
To me the second option is an extremely bizarre take and I cannot imagine why anyone would even consider it.
> Hence neither a man's contemporaries nor the man himself can form any final estimate of him or of his fitting position, because their knowledge is too imperfect. History often reverses the decision of contemporaries.
It’s a metaphor (ironically originating _on_ Twitter, not _about_ Twitter, pre-Musk); essentially, once you allow Nazis in a bar, they metastasize, and pretty soon you’re a Nazi bar. It’s perfectly applicable to the current state of twitter.
what metaphor was it called back when they allowed far-left hate speech but censored, shadow-banned or otherwise slowed stories that their secret thought-control departments didn't like? what would you call that?
Is it a political statement if it's also a statement of fact? Sure, the comment has some color to it, I'll concede that, but one can no longer post these kinds of things on Twitter and get the honest engagement from community members one used to. It's no longer a welcoming place for this kind of discussion.
With no account, I can no longer read comment chains on Twitter. It will only show the direct comment linked to. If you go to the user's page, all you see are the promoted tweets. There's no way to access the timeline sequentially anymore.
With those restrictions, you're writing only to a captive audience if you post on Twitter.
So you are technically correct, you literally cannot post these things on Twitter.
Offtopic but oh boy was that page difficult to scroll. Up/down arrows jump to the next post and page up/down isn't too helpful for reading. I have the keyboard overrides forbidden in my browser but obviously the web page can still bind events to keys not usually reserved for browser shortcuts... So, the usual navigation breaks up, leaving me to learn one particular site's idiosyncratic behaviour in the user interface space. No thanks, I just left.
Some people saw this right from the start. I remember the time when disallowing javascript would mostly spare you from unnecessary and irritating opt-ins, and you could still consume the actual content of the page using the browser as basically a text reader with hyperlinks, like originally intended. Now you can no longer, in effect, do that as pages consider the browser a VM to present themselves, and this just leads to a tug of war between the browser and its users vs the page creators. Both assume a level of control of a more than Turing complete medium and there's no compromise into that. The working solutions I see are either you write programs that run in the browser-VM to implement web stores etc. or you write effectively HTML 1.0 level structured documents to deliver information and leave the presentation to the browser-reader.
Back in the old days HTML was a huge step up from text files and proprietary hypertext documents but these days I'm more like hoping everything was ultimately, mostly plaintext.
The issue existed from me in both firefox and chrome. Click on outside columns will result in normal scroll. Click or highlight in the center column will result in the jumpy scroll that does not quite scroll one comment at a time with up/down arrow.
It kinda happens to me on firefox, one press of the down arrow scrolls so "Here's a formula for the lemniscate in polar coordinates" in the first reply is at the top of the screen, not helpful.
no idea why i even go for bait like this. because i like doing unpaid support work i guess. i tested in firefox and chrome. both work fine and don't do it like op decribes - no keybinds, keys behave normal.
maybe one of the dudes from yesterdays thread that had his own chatgpt programmed browser extensions installed that break the web for him.
Oddly enough, “never fails to disappoint” can have the meaning “never disappoints” as well as “routinely disappoints”. I’ve never thought about that one before
Native EN parser here. I would never consider this usage correct except as a rhetorical (facetious) insult. People may well repeat it without understanding the original nor their mistake. Although if enough people bust the syntax, it may attract descriptivist reporting, as with the widely observed malapropism "irregardless".
It’s not a matter of correctness, but of understanding. OP definitely intended to imply the content does not disappoint, and used a colloquialism most native speakers would understand
I am a native speaker and got the gist and saw the paradox, and found the phrasing a bit tortured by the triple negative. Thank you for explaining that this was a colloquialism. Now I have to go look up the etymology... And upon further inspection, this usage is actually a misnegation.
"It is a veiled insult: an ironic form of insult delivery which is misinterpreted as flattery to the buffoon who is targeted by it, much to the entertainment of anyone else within earshot who understands the true meaning."
There are East European languages, mostly Slavic ones, that have these weird double negatives which are grammatically correct and mean the opposite. A sentance such as: "I haven't never been there" means you've never been there.
This is the first time I come across this mistake / non-mistake so I misunderstood your comment. Are you sure it’s a common enough misnegation for people to understand what you meant ?
I didn’t use the expression, I don’t think I would have myself, but it didn’t even strike me as odd until I read the comment by hinkley. Did you read the original comment and think BeetleB follows John and thinks all of his content is disappointing?
Never heard that one, but maybe it's like 'could care less', which has acquired the opposite of it's actual meaning (the phrase should be 'could not care less') by repeated incorrect use.
A complete stranger who has nothing whatsoever to do with you, who has never tried to do anything for you, nor has been expected to, has never disappointed you. They've also never failed to disappoint you, because they have not failed in any regard whatsoever.
This is an example of a vacuous truth.
I've never failed an airliner landing. While that may sound like I'm boasting of being a good pilot, in fact I'm not a pilot at all, and I've never attempted such a thing.
Another vacuous truth.
Every crow in an empty set of crows is white.
Also, every crow in an empty set of crows is black.
Propositions universally quantified over an empty set are all vacuously true.
Statements with always and never are universally quantified over some set of events. If that set is empty it leads to vacuous truths.
"Every time I've seen a crow, it has always been white" is vacuously true if I've never seen a crow. I.e. the set of crows I've seen is empty, and consequently is a true statement that they're all white.
> A complete stranger who has nothing whatsoever to do with you, who has never tried to do anything for you, nor has been expected to, has never disappointed you. They've also never failed to disappoint you, because they have not failed in any regard whatsoever.
Nobody who uses the phrase ever means it in this way. The point of using the statement is to convey that you are familiar with the person’s history.
As another commenter has already pointed out, “has never failed to disappoint” is not the same statement as “never fails to disappoint”. The habitual present can’t refer to empty sets, as it is only used to refer to repeated actions.
> Nobody who uses the phrase ever means it in this way.
That is true. Outside of formal logic situations, deliberately uttered vacuous truths are only ever used by nerds to be clever, or for sarcasm, or insult and such.
Someone habitually using "never fails to disappoint" intended as a compliment has somehow latched onto an incorrect idiom; they likely intend something slightly funny like "never manages to disappoint" (tries hard to disappoint, but never does, due to being so good!). Or maybe it's supposed to be a deliberately funny mixup of "never fails" and "never disappoints".
> A complete stranger who has nothing whatsoever to do with you, who has never tried to do anything for you, nor has been expected to, has never disappointed you. They've also never failed to disappoint you, because they have not failed in any regard whatsoever.
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