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?
> 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: (...)
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.
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!
The wikipedia link has this occurring in the 12th century. But Hellenistic astronomers were already working with sine tables. What did they call the concept?
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'.
It's the opposite of zenith, another word ultimately derived from Arabic.
The difference between an apex and a zenith is that an apex exists as a point in space, while a zenith is a direction, with no fixed point which may be said to be "the" zenith. There are other differences given that apex has a few related meanings, but this is the main one.
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.
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.
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.
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:
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.
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.
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.
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.
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.
This discussion helped me discover my new favorite map. https://en.wikipedia.org/wiki/File:Peirce_Quincuncial_Projec...
More projections in a friendly pdf: “An Album of Map Projections”[1], the one above is on page 190.
For a more festive example see Berghaus star projection on page 156.
[1]: https://pubs.usgs.gov/pp/1453/report.pdf (1989)
OK, that's fucking awesome. Thanks
oh wow that's a lot like a maximally extended penrose diagram
I believe I have seen the same projection used in a mod for quake 3 for dramatically increased POV.
And to protect you from it, you can use the following lucky clover charm (polar plot r=cos(2theta) ): https://www.wolframalpha.com/input?i=+plot+r%3Dcos%282theta%... whose perimeter can also define a constant 4*E(-3) ~ 4 * 2.4221
https://www.wolframalpha.com/input?i=plot+r%3Dcos%282theta%2...
> ” This ∞-shaped curve is called a 'leminscate', and ϖ is called the 'lemniscate constant'. I'll show you the leminiscate in my next post.”
This got me confused, so I went to check. Apparently ”lemniscate” is the correct spelling.
Fixed - thanks.
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:
https://en.wikipedia.org/wiki/Elliptic_integral
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:
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.
π is derived from the circle, which is defined by distance from a single point.
ϖ is derived from the lemniscate of Bernoulli, which is defined by distances from two points.
Is there an analogous constant that is derived from a shape defined by distances from three points?
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.)
https://www.desmos.com/3d/dl9v2vqbqb
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.
https://en.wikipedia.org/wiki/Cassini_oval#Generalizations
https://en.wikipedia.org/wiki/Polynomial_lemniscate#Erd%C5%9...
It's easy to generalize this to more points.
https://www.desmos.com/calculator/fo7tqlfjgo
it sounds like you are suggesting it might be turtles all the way down?
> 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.
https://www.scientificamerican.com/article/a-natural-log/
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)
fibonacci retrace shows up in liquid markets a lot
Humans have quite a few logarithmic responses: Brightness of light, loudness of sound, musical octaves and relative pitch.
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!
https://imgur.com/a/UkxkPzW
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.
Note that the arithmetic-harmonic mean I think you were going for is just the geometric mean (not the arithmetic-geometric mean, just the geometric mean simpliciter; see https://mathworld.wolfram.com/Arithmetic-HarmonicMean.html).
Other notable constants and where they show up:
Euler–Mascheroni Constant (integrals and sums involving the harmonic series, Gamma functions)
Catalan’s Constant (certain trigonometric series, lattice Green’s function)
Feigenbaum Constants (logistic map, chaos in dynamical systems)
Khinchin’s Constant (partial quotients in simple continued fractions)
Glaisher–Kinkelin Constant (asymptotic expansions of the Barnes G-function, combinatorial limits and certain product expansions)
Ramanujan’s Constant (complex multiplication of elliptic curves)
Omega Constant (Omega times e to the power of Omega = 1, Lambert W function, x^x^x^... = 2)
What do you mean by x^x^x^... = 2? Isn't the solution to that sqrt(2)?
f(x,n) = f(x,n-1) ^ x
f(x,1) = x
c_n is such that f(c_n,n) = 2
c = lim c_n
Do you mean f(x,n-1)^x or x^f(x,n-1)? With the first definition, c=1, the second, c=sqrt(2). I'm still not seeing the connection to omega.
please explain how Ramanujan’s Constant is relevant to operations on elliptic curves
https://www.sciencedirect.com/science/article/pii/S089812211...
How do you even know this?
Spend some time doing math and you'll pick up a few without noticing. Somewhat path dependent, but they do crop up.
See also: https://en.wikipedia.org/wiki/List_of_mathematical_constants
Some of those are quite obscure
And some aren't; it's not a list of "commonly encountered" constants, just a list.
It seems obvious that these are not twins. We can only say that π and ϖ are two among the infinite multitude of siblings ϖₙ.
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?
There is a generalization:
> 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: (...)
Yes. But the question remains: is there a geometrical analogue?
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.
> This ∞-shaped curve is called a 'leminscate', and ϖ is called the 'lemniscate constant'. I'll show you the leminiscate in my next post.
Two of these...do not belong?
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.
"It is a damn poor mind that can think of only one way to spell a word."
Unfortunately Daniel Webster ruined that for the rest of us.I find it hard to believe that Shakespeare would spell the same wird dyfferntli as if heez noom is Sheikhspier een uh deefirind koontri.
This might feed the “Shakespeare was not one person” theory
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?
You wouldn't download Hamlet's pirate story!
I guess you mean:
first published English dictionary
and
It wasn't uncommon / It was common
Yes, I was rather tired and typing on my phone required more correcting of the autocorrect feature than I could manage.
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.
"Lemniscate" is the correct spelling. All the other variants are mistyped.
It's quite funny imo that someday english people were like "forget about latin or german, greek is lit! Let's use greek"
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!
1. https://en.m.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_...
2. https://en.m.wikipedia.org/wiki/Sine_and_cosine#Etymology
The wikipedia link has this occurring in the 12th century. But Hellenistic astronomers were already working with sine tables. What did they call the concept?
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!
Arabic is also popular, particularly in maths.
> Television? The word is half Greek and half Latin. No good will come of this device.
―C. P. Scott
Is it? I can only think of (the very frequently noted) ‘algebra’ and ‘algorithm’.
Also ‘zero’, and ‘cipher’ (which, oddly, derive from the same word). And ‘average’. There are a few of them.
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.
Not math but I just learned alkali is the word for "ash" in Arabic.
And “alcohol”, frequently consumed at science and math conferences
"Alcohol" has a very interesting etymology, too.
As others have said, there are a few celestial terms that come to mind:
Also some chemistry terms, again just from top of brain, might be wrong: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'.
It's the opposite of zenith, another word ultimately derived from Arabic.
The difference between an apex and a zenith is that an apex exists as a point in space, while a zenith is a direction, with no fixed point which may be said to be "the" zenith. There are other differences given that apex has a few related meanings, but this is the main one.
Ah, yes. I knew it wasn’t quite the right word.
Sofa!
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.
They're asking about Arabic loanwords.
If you add all Latin words with Greek origins, most European languages are really forms of Greek
azimuth is the only other one I can think of off the top of my head
You'll find "zenith" at your feet.
You're thinking of nadir, also Arabic. The zenith is in the opposite direction.
Gah! So much for my wit.
The sheriff says "hold my beer".
A funny false or convergent etymology - shire reeve not sharif.
Here's another false trail from a real conversation:
Of course, the correct answer is California from Khalifa transliterated through a Spanish novel:https://en.wikipedia.org/wiki/Etymology_of_California#Las_Se...
https://en.wikipedia.org/wiki/Calafia
Don't most European languages use Landis loan words from both Latin and Greek? Both used to be taught in classical education.
What makes you think it was the English? I am pretty sure it comes from continental Europe.
Latin is lemniscus, so someday Latin people were like “let’s use Greek”
Latin writers have been like "let's use Greek" at least since Virgil, so modern writers can be excused for getting their roots mixed up.
It's so evil that it defies spelling
Lemniscate.
https://en.wiktionary.org/wiki/%CE%BB%CE%B7%CE%BC%CE%BD%CE%A...
Even the word has evil twins
I understand the confusion. Lemons smell good. The second root, on the other hand, far less pleasant.
Not to be confused with the "lemonscape", a hallucinated world you enter when you've eaten too many lemons.
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.
The Anvil of the Stars, by Greg Bear, featured a race of aliens whose mathematics weren't based on integers.
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.
Sounds like a Greg Egan writing prompt.
Baez and Egan are close friends, so don’t be surprised if you see it pop up.
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")
People just prompt themselves
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.
Kindle Edition: £2.99
Sounds like at least £2.99's worth of fun to me from the blurb, so it's now queued up.
I swear I'll get to it eventually.
... honest.
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Please don't do this here.
Can you really call yourself a hacker if you’ve never spelled 80085 on a calculator?
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Lots of things are right.
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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
I guess these are the letters?
https://en.wikipedia.org/wiki/Archaic_Greek_alphabets
Ϝ Digamma
Ͱ Heta
Ϻ San
Ϙ Koppa
Ͷ Tsan, Digamma
Ͳ Sampi
Considerate of them; very helpful for future mathematicians.
Any actual Greeks around here? I always wondered what π looks like when jotted down in, say, a shopping list...
This is actually a normal manuscript π as taught in Greek school. See https://www.typotheque.com/articles/modern-handwriting-a-his...
As psted now and then on HN:our alphabet has a variant as well: ampersand (per se: and).
The Fourier transform is composed of trigonometric sines and cosines.
There must be an analogous transform composed of lemniscate sines and cosines?
You could try to make a transform based on a sum of lemniscates in the complex plane
Infinity symbol with Lissajous curve:
x = Asin(at + delta)
y = Bsin(bt)
https://ericfortis.github.io/lissajous/?preset=Infinity
Interesting! I can see it in two ways: (1) as elongated U-shaped ellipsis that rotates sideways and (2) as bent lemniscate that rotates vertically.
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.
I read it as “omega-bar.”
"figure of eight" curves .... perhaps the simplest is the lemniscate of Gerono, which has the parametrization:
x = cos(t); y = sin(2t) / 2; and looks like this:
Lemniscate of Gerono animation https://i.sstatic.net/VKBgs.gif
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:
Lemniscate of Bernoulli animation https://i.sstatic.net/nOPMx.gif
per: https://gamedev.stackexchange.com/questions/43691/how-can-i-...
You just blew my mind. I'm taking a dive on this.
Wish people wouldn’t inject weird social jabs into stuff like this
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.
The author isn’t going to convince anyone or change anything with a random jab and it detracts from what they’re communicating
Sounds like his problem
Change pi to ϖ in this setup.
2022 - Non-Euclidean Doom: What happens to a game when pi is not 3.14159… https://youtu.be/_ZSFRWJCUY4?t=406
Such a promising yet disappointing talk.
Is there an evil twin to the set of prime numbers?
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.
The opposite of prime numbers are
https://en.wikipedia.org/wiki/Highly_composite_number
But I'm not sure if these ones are evil.
There are the anti prime numbers (also called highly composite numbers).
Every even number?
2?
Well, this one is pretty evil as prime numbers go.
> This ∞-shaped curve is called a 'leminscate', and ϖ is called the 'lemniscate constant'. I'll show you the leminiscate in my next post.
I think others have commented, but this three-way spelling certainly got a chuckle from me.
So are there an infinite amount of constants like this? In terms of pi, e and this number?
Just wondering, there are an infinite number of shapes I suppose? But does that mean there is an infinite amount of constants?
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!
———
[1] At least, that’s what they tell us… :p
Is there an abstraction of a leminscate/consonant with 3+ center points?
Yes, the Lissajous https://en.wikipedia.org/wiki/Lissajous_curve which also can be turned vertical (to look like an 8 instead of ∞).
But this doesnt have the property that the product of the distances to the focal points is constant no?
Heres two examples for 3 and 6 points in 2D, 3D respectively: https://en.wikipedia.org/wiki/Cassini_oval#Generalizations
These are symmetric as well though.
Kinda looking like hydrogen orbitals.
The lemniscate really looks like a homoclinic orbit in a 2d dynamics problem
I thought it might be e. e is often used to model unbounded growth, so it's chaotic, while pi is harmonic.
Plus, evil starts with 'e', so why not.
"Laugh with me Jocko!" "Eeeeeeeeeeeeee!"
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.
https://en.wikipedia.org/wiki/Bizarro
Side by side, there is a clear parallel to monopolar and bipolar fields. Is this found in any version of Maxwell’s equations?
>On our planet, it was Bernoulli, Euler and Gauss who discovered this math.
You don't say. Newton must have been sick that day.
I thought this was going to be about tau, which is not pi's evil twin, but rather, the One True Circle Constant.
https://tauday.com/tau-manifesto
upvote for tau, the one really running the show while pi gets the fame & fortune
Is there something like ThreadReaderApp for Mastodon?
Am I the only one who expected the evil twin to be 3 ?
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.
Wow, pomega is such a terrible name for it!
Woah it even has a w for wa-pi-rio.
mupi (mutant pi) or piet (pi evil twin) would be better names
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
Seems like a fine number, but I bet there's quite a few more irrational computable numbers out there.
how close is ϖ to e?
> I'm not enough of a cultural relativist to believe there's a civilization that cares more about the shape ∞ than the shape ◯.
Rumor has it there is one civilization of lizard-people out there. One is in fact running a company here on Earth with this shape as a logo!
/s
You mean Arduino is ran by the Illuminate?
dun-dun-dunnn!
But I was actually alluding to Meta and the memes about Mark Zuckerberg being a lizard: https://www.youtube.com/watch?v=jiudBq7z8wk
yes you can blink leds
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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?
I also found it extremely helpful that the author virtue signalled to agree with me, so I know whether I am supposed to like it or not.
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.
Interpretation 1 is more likely and made the dry mathematician more relatable/human which made the writing better imo.
> 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.
Probably true about Elon.
But I think mainly in the direction of demotion. Offhand I can't think of examples of someone ... oh, wait Van Gogh.
Baruch Spinoza is another. He was excommunicated.
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?
Twitter already contained a lot of hateful speech before Elon acquired it.
Those tweets would typically be demoted instead of promoted.
I absolutely agree, a dead 1920's German nationalist movement is exactly why we shouldn't allow free speech online.
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I was a Twitter user from 2007 to late 2022. That idiot wasted no time in ruining it; by Dec 2022 it was very clearly time to go.
Nobody uses twitter these days.
What a daft claim to make. A simple web search would have informed you that Twitter has over half a billion monthly active users.
Hmm you mean X right?
Same thing.
Since it's an "evil twin" should we not expect to find it in an alleged Nazi bar?
Yes, that is when I stopped reading. I left Twitter recently, but I would not call it a Nazi bar. It is just not for me, any longer.
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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.
> Why does this guy think it's acceptable to bitterly insult so many people,
Won’t someone think of the people?
My mom is on X - I don’t see how that offhand remark insults her
> I think HN should have a policy for submitted content that is along the lines of the policies in place for comments
We can already flag and vote - what more censorship do you want ?
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Same here. I have to install a userscript to restore usable scrolling:
> Up/down arrows jump to the next post and page up/down isn't too helpful for reading.
I didn't experience this at all on Firefox, up/down and page up/down scrolled in the normal way.
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.
here's a nickel, get a new browser.
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.
I follow John on Mastodon. He never fails to disappoint.
Then why do you follow 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".
https://english.stackexchange.com/questions/139448/never-fai...
https://en.m.wikipedia.org/wiki/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."
Only in the same sense that “could care less” is understandable but also means the opposite of the intention.
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.
I have only ever heard it used as a high brow burn, and a wickedly hard one at that.
I've never failed to win a game of mahjong against a bunch of grannies a Chinatown back room joint.
I've never tried such a thing; therefore, I've never failed.
Frankly, I could care less
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?
Here's a StackExchange thread on this exact mix-up (a "misnegation"):
https://english.stackexchange.com/questions/139448/never-fai...
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.
I’d say that’s more tolerated than embraced. We know what you meant, you just didn’t say what you meant. Not everyone tolerates it.
> can have the meaning “never disappoints”
How?
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.
"never failed" != "never fails"
They said they were a follower of them though.
¯\_(ツ)_/¯ linguistic drift. Technically it means you routinely disappoint, but it’s often used idiomatically to mean the opposite
What a country!
Heh. This comment blew up on me. Yes, it was a typo.