johncarlosbaez

johncarlosbaez, 9 months ago (edited 9 months ago) to random

I have so many questions about what just happened with Voyager 2. But let's review:

On August 20, 1977, Voyager 2 was launched from Earth.

In December 1977, it entered the asteroid belt.

In June 1978, its main radio receiver failed. Since then it's been using the backup receiver!

On July 9, 1979, it flew past many of Jupiter's moons, made its closest approach to Jupiter, and took tons of beautiful pictures.

On August 26, 1981 it shot past Saturn and took tons of beautiful pictures.

On August 25, 1989 it shot past Neptune and took tons of beautiful pictures.

On November 5, 2018 it crossed the heliopause and entered interstellar space, 120 times farther from the Sun than we are.

On July 18, 2023, it overtook Pioneer 10 and became the second farthest man-made object from the Sun.

3 days later, some idiot sent a command that pointed its high gain antenna 2 degrees away from Earth. HOW EXACTLY DID THIS HAPPEN?

On August 4, 2023, NASA used its most high-powered transmitter to successfully command Voyager 2 to reorient towards Earth, resuming communications. HOW WAS THAT POSSIBLE?

Voyager 2 is now 133 AU away. How can you "shout" across such a distance and attract the attention of someone who is not looking in your direction? That's very far. It takes light about 18 hours to travel that far.

johncarlosbaez, 7 months ago (edited 7 months ago) to random

If you draw all roots of all polynomials whose coefficients are ±1, you get an amazing picture that raises lots of challenging puzzles!

I really hope someone reads our short article:

https://www.ams.org/journals/notices/202309/rnoti-p1495.pdf

and solves the main puzzle: why do the fractal regions of this set look so much like "dragon sets"? We have a good heuristic explanation, but no proof yet.

People sometimes get excited about math when they learn about fractals, and then disappointed when they discover rather few professional mathematicians prove theorems about fractals. If you ever wanted to prove a cool theorem about fractals, this could be your chance!

If you read part 2, I'll show you what I'm talking about.

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johncarlosbaez, 3 months ago (edited 3 months ago) to random

If our civilization collapses, extraterrestrial archeologists can look at this and be impressed. Three satellites following the Earth in an equilateral triangle, each 2.5 million kilometers from the other two. Each contains two gold cubes in free-fall. The satellites accelerate just enough so they don't get blown off course by the solar wind. The gold cubes inside feel nothing but gravity.

Lasers bounce between each cube and its partner in another satellite, measuring the distance between them to an accuracy of 20 picometers: less than the diameter of a helium atom! This lets the satellites detect gravitational waves — ripples in the curvature of spacetime — with very long wavelengths, and correspondingly low frequencies.

It should see so many binary white dwarfs, neutron stars and black holes in the Milky Way that these will be nothing but foreground noise. More excitingly, it should see mergers of supermassive black holes at the centers of galaxies as far as... the dawn of time, or whenever such black holes were first formed. (The farther you look, the older things you see.)

It may even be able to see the "gravitational background radiation": the thrumming vibrations in the fabric of spacetime left over from the Big Bang. These gravitational waves were created before the hot gas in the Universe cooled down enough to become transparent to light. So they're older than the microwave background radiation, which is the oldest thing we see now.

It's called LISA - the Laser Interferometric Satellite Antenna. And we're in luck: ESA has just decided to launch it in 2035.

johncarlosbaez, 7 months ago (edited 7 months ago) to random

The University of Pennsylvania is acting proud of Katalin Karikó now that she's won a Nobel. But they kicked her out of her research assistant professor job when she insisted on doing the work that won her that prize:

"She recalls spending one Christmas and New Year’s Eve conducting experiments and writing grant applications. But many other scientists were turning away from the field, and her bosses at UPenn felt mRNA had shown itself to be impractical and she was wasting her time. They issued an ultimatum: if she wanted to continue working with mRNA she would lose her prestigious faculty position, and face a substantial pay cut.

”It was particularly horrible as that same week, I had just been diagnosed with cancer,” said Karikó. “I was facing two operations, and my husband, who had gone back to Hungary to pick up his green card, had got stranded there because of some visa issue, meaning he couldn’t come back for six months. I was really struggling, and then they told me this."

"While undergoing surgery, Karikó assessed her options. She decided to stay, accept the humiliation of being demoted, and continue to doggedly pursue the problem. This led to a chance meeting which would both change the course of her career, and that of science."

Elsewhere she recalled:

“I thought of going somewhere else, or doing something else. I also thought maybe I’m not good enough, not smart enough."

She's now an adjunct in UPenn's neurosurgery department. Will they fast-track her for tenure now that she has a Nobel, or just live with the shame?

Both quotes here come from interesting stories. The first is from here:

https://www.wired.co.uk/article/mrna-coronavirus-vaccine-pfizer-biontech

The second is from here:

https://billypenn.com/2020/12/29/university-pennsylvania-covid-vaccine-mrna-kariko-demoted-biontech-pfizer/

johncarlosbaez, 7 months ago (edited 7 months ago) to random

NOTE: I quoted a report from an executive of DuckDuckGo attending the antitrust lawsuit against Google. This article has now been retracted from Wired:

"After careful review of the op-ed, "How Google Alters Search Queries to Get at Your Wallet," and relevant material provided to us following its publication, WIRED editorial leadership has determined that the story does not meet our editorial standards. It has been removed."

I hope we'll learn more about what Google actually does, since September 28, the court established a process allowing the Justice Department to publish more information about this case.

https://www.wired.com/story/google-antitrust-lawsuit-search-results/

johncarlosbaez, 1 month ago (edited 1 month ago) to random

The 'hexagonal tiling honeycomb' is a beautiful structure in 3-dimensional hyperbolic space. I'm trying to figure out something about it.

It contains infinitely many sheets of hexagons, tiling planes in the usual way hexagons do. These are flat Euclidean planes in 3d hyperbolic space, called 'horospheres'. I want to know the coordinates of the vertices of these hexagons. I have some clues.

The hexagonal tiling honeycomb has Schläfli symbol {6,3,3} . The Schläfli symbol is defined in a recursive way. The symbol for the hexagon is {6}. The symbol for the hexagonal tiling of the plane is {6,3} because 3 hexagons meet at each vertex. Finally, the hexagonal tiling honeycomb has symbol {6,3,3} because 3 hexagonal tilings meet at each edge!

The symmetry group of the hexagonal tiling honeycomb is the Coxeter group {6,3,3}. This is a discrete subgroup of the Lorentz group O(3,1), which acts on 3d hyperbolic space because that space is the set of points (t,x,y,z) in Minkowski spacetime with

t² − x² − y² − z² = 1 and t > 0

The Coxeter group {6,3,3} is generated by reflections, but its 'even part', generated by pairs of reflections, is a discrete subgroup of PSL(2,ℂ), because this is the identity component of the Lorentz group. In fact, this Coxeter group is almost PSL(2,𝔼), where 𝔼 is the ring of 'Eisenstein integers'. These are complex numbers of the form

a + bω

where a,b are integers and ω is a nontrivial cube root of 1. So there should be a nice description of the hexagonal tiling honeycomb using Eisenstein integers! And this is what I'm trying to find... quickly, before May 1st because I'm have a column due then. 😧

I should ask @roice3, who drew this....

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johncarlosbaez, 1 year ago (edited 1 year ago) to random

In computer science it's against the rules to publish your work in many prestigious venues unless you physically travel to present that work at a conference!

They seem to be forgetting that there's a thing called the "internet" that makes this unnecessary. Pretty weird for computer scientists.

Here's the famous computer scientist Moshe Vardi on how this needs to change:

"As I argued in January 2020, in view of the worsening climate crisis, our discipline's practice that every publication requires travel, often trans-oceanic, is inconsistent with the public good. A report released in March 2023 by the Intergovernmental Panel on Climate Change paints an even bleaker picture of the worsening climate crisis. The world is on brink of catastrophic warming, the report warned. A dangerous climate threshold is near, but 'it does not mean we are doomed' if swift action is taken, the scientists said."

"Going back to the pre-pandemic conference-travel culture is simply not morally acceptable, I believe. Yet many conferences have gone back to a full in-person model, and authors are required to present in person. This requirement is drawing criticism. A recent article by Theoretical Computer Scientists for Future (TCS4F) concluded, 'Coupling formal publications with an in-person gathering no longer makes sense for everyone.'"

https://cacm.acm.org/magazines/2023/5/272297-acm-for-the-public-good/fulltext

johncarlosbaez, 9 months ago (edited 9 months ago) to random

Suppose you were trying to invent a bright orange powder that could easily dye clothes and be hard to wash off. Using your knowledge of quantum mechanics you'd design this symmetrical molecule where an electron's wavefunction can vibrate back and forth along a chain of carbons at the frequency of green light. Absorbing green light makes it look orange! And this molecule doesn't dissolve in water.

Yes: you'd invent turmeric!

Or more precisely 'curcurmin', the molecule that gives turmeric its special properties.

The black atoms are carbons, the white are hydrogens and the red are oxygens.

Read on and check out what pure curcurmin looks like.

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johncarlosbaez, 2 months ago (edited 2 months ago) to random

This curve is not an elliptic curve - because even though you can write it in as

y² = P(x)

with P a cubic polynomial, elliptic curves need to be smooth! We say this curve is 'singular', not smooth everywhere, because it crosses itself at one point, making a kind of X shape. Mathematicians call this point a 'node'. So this curve, which I'd rather write as

y² = x³ - x²

is called a 'nodal cubic'.

It's still fun to count the solutions of this equation in a finite field. Let's do it!

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johncarlosbaez, 1 year ago (edited 1 year ago) to random

The surprising part is not that math grad students named Cox and Zucker would come up with the idea of writing a paper together just as a joke.

It's that they followed through after they became professors, and wrote a paper that was actually rather significant.

https://en.wikipedia.org/wiki/Cox%E2%80%93Zucker_machine

johncarlosbaez, 7 months ago (edited 7 months ago) to random

Though part of me - the worst part - would like to join the clever crowd who endlessly pontificate and interview each other, I'm held back by my intense aversion to publicly talking about:

1. consciousness
2. free will
3. string theory and other theories of everything
4. are mathematical objects real?
5. is reality a simulation?
6. interpretations of quantum mechanics
7. quantum computers
8. large language models, machine learning, AI

and most other topics that the "digiterati", the "intellectual dark web", and other quasi-scientific talking heads enjoy bloviating about. I'd much rather curl up with a good solid book on the life cycle of lichens, or the organizational structure of car repair shops.

johncarlosbaez, 8 months ago to random

Who drew this picture for me? - without the purple and yellow. Now I want to use it, and I forget who to credit!

Recently Quanta came out with an article explaining modular forms:

https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/

It does a heroically good job. One big thing it doesn't do is explain the funny looking 'fundamental domains' in the upper half-plane.

By sheer coincidence, I just wrote a little article explaining the concept of 'moduli space' through an example which does touch on these fundamental domains:

https://johncarlosbaez.wordpress.com/2023/09/23/the-moduli-space-of-acute-triangles/

It's due October 1st so I'd really appreciate it if you folks could take a look and see if it's clear enough. It's really short, and it's written for people who know more math than your typical Quanta reader, but not necessarily anything about moduli spaces.

The cool part is the connection between the moduli space of acute triangles — that is, the space of all shapes an acute triangle can have — and the more famous moduli space of elliptic curves!

johncarlosbaez, 8 months ago (edited 8 months ago) to random

People say "power corrupts", but most of us - I hope - realize there's more to it. Positions of power actually attract people with the "dark triad" of personality traits: narcissism, Machiavellianism and psychopathy. And in many systems, these people are better than average at getting into power.

None of this is news. Nor is it news that this is one of the biggest problems in the world, with many states and organizations run by people like this, and others locked in struggles against - or between - such people.

The question is, what to do about it? Brian Klass has some ideas:

https://youtu.be/3eBN_9rMoVI

Mainly, he wants us to set up systems that tend to filter out people with bad traits as they rise to power in companies, bureaucracies, governments, etc. - and also scrutinize people who have achieved power.

What I don't see him saying is how to get out of situations where these people have already achieved power. I guess he's saying we should take full advantage of the places and moments where there's some room to set up better systems. We tend to focus our energies on the crises and fights of the moment rather than putting work into setting up good systems.

Haven't seen the "dark triad" before? Check out part 2.

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johncarlosbaez, 4 months ago (edited 4 months ago) to random

Big news in bug math. This is the first year since 1803 when both 13-year cicadas and 17-year cicadas will emerge from the ground simultaneously in the US!

13 and 17 are both prime. It's believed cicadas evolved to have prime-number life cycles, thus avoiding predators that emerge more frequently, like once every 4 years or 5 years or... whatever. By showing up infrequently, with a prime number life cycle, cicadas can starve out those predators.

And since 13 and 17 are both prime and 13 × 17 = 221, both kinds of cicadas emerge simultaneously only once every 221 years. And

1803 + 221 = 2024

so now they'll both emerge simultaneously and we'll have 𝑙𝑜𝑡𝑠 of cicadas!

Also, this year the two kinds can interbreed! Maybe they're be fruitful and multiply... and we'll get 221-year cicadas. 😆

The last time the Northern Illinois Brood’s 17-year cycle aligned with the Great Southern Brood’s 13-year cycle, Thomas Jefferson was president.

https://www.nytimes.com/2024/01/19/science/cicadas-emergence-broods.html

johncarlosbaez, 1 month ago (edited 1 month ago) to random

So you wake up one day wanting to invent a 2-dimensional number system. This requires a new number 𝑖 that's at right angles to 1. So you figure multiplying by 𝑖 must rotate numbers by 90°. So multiplying by 𝑖² rotates by 180°, so

𝑖² = -1

Cool!

Then you notice something else. The derivative of a function in the 𝑦 direction must be 𝑖 times its derivative in the 𝑥 direction, because the derivative is linear and you get the 𝑦 direction by rotating the 𝑥 direction by 90°: that is, multiplying it by 𝑖. So you get this equation:

[ \frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x} ]

Cool!

Then you notice something else. If you use this equation twice you get

[ \frac{\partial^2 f}{\partial y^2} = i \frac{\partial f}{\partial x\partial y} = i^2 \frac{\partial^2 f}{\partial x^2} = - \frac{\partial^2 f}{\partial x^2} ]

so

[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 ]

Wow! Every function with a second derivative obeys the Laplace equation!

You decide this one is a keeper.

https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations

johncarlosbaez, 4 months ago (edited 4 months ago) to random

Camembert is an endangered species! It relies on a mold that had lost the ability to reproduce sexually... and now, thanks to inbreeding, this mold has also lost the ability to reproduce using spores. Roquefort is also in trouble, but for Camembert it's worse:

"The world over, this other symbol of French gastronomy is inoculated exclusively with one single strain of Penicillium camemberti, a white mutant that was selected for Brie cheeses in 1898 and Camemberts in 1902.

The problem is that ever since then the strain has been replicated by vegetative propagation only. Until the 1950s, Camemberts still had grey, green or in some cases orange-tinged moulds on their surface. But the industry was not fond of these colours, considering them unappealing, and staked everything on the albino strain of P. camemberti, which is completely white and moreover has a silky texture. This is how Camembert acquired its now-characteristic pure white rind.

Year after year, generation after generation, the albino strain of P. camemberti, which was already incapable of sexual reproduction, lost its ability to produce asexual spores. As a result it is now very difficult for the entire industry to obtain enough P. camemberti spores to inoculate their production of the famous Norman cheese.

Worse still, while the Roquefort PDO (Protected Designation of Origin) standard retains a degree of microbial biodiversity, the PDO specifications for Camembert require farmers and other producers to use P. camemberti exclusively."

What to do about it? Switch to eating American cheese, or Velveeta? Read on....

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johncarlosbaez, 11 months ago (edited 11 months ago) to random

I'm very glad I left Twitter. But if Twitter was like heroin for me, Mastodon is like methadone. It doesn't give me enough of a high to keep me addicted. But it could be a useful step on my road to recovery!

With not enough people to talk to here, I've been spending less time on general-purpose social media and more time on specialized sites like the nLab and the Category Theory Community Server. I'm getting into more conversations that actually help me with my work, and I hope I'm helping students more. I miss the lively diversity of community, but I don't know what to do about that.

Mind you, I'm very grateful for all the nice replies I get to my posts here! I'm especially happy that some experts are replying to my amateur posts about music theory. But these replies feel few and far between.

To use another analogy: if Twitter felt like an echo chamber, Mastodon feels like an anechoic chamber... or padded cell. That's probably why most of the friends I eagerly welcomed here are gone now:

https://mathstodon.xyz/@johncarlosbaez/109262546588286591

I'm not sure where they went. Back to Twitter? That's not an acceptable option for me. Back to real life? That would be interesting.

I'm not planning to "quit" Mastodon. But if you've noticed my posts here slacking off, this is why.

johncarlosbaez, 1 year ago (edited 1 year ago) to random

A common name for the inverse sine function is

arcsin 𝑥

The usual name for inverse hyperbolic sine function is

arcsinh 𝑥

Now some Wikipedians have decided to start calling it

arsinh 𝑥

I've never seen this before and it makes me think "arse". I don't like it!

A more common alternative is

sinh⁻¹ 𝑥

Some computer scientists use

asinh 𝑥

Some pedants, claiming that hyperbolic trig functions aren't connected to arcs, argue for

argsinh 𝑥

Who came up with "arsinh" and why? Is it too late to kill off this notation? It seems like a case of Wikipedia editors run amok. It reminds me of how some of them made up their own very precise definition of "order of magnitude" - and now run around correcting people who say "to within an order of magnitude" if it doesn't match their made-up definition.

When I want to be understood I'll use "arcsinh". When I want to act mathematical I'll use "sinh⁻¹". If I wanted to save space I'd write "asinh". I might use "argsinh" if I were pretending to be a pirate. But I'd only use "arsinh" if I wanted to be an arse.

By the way, I don't like debates about notation. So I don't know why I'm talking about this when I have more interesting things to talk about. Just blowing off steam, I guess.

https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions

johncarlosbaez, 9 months ago (edited 9 months ago) to random

Musical tuning systems is the subject where you get mad at irrational numbers. Nothing works perfectly - and it's not your fault: it's math's fault. It's all about pushing around lumps in the carpet.

An 'octave' is the chord where the high note vibrates 2 times as fast as the low note. In a 'perfect fifth' it vibrates 3/2 times as fast. In a 'perfect fourth' it vibrates 4/3 as fast. In a 'major third' it vibrates 5/4 as fast. Our ears love these simple fractions.

But if you go up 4 perfect fifths, it's not quite the same as going up 2 octaves and a major third, since

3/2 × 3/2 × 3/2 × 3/2 = 81/16

is not quite

2 × 2 × 5/4 = 5 = 80/16

AARGH! 😠

The difference between these is called the 'syntonic comma'. Well, actually the ratio

81/80 = 1.0125

is called the syntonic comma. Listen to two notes with this frequency ratio:

https://en.wikipedia.org/wiki/Syntonic_comma

You can hear they aren't in tune, and it probably sounds annoying. This is why we can't have nice things.

Another problem is that if you go up 7 octaves it's almost but not quite 12 perfect fifths, since

2⁷ = 128

is not

(3/2)¹² = 129.746337890625

The ratio of these is called the 'Pythagorean comma':

531441/524288 = 1.0136432647705078125

This is why a 12-tone scale with all the notes equally spaced can't have perfect fifths. But for vocal music, the syntonic comma is more urgent problem, since it involves simpler fractions. It shows up in lots of different ways: two people can sing two different parts starting in tune, each singing beautifully, and wind up out of tune.

johncarlosbaez, 4 months ago (edited 4 months ago) to random

The muon is a particle very much like an electron but about 207 times heavier. You can make an atom like hydrogen with a muon orbiting a proton! It's called 'muonic hydrogen'.

Muonic hydrogen is 207 times smaller than hydrogen. But the muon moves at the same average speed as the electron does in hydrogen: about 1/137 times the speed of light. This is pretty fast by human standards.

Unfortunately a muon doesn't live for long: its half-life is roughly a microsecond, and then it decays into an electron and some junk.

But relativity says that time passes more slowly for a fast-moving object. So a muon must live a bit longer when it's in muonic hydrogen. So I wondered: has anyone detected this effect?

Unfortunately the answer is "not yet, because the effect is too small". But we're getting close to being able to do it.

Interestingly, the lifetime increase due to relativity is 60 times smaller than the lifetime decrease due to the muon being captured by the proton it's orbiting! And that is an effect we can measure:

For details, check out this:

https://physics.stackexchange.com/questions/689478/does-a-muon-in-muonic-hydrogen-have-a-longer-half-life-due-to-time-dilation?noredirect=1#comment1787992_689478

johncarlosbaez, 10 months ago to random

Boris Zilber is a logician at Oxford. He notified me about a paper of his with some radical ideas. As always, I am skeptical about new theories of physics, especially before they make quantitative predictions that match anything we've seen. But at least his ideas involve some interesting math I hadn't known!

It starts out like this:

"Since it is now accepted that there is a minimal length, the Planck length, assumption of the spatial finiteness of the universe implies the assumption of the finiteness of the number of its elements. In [1] we discussed the concept of approximation in physics and the suggestion that the physical universe is co-ordinatised by a huge finite field 𝔽. It was proved (Proposition 5.2 of [1]) that the only metric field (locally compact field) that can be approximated by finite fields is the field ℂ of complex numbers. Thus “seen from afar” the huge finite field looks like a field of complex numbers ℂ."

That's strange. But then it gets stranger and more interesting: he actually takes his huge finite field to be what logicians call a "pseudo-finite field": an infinite model of the theory of finite fields. And to get ahold of this field, he uses the nonstandard integers mod a nonstandard prime p.

"Thus, when we work with feasible numbers 1, 2, . . . inside 𝔽 we can think on these as the usual integers, but when our integers become much much bigger then, according to the approximation theorem of [1], they start to behave like complex numbers, e.g. if an integer i satisfies i² + 1 = 0 mod p it takes the role of √−1."

I have so many questions I don't know where to start!

Here's the paper:

https://arxiv.org/abs/2306.15698

johncarlosbaez, 6 months ago (edited 6 months ago) to random

Last night Lisa and I went to a Chinese restaurant with the computer scientist Mike Fourman, his partner Jeanne, and her son and his wife. The bill was £98. To add a tip and make the bill divisible by 3, Mike rounded it up to £108. I instantly recognized that this number is divisible by 3, since the digits add up to 9. But to my annoyance I didn't instantly remember what 108 divided by 3 is. That is, I didn't remember thinking about this question before.

Later, in the middle of the night, I realized I had run into this question!

Do you see what I mean?

If not, read on....

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johncarlosbaez, 10 months ago (edited 10 months ago) to random

Today it's so windy near Edinburgh that the price of power has gone negative! So we're helping out the electric company by turning on more lights.

Why does the price go negative? Apparently because it costs more to slow the turbines than to let them spin and generate power. And it costs more to dump the power somewhere than to have customers use it.

We get the pricing data in real time from here:

https://energy-stats.uk/octopus-agile-southern-scotland/

johncarlosbaez, 1 year ago to random

I was walking through downtown Santa Fe with Lisa. She said "Hey!"

I looked - at first I saw Joan Baez's name and wondered why she was giving a concert here. Then I realized this is where I'm giving a talk on Tuesday.

It feels strange seeing my name on a movie marquee. Right after Don Giovanni, no less!

johncarlosbaez, 26 days ago (edited 26 days ago) to random

@TruthSandwich got me interested in the vibrational modes of bells. They're not harmonics with frequencies 1, 2, 3, 4, ... times the lowest frequency: they're much more complicated! That's why bells sound clangy. This chart shows how they sometimes work.

The lowest frequency vibrations are called:

• the 'hum' (the lowest frequency)

• the 'prime' (with frequency roughly 2 times that of the hum)

• the 'tierce' (roughly 2.4 times the hum, so a minor third above the prime)

• the 'quint' (roughly 3 times the hum, so a major fifth above the prime)

• the 'nominal' (roughly 4 times the hum, so an octave above the prime)

and so on. If you think these names are illogical, join the club! One reason it's tricky is that the loudest vibration is not the lowest one: it's the 'prime'.

The numbers I just gave you should be taken with a big grain of salt. They really depend on the shape of the bell, and you'd have to be great at designing bells to make them come out as shown here. It's not like a violin string or flute, where the math is on your side.

This quote helps explain the chart:

"Modern theory separates the modes of vibration into those produced by the "soundbow" and those produced by the remaining bell "shell". The bell vibrates both radially and axially and the principal vibrational modes are shown in the diagram together with their classification using the scheme proposed by Perrin et al. This scheme consists of the mode of vibration (RIR - Ring Inextensional Radial, RA - Ring Axial, R=n - Shell driven), the number of meridians (where “m” is half the number of meridians) and the number of nodal circles (n)."

Starting to sound like orbitals in quantum mechanics!

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