An epidemiologist having a category-theoretic revelation. My colleague Nathaniel Osgood, discovering how the process of converting stock and flow diagrams into causal loop diagrams can be captured by a left adjoint functor between presheaf categories. These two kinds of diagrams are both important in the modeling tradition called 'system dynamics', which is used in epidemiology as well as economics and other disciplines.
https://media.mathstodon.xyz/media_attachments/files/112/495/874/881/338/846/original/6115c8eac71e8722.jpg
If our civilization collapses, extraterrestrial archeologists can look at this and be impressed. Three satellites following the Earth in an equilateral triangle, each 25 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. This radiation was created before the hot gas in the Universe cooled down enough to become transparent to light. So it's 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 2034.
https://media.mathstodon.xyz/media_attachments/files/111/829/191/809/166/973/original/14a019c7ca47beab.mp4
@3ecc8fb4 - great! I remember back in the late 1980s when braided monoidal categories first became important in mathematical physics, and they seemed very far-out and cool! Now Microsoft is trying to use them to build quantum computers.
https://physicsworld.com/a/from-electronics-to-anyonics/
@133234fb - oh, the talk I gave on August 24th had nothing to do with fields. But it's on YouTube here:
https://math.ucr.edu/home/baez/8_and_24/
Two of My Favorite Numbers: 8 and 24
The numbers 8 and 24 play special roles in mathematics. The number 8 is special because of Bott periodicity, the octonions and the E8 lattice, while 24 is special for many reasons, including the binary tetrahedral group, the 3rd stable homotopy group of spheres, and the Leech lattice. The number 8 does for superstring theory what the number 24 does for bosonic string theory. In this talk, which is intended to be entertaining, I will overview these matters and also some connections between the numbers 8 and 24.
@133234fb - lesser known algebraic gadgets as motivating examples? Which lesser known gadgets do you mean? I try to give well-known things as motivating examples. 😳
I don't know what talk about fields you mean, either! I have a talk about "Life's struggle to survive" that I'm struggling to edit, that's taking a while.
@c503a79c - I quit instantly because I disliked him too much to contribute my work to him for free. I'd rather help build a community that has more of a sense of democracy and fairness.
What happens in a world without equality? All you have are things, processes that turn one thing into another, meta-processes that turn one process to another, and so on... forever!
If this is too scary you can truncate it at the nth level. Then you're dealing with an 'n-category'. This has things (called 'objects'), processes (called 'morphisms'), meta-processes (called '2-morphisms') and so on up to n-morphisms.
In this talk I explain the periodic table of n-categories - a fundamental structure that organizes our understanding of these .
I put a lot of work into making it fun and easy to follow... and I think it worked!
(Alas, the video quality is still not great, but it's better than last week's lecture where I introduced n-categories. The volume is low so you have to really crank up your speaker... and the only way I have to boost the volume of a video also makes the file a lot bigger.)
https://www.youtube.com/watch?v=X1PkkqDwf8Y
@3132ba89 - that's fascinating! I really like that graph and should someday think about various measures of dissonance or 'resonance' between frequencies. (I'm already curious about celestial mechanics and how resonances sometimes stabilize orbits and sometimes destabilize them; I've never had time to really understand it.)
I don't really myself find the fifth to be 'sweet'; I was just trying take the attitude of common practice harmony theory, roughly 1650-1850, where you often read things like:
The perfect fifth and the perfect octave are considered perfect consonances. The unison is a consonance insofar as it can be considered an interval at all (many say it cannot).
The major third and sixth, as well as the minor third, sixth, are considered to be imperfect consonances.
The perfect fourth is dissonant in some contexts but consonant in others (see below). Specifically, the perfect fourth is dissonant when it is formed with the bass note of any sonority.
https://en.wikibooks.org/wiki/Music_Theory/Consonance_and_Dissonance
If I'd wanted to be more technical I would have said "the fifth was the only interval besides the octave to be considered a 'perfect consonance' in the common practice era." But that would have killed off half my readership!
@6f0f8100 - Yeah, I screwed up the diagram. Each arrow should point to a frequency that's 3/2 times as high.
Thanks - I'll fix it! You're apparently the first person who actually looked at it.
To draw the diagram correctly it's also important to know which is bigger: 729/64 or the appropriate power of 2 (namely 128) time 64/729. That's what tells us which circle to draw to the left, and which to the right!
I think I got that part right, but just drew the arrows wrong.
@2c8b8664
@6f0f8100 -by the way, part of my confusion was due to the fact that the augmented 4th is higher, not lower, than the diminished 5th.
If you don't get that, never mind - it's just somewhat confusing stuff!
@2c8b8664
@c7c371fb - I think this is all we've got right now. That's why I said
"I hope we'll see more evidence proving or disproving this, since September 28, the court established a process allowing the Justice Department to publish more information about this case."
@6f0f8100 - Yeah, I screwed up the diagram. Each arrow should point to a frequency that's 3/2 times as high.
Thanks - I'll fix it! You're apparently the first person who actually looked at it.
To draw the diagram correctly it's also important to know which is bigger: 729/64 or the appropriate power of 2 (namely 128) time 64/729. That's what tells us which circle to draw to the left, and which to the right!
I think I got that part right, but just drew the arrows wrong.
@2c8b8664
@e89610ab@7edccf3d - if all known transmissible cancers are this young, we either haven't looked hard enough or there must be some reason they "burn out" - which would be very interesting to understand.
Theoretically there could be some reason *all* transmissible cancers have arisen only recently, but I can't believe that.
@8a3ca29f - things get really complicated! In the Pythagorean scale here we get two different notes: the augmented 4th which is a bit lower than the tritone, and the diminished 5th which is a bit higher. The augmented 4th is
1024/729 ≈ 1.404664
the tritone is
√2 ≈ 1.414214
and the diminished 5th
729/512 ≈ 1.423828
I sure don't claim I can hear the difference! But if two were played simultaneously they would sound out of tune.
@61df5b23 - has someone done experiments to show that? I'd be interested.
I'm constantly torn, when improvising on the piano, between feeling like the perfect fifth is halfway between the tonic and an octave up, and seeing that it's not on the chromatic scale. (I actually use tritones a fair amount, treating that as the midpoint.)
In Pythagorean tuning, we try to force all frequency ratios to be powers of 3/2. In music, 3/2 is the 'perfect fifth': the sweetest of intervals except for the octave.
If we start with some frequency and go up and down by powers of 3/2, we create the 'circle of fifths' shown here. It's almost a 12-pointed star, with one point for each note in the 12-tone equal-tempered scale.
Almost - but not quite! When we go up 12 perfect fifths, we get a note that's almost but not quite 2⁷ times the frequency we started with. In other words, it's almost but not quite 7 octaves higher. So there's a glitch.
Here I've stuck that glitch at the opposite from the note labeled 1. The spot directly opposite 1 is called the 'tritone', or sometimes 'diabolus in musica' - the devil in music. 😈
The size of the glitch is called the 'Pythagorean comma'. It's
(3/2)¹² / 2⁷ ≈ 129.74633789 / 128 ≈ 1.01364326477
https://en.wikipedia.org/wiki/Pythagorean_tuninghttps://en.wikipedia.org/wiki/Pythagorean_commahttps://media.mathstodon.xyz/media_attachments/files/111/181/039/976/449/841/original/6423e662f54d0985.jpg
@3f3ce2f5 - So far I've been scared of that septimal stuff. I try to understand things really systematically, and I think there's a lot left for me to understand about 3-limit and 5-limit tuning. But someday....
@00b2f577 - I've been trying DuckDuckGo and am not thrilled. For example if I want to see something I wrote about modular forms I type in
+baez modular forms
and it returns a lot of results that apparently don't contain the word "baez". Just an example - it's not that I'm so self-centered, I just want a search engine where you can fine-tune a search.
Some information coming out of the antitrust lawsuit against Google:
"Google likely alters queries billions of times a day in trillions of different variations. Here’s how it works. Say you search for “children’s clothing.” Google converts it, without your knowledge, to a search for “NIKOLAI-brand kidswear,” making a behind-the-scenes substitution of your actual query with a different query that just happens to generate more money for the company, and will generate results you weren’t searching for at all. It’s not possible for you to opt out of the substitution. If you don’t get the results you want, and you try to refine your query, you are wasting your time. This is a twisted shopping mall you can’t escape.
Why would Google want to do this? First, the generated results to the latter query are more likely to be shopping-oriented, triggering your subsequent behavior much like the candy display at a grocery store’s checkout. Second, that latter query will automatically generate the keyword ads placed on the search engine results page by stores like TJ Maxx, which pay Google every time you click on them. In short, it's a guaranteed way to line Google’s pockets."
https://www.wired.com/story/google-antitrust-lawsuit-search-results/
@3b0cd0b9 - did you read how Google secretly alters your queries, adding extra words to them? It's come out in the antitrust lawsuit.
"Google likely alters queries billions of times a day in trillions of different variations. Here’s how it works. Say you search for “children’s clothing.” Google converts it, without your knowledge, to a search for “NIKOLAI-brand kidswear,” making a behind-the-scenes substitution of your actual query with a different query that just happens to generate more money for the company, and will generate results you weren’t searching for at all. It’s not possible for you to opt out of the substitution. If you don’t get the results you want, and you try to refine your query, you are wasting your time. This is a twisted shopping mall you can’t escape.
Why would Google want to do this? First, the generated results to the latter query are more likely to be shopping-oriented, triggering your subsequent behavior much like the candy display at a grocery store’s checkout. Second, that latter query will automatically generate the keyword ads placed on the search engine results page by stores like TJ Maxx, which pay Google every time you click on them. In short, it's a guaranteed way to line Google’s pockets."
Reference:
https://www.wired.com/story/google-antitrust-lawsuit-search-results/
@af6bf358 - you get a circle of fifths if you move up the red arrows. Each circle of fifths includes all 12 notes.
You get a circle of major thirds if you move along the blue arrows - but each circle of major thirds includes only 3 notes, e.g.
C E G# C E G# ...
And you get a circle of minor thirds if you move along the red arrows - but each circle of minor thirds includes only 4 notes, e.g.
C E♭ F♯ A C E♭ F♯ A ....
Did you ever think of music as 2-dimensional?
• Following a red arrow, the frequency goes up by a factor of 3/2 (called a perfect fifth).
• Following a blue arrow, the frequency goes up by a factor of 5/4 (called a major third).
• Following a black arrow, the frequency goes up by a factor of 6/5 (called a minor third).
This setup is used for some keyboards - like on accordions - but it's also useful in '5-limit tuning', where frequency ratios only involve powers of 2, 3, and 5. Just intonation, which I talked about yesterday:
https://mathstodon.xyz/@johncarlosbaez/111170247654517571
is the best known 5-limit tuning system. But there are others!
We often use logarithms to turn frequency ratios into numbers we can add instead of multiply. Since ln(2), ln(3) and ln(5) are linearly independent over the rational numbers, the numbers ln(2ⁱ3ʲ5ᵏ) actually form a 3-dimensional structure: mathematicians call it a free abelian group of rank 3. This chart just shows a 2-dimensonal 'slice' of that.
For example, following the arrows on this chart (which extends indefinitely) you'll never get a note with exactly 2 times the frequency of your starting note. That means you never get a perfect octave! 😿
But the perfect octave does lurk in the 3-dimensional structure of which this chart is a slice.
https://media.mathstodon.xyz/media_attachments/files/111/176/028/949/590/278/original/97fda10dc4b002b4.jpg
@f6878bd4@365dd660 - very little is known for sure about what Pythagoras did. I read a book-long biography of him which starts by listing everything known about him in the first paragraph: he was the son of Mnesarchus, a gem-engraver, and around 530 BC he travelled to Croton in southern Italy. The rest is echoes of echoes of echoes.
@2c46aa33 - it turns out that article was written by someone who doesn't quite know academia. I've now learned a few details and they seem a bit different: she was a research assistant professor paid for by soft money, apparently from other people's grants, and when her research seemed to not be going anywhere those people said she should work on something else or she'd have get a job as an adjunct. That's my impression, anyway.
@94babfe9 - yes, she wrote:
“If you are working to please a company or boss then get ready for disappointment. You cannot make a goal to please someone else. You have to tell yourself you want to understand something – because then you’ll never get disappointed. Even if someone publishes something related to the work you’re doing, that’s ok. Because that will help you understand that particular part of your work. It just takes a small change in perspective and you could be so much happier.”
https://inews.co.uk/news/science/katalin-kariko-covid-19-vaccine-pioneer-urges-more-girls-and-young-women-to-take-up-science-1460380
@3b0cd0b9 - really bummed.
🙃 Seriously, I wasn't ever hoping for warp drives. They would screw up the spacetime continuum and also help invasive, obnoxious civilizations (like ours).
"Just intonation" is a tuning system where the tones have simple fractions as their frequency ratios. In the key of C it looks like this. The notes C E G have frequency ratios of 1, 1.25 and 1.5. This is called a "major triad". The notes G B D form another major triad with the same frequency ratios, and so do the notes F A C. This determines the whole tuning system!
Notice here we are working up from C to D, and down from C to F. We're doing that because it's easiest to describe the math that way.
Just intonation may go back to Mesopotamia, but it was carefully described by Claudius Ptolemy - the guy famous for his work on astronomy - in his "Harmonikon" around 140 AD. In fact the term "just intonation" is a bit vague and a more precise term is "Ptolemy's intense diatonic scale":
https://en.wikipedia.org/wiki/Ptolemy%27s_intense_diatonic_scale
I don't know why it's called "intense". Anyway, this scale was advocated by some very influential music theorists in the Renaissance, like Zarlino and Tartini, but it later fell out of fashion for a couple of good reasons.
The fractions in this scale only involve powers of 2, 3 and 5, so it's also called "five-limit tuning":
https://en.wikipedia.org/wiki/Five-limit_tuning
In this scale, multiplying the frequency by 2 takes you up an octave. Multiplying by 3 takes you up an octave and a fifth. And multiplying by 5 takes you up two octaves and a major third.
So: multiplying by 3/2 takes you up an fifth, and multiplying by 5/4 takes you up a third!
A major triad consists of a tone, the tone a third up, and the tone a fifth up. This is why the major triads in this chart get the frequency ratios 1, 5/4 and 3/2, or in other words: 1, 1.25 and 1.5.
(1/n)
https://media.mathstodon.xyz/media_attachments/files/111/170/054/156/928/425/original/9e57303cdb6ffdcb.png
One downside is that it's impossible to get a 12-tone scale where *all* major triads have frequency ratios 1, 5/4, 3/2. So something has got to break when we try to switch keys and stay in just intonation. This video by the wonderful Elam Rotem explains what breaks:
https://www.youtube.com/watch?v=XhY_7LT8eTw
(2/2)
@f28e93c3 - okay, that makes sense. Thanks! The Wired story says
"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"
which sounds like it's written from and for people outside academia. Who were her "bosses", exactly? Maybe she was existing on soft money from some team of PIs?
I imagine people will start investigating and writing up this story more carefully now that she's so famous.
@e552091f - indeed; she can write her own ticket now. Since 2019 was senior vice president of BioNTech RNA Pharmaceuticals, but in 2022 she left BioNTech to devote more time to research. So she obviously still wants to do research.
@b80dcf15 - as you might expect, the English Wikipedia article has now been fixed.
She has a professorship in the University of Szeged, in Hungary, where she got her BS and PhD. But I don't know if she actually works there: it may be a kind of honorary thing.
@0269c0e4 - Since 2019 she was senior vice president of BioNTech RNA Pharmaceuticals, but in 2022 she left BioNTech to devote more time to research. It'll be interesting to see what she does now.
Another fun thing: Euler's formula
𝑛² + 𝑛 + 41
which gives primes from 𝑛=1 to 𝑛 = 39 is actually related to the fact that 41×4-1 = 163 is a Heegner number. And this fact has (less impressive) relatives for the smaller Heegner numbers!
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