johncarlosbaez

johncarlosbaez, 2 hours ago (edited 1 hour ago) to random

Robert Recorde introduced the equal sign in 1557. He used parallel lines because "no two things can be more equal". And his equal sign was hilariously looooooong.

This is from @mjd's excellent blog article:

https://blog.plover.com/math/recorde.html

and I recommend following him here on Mastodon.

It's fun to fight your way through Recorde's text, with its old font and spellings. But if you give up, @mjd has transliterated it:

Howbeit, for easie alteration of equations. I will propounde a fewe exanples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes "is equalle to" I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lengthe, thus: =====, bicause noe 2 thynges, can be moare equalle.

The only real mystery here is "Gemowe", which means "identical" and comes from the same root as "Gemini": twins.

In the same book Robert Recorde introduced the mathematical term "zenzizenzizenzike", but I'm afraid for that you'll have to read @mjd's article!

johncarlosbaez, 20 hours ago to random

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 pushforward 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.

System dynamics:

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

"Left pushforward" as a method of data migration:

David Spivak, Functorial data migration, https://arxiv.org/abs/1009.1166

johncarlosbaez, 1 day ago to random

I want to read this book: A Darwinian Survival Guide. Sounds like a realistic view of what we need to do now. You can read an interview with one author, the biologist Daniel Brooks. A quote:

...

Daniel Brooks: What can we begin doing now that will increase the chances that those elements of technologically-dependent humanity will survive a general collapse, if that happens as a result of our unwillingness to begin to do anything effective with respect to climate change and human existence?

Peter Watts: So to be clear, you’re not talking about forestalling the collapse —

Daniel Brooks: No.

Peter Watts: — you’re talking about passing through that bottleneck and coming out the other side with some semblance of what we value intact.

Daniel Brooks: Yeah, that’s right. It is conceivable that if all of humanity suddenly decided to change its behavior, right now, we would emerge after 2050 with most everything intact, and we would be “OK.” We don’t think that’s realistic. It is a possibility, but we don’t think that’s a realistic possibility. We think that, in fact, most of humanity is committed to business as usual, and that’s what we’re really talking about: What can we begin doing now to try to shorten the period of time after the collapse, before we “recover”? In other words — and this is in analogy with Asimov’s Foundation trilogy — if we do nothing, there’s going to be a collapse and it’ll take 30,000 years for the galaxy to recover. But if we start doing things now, then it maybe only takes 1,000 years to recover. So using that analogy, what can some human beings start to do now that would shorten the period of time necessary to recover?

https://thereader.mitpress.mit.edu/the-collapse-is-coming-will-humanity-adapt/

johncarlosbaez, 3 days ago (edited 2 days ago) to random

When global warming gets bad enough to create millions of refugees, just imagine how nasty the EU and US will become. They may just send people out to the desert to die.

Oh wait, you don't have to imagine! It's happening already:

"A year-long joint investigation by The Washington Post, Lighthouse Reports and a consortium of international media outlets shows how the European Union and individual European nations are supporting and financing aggressive operations by governments in North Africa to detain tens of thousands of migrants each year and dump them in remote areas, often barren deserts.

European funds have been used to train personnel and buy equipment for units implicated in desert dumps and human rights abuses, records and interviews show. Migrants have been pushed back into the most inhospitable parts of North Africa, exposing them to abandonment with no food or water, kidnapping, extortion, sale as human chattel, torture, sexual violence and, in the worst instances, death."

The full story is here: https://archive.is/1tEgl

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

A while back @pschwahn raised an interesting puzzle here:

You can define a well-behaved cross product of vectors only in 3 and 7 dimensions. The 7d cross product is weird because it's not preserved by all rotations of 7d space. But very smart people have told us there's a way to get the group of 𝟯𝗱 rotations to act on 7d space while preserving the 7d cross product. In fact, you can do it while also getting this group to act 'irreducibly', meaning the only subspaces of 7d space preserved by this action are {0} and the whole space!

The puzzle is: 𝗰𝗮𝗻 𝘆𝗼𝘂 𝘄𝗿𝗶𝘁𝗲 𝗱𝗼𝘄𝗻 𝘁𝗵𝗲 𝗳𝗼𝗿𝗺𝘂𝗹𝗮𝘀 𝗳𝗼𝗿 𝗵𝗼𝘄 𝘁𝗵𝗶𝘀 𝘄𝗼𝗿𝗸𝘀?

I got stuck on this so I asked some of my friends, and now Layra Idarani has outlined a nice way to do it:

https://golem.ph.utexas.edu/category/2024/05/3d_rotations_and_the_cross_pro.html#c063185

Interestingly he doesn't actually give the formulas; he just tells you how to get them. So I will need to do some work to check his answer! If you want to help out, that would be great.

Layra said "The devil of the details is in the eating". I thought the proof was in the pudding. Now I'm hungry for devil's food cake.

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

Since May 1st, a small team of category theorists, computer scientists and epidemiologists have been meeting daily in James Clerk Maxwell’s childhood home in Edinburgh. We’re hard at work on our project called New Mathematics and Software for Agent-Based Models.

We're creating a general software framework for agent-based models in epidemiology. By now we've really entered the zone where all these ideas come together. We're equally likely to talk about details of opioid abuse models or using coproducts of representables to make our software more efficient. It's exciting!

First we came up with a general framework for 'stochastic C-set rewriting systems'. These are models where graphs or more complicated combinatorial structures change in a random way through local 'rewrite rules'. Each rewrite rule says that when a bit of your structure matches some pattern, you can 'rewrite' it to some specific new pattern. See the pictures below for a couple examples.

'Stochastic' is a fancy word for 'random'. In our models you specify the randomness in a carefully crafted way by associating to each rewrite rule a 'timer'. The timer says the probability with which the rule is applied - as a function of time. A timer starts whenever a new match to the rule appears.

Kris Brown has already created a program that lets you run these stochastic C-set rewriting systems in AlgebraicJulia. This is a Julia package for scientific computing with categories. But we're just getting started!

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johncarlosbaez, 6 days ago (edited 5 days ago) to random

When Maxwell realized in 1862 that light consists of waves in the electromagnetic field, why didn't anyone try to use electricity to make such waves right away? Why did Hertz succeed only 24 years later?

According to 𝘛𝘩𝘦 𝘔𝘢𝘹𝘸𝘦𝘭𝘭𝘪𝘢𝘯𝘴:

"Since he regarded the production of light as an essentially molecular and mechanical process, prior, in a sense, to electromagnetic laws, Maxwell could elaborate an electromagnetic account of the propagation of light without ever supposing that ether waves were produced purely electromagnetically."

In 1879, a physicist named Lodge realized that in theory one could make "electromagnetic light". But he didn't think of creating waves of lower frequency:

"Send through the helix an intermittent current (best alternately reversed) but the alternations must be very rapid, several billion per sec."

He mentioned this idea to Fitzgerald, who believed he could prove it was impossible. Unfortunately Fitzgerald managed to convince Lodge. But later he realized his mistake:

"It was FitzGerald himself who found the flaws in his "proofs." He then proceeded to put the subject on a sound theoretical basis, so that by 1883 he understood quite clearly how electromagnetic waves could be produced and what their characteristics would be. But the waves remained inaccessible; FitzGerald, along with everyone else, was stymied by the lack of any way to detect them."

In 1883, Fitzgerald gave a talk called "On a Method of Producing Electromagnetic Disturbances of Comparatively Short Wavelengths". But he couldn't figure out how to 𝘥𝘦𝘵𝘦𝘤𝘵 these waves. Hertz figured that out in 1886.

johncarlosbaez, 6 days ago to random

Is there a chance that the physicist Oliver Heaviside was really Wolverine?

image/jpeg

johncarlosbaez, 8 days ago (edited 7 days ago) to random

The precise location of the boundary between the knowable and the unknowable is itself unknowable. But we 𝑑𝑜 know some details about 𝑤ℎ𝑦 this is true, at least within mathematics. It's being studied rigorously in a branch of theoretical computer science called 'meta-complexity theory'.

For some reason it's hard to show that math problems are hard. In meta-complexity theory, people try to understand why.

For example, most of us believe P ≠ NP: merely being able to 𝑐ℎ𝑒𝑐𝑘 the answer to a problem efficiently doesn't imply you can 𝑠𝑜𝑙𝑣𝑒 it efficiently. It seems obvious. But despite a vast amount of work, nobody has been able to prove it!

And in one of the founding results of meta-complexity theory, Razborov and Rudich showed that if a certain attractive class of strategies for proving P ≠ NP worked, then it would be possible to efficiently crack all codes! None of us think 𝑡ℎ𝑎𝑡'𝑠 possible. So their result shows there's a barrier to knowing P ≠ NP.

I'm simplifying a lot of stuff here. But this is the basic idea: they proved that it's probably hard to prove that a bunch of seemingly hard problems are really hard.

But note the 'probably' here! Nobody has 𝑝𝑟𝑜𝑣𝑒𝑑 we can't efficiently crack all codes. And this too, seems very hard to prove.

So the boundary between the knowable and unknowable is itself shrouded in unknowability. But amazingly, we can prove theorems about it!

https://www.quantamagazine.org/complexity-theorys-50-year-journey-to-the-limits-of-knowledge-20230817/

johncarlosbaez, 9 days ago to random

Chemistry is like physics where the particles have personalities - and chemists love talking about the really nasty ones. It makes for fun reading, like Derek Lowe's column "Things I Won't Work With". For example, bromine compounds:

"Most any working chemist will immediately recognize bromine because we don't commonly encounter too many opaque red liquids with a fog of corrosive orange fumes above them in the container. Which is good."

And that's just plain bromine. Then we get compounds like bromine fluorine dioxide.

"You have now prepared the colorless solid bromine fluorine dioxide. What to do with it? Well, what you don't do is let it warm up too far past +10C, because it's almost certainly going to explode. Keep that phrase in mind, it's going to come in handy in this sort of work. Prof. Seppelt, as the first person with a reliable supply of the pure stuff, set forth to react it with a whole list of things and has produced a whole string of weird compounds with brow-furrowing crystal structures. I don't even know what to call these beasts."

https://www.science.org/content/blog-post/higher-states-bromine

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

Fun article by John Psmith featuring some ferociously competitive mathematicians and physicists. A quote:

.....

In the 1696 edition of Acta Eruditorum, Johann Bernoulli threw down the gauntlet:

"I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.

Given two points A and B in a vertical plane,
what is the curve traced out by a point acted on only by gravity,
which starts at A and reaches B in the shortest time."

This became known as the brachistochrone problem, and it occupied the best minds of Europe for, well, for less time than Johann Bernoulli hoped. The legend goes that he issued that pompous challenge I quoted above, and shortly afterward discovered that his own solution to the problem was incorrect. Worse, in short order he received five copies of the actually correct solution to the problem, supposedly all on the same day. The responses came from Newton, Leibniz, l’Hôpital, Tschirnhaus, and worst of all, his own brother Jakob Bernoulli, who had upstaged him yet again.

(1/2) (The fun part about Newton comes in part 2.)

https://www.thepsmiths.com/p/review-the-variational-principles

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

There's a dot product and cross product of vectors in 3 dimensions. But there's also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There's nothing really like this in other dimensions.

We can get the dot and cross product in 3 dimensions by taking the imaginary quaternions and defining

v⋅w= -½(vw + wv), v×w = ½(vw - wv)

We can get the dot and cross product in 7 dimensions using the same formulas, but starting with the imaginary octonions.

The following stuff is pretty well-known: the group of linear transformations of ℝ³ preserving the dot and cross product is called the 3d rotation group, SO(3). We say SO(3) has an 'irreducible representation' on ℝ³ because there's no linear subspace of ℝ³ that's mapped to itself by every transformation in SO(3).

Much to my surprise, it seems that SO(3) also has an irreducible representation on ℝ⁷ where every transformation preserves the dot product and cross product in 7 dimensions!

It's not news that SO(3) has an irreducible representation on ℝ⁷. In physics we call ℝ³ the spin-1 representation of SO(3), or at least a real form thereof, while ℝ⁷ is called the spin-3 representation. It's also not news that the spin-3 representation of SO(3) on ℝ⁷ preserves the dot product. But I didn't know it also preserves the cross product on ℝ⁷, which is a much more exotic thing!

In fact I still don't know it for sure. But @pschwahn asked me a question that led me to guess it's true:

https://mathstodon.xyz/@pschwahn/112435119959135052

and I think I almost see a proof, which I outlined after a long conversation on other things.

The octonions keep surprising me.

https://en.wikipedia.org/wiki/Seven-dimensional_cross_product

johncarlosbaez, 12 days ago (edited 12 days ago) to random

We were listening to bouzouki music, and the conversation naturally turned to bazookas. It turns out that there's also bazooka music!

The comedian Bob Burns invented a horn-like instrument in the 1910s - he's shown with it below. It actually caught on in jazz in the 1930s. Someone jokingly called it the "bazooka" after the word "bazoo", which was slang for "mouth".

Later, in World War II, "bazooka" became the name for a new American anti-tank weapon, because it looked like this instrument.

Ironically, the slang word "bazoo", for "mouth", probably came from the word "buisine", which was the name of a medieval trumpet! And that comes from "buccina", a brass horn used by the Roman army.

In case you're wondering, he word "bouzouki" is unrelated. It comes from the Turkish word "bozuk", meaning "broken" or "modified", which refers to a particular way of tuning a string instrument where the notes are not arranged from low to high.

johncarlosbaez, 13 days ago (edited 13 days ago) to random

Hardcore math puzzle:

Suppose raindrops are falling on your head, randomly and independently, at an average rate of one per minute. What's the average of the 𝑐𝑢𝑏𝑒 of the number of raindrops that fall on your head in one minute?

The probability that (k) raindrops fall on your head in a minute is given by the Poisson distribution of mean 1, so it's
[ \frac{1}{ek!} ]
I could explain this but let's move on. The puzzle asks us to compute the expected value of (k^3) for this probability distribution, which is
[ \sum_{k=0}^\infty \frac{k^3}{ek!} ]
The heart of the puzzle is to figure out this sum. It turns out that
[ \sum_{k = 0}^\infty \frac{k^n}{k!} = B_n e ]
where (B_n) is the (n)th 'Bell number': the number of partitions of an (n)-element set into nonempty subsets. This is called 'Dobiński's formula'. I'll prove it in my next post. Now let's just use it!

We're interested in the case (n = 3). There are 5 partitions of a 3-element set
[ {{1,2,3}}, ]
[ {{1,2}, {3}}, ; {{2,3}, {1}}, ; {{3,1}, {2}}, ]
[ {{1}, {2}, {3}} ]
so (B_3 = 5).

So, the average of the cube of the number of raindrops that fall on your head in one minute is 𝟓.

Wild, huh? From probability theory to combinatorics.

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johncarlosbaez, 14 days ago to random

The rise of book bans in Florida made Lauren Groff start a new bookstore there. It 𝑠𝑝𝑒𝑐𝑖𝑎𝑙𝑖𝑧𝑒𝑠 in banned books. It's called The Lynx.

More here: https://archive.is/fD0uF

johncarlosbaez, 14 days ago (edited 14 days ago) to random

Costa Talalaev gave me a fractal: a 3-dimensional Sierpinski gasket!

He made it using the 3d printer at the Hacklab, a makerspace here in Edinburgh. It was a bit hard to make since it's held together only at tiny spots. He had to build something with more plastic, a kind of scaffolding, and then tear that off. The end result is very light yet sturdy.

johncarlosbaez, 15 days ago to random

This is an ORC - an "odd radio cluster". It's a faint circle of radio emissions surrounding a distant galaxy. 5 ORCs have been found. This one is about a million light years in diameter, roughly 10 times the size of a galaxy like ours. So it was probably formed by some sort of explosion, and took a long time to get this big. But the details remain a mystery!

Astronomers have just discovered that one ORC is emitting X-rays. So at least in that one ORC, the diffuse gas must be hot: about 8 million Celsius.

Here are some theories of what an ORC might be:

• it's a spherical shock wave from a cataclysmic event in the host galaxy, such as a merger of two supermassive black holes

• it's the shock wave formed by a 'starburst wind' created by a burst of star formation in the host galaxy

• it's the jet produced by a supermassive black hole, seen head on.

• it's a wormhole

As often the case, the most exciting theory is the least likely unless the others get ruled out.

But we can still have some fun. So watch an animated gif of what the birth of an ORC might look like!

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johncarlosbaez, 16 days ago to random

Last year, for the first time, 30% of electricity produced worldwide was from renewable sources. Wind and solar are growing. But notice that the biggest is hydroelectric, and it's going down! One reason is droughts in India, China, North America and Mexico. Climate change is causing droughts.

We're in a race against time. But at least we're running.

https://apnews.com/article/renewable-energy-climate-solar-wind-fossil-fuels-2718fce0ed37232dc25dbf46fff87955

johncarlosbaez, 17 days ago (edited 17 days ago) to random

Tolstoy: "Happy families are all alike; every unhappy family is unhappy in its own way."

Mathematics: "Real tori are all alike; every complex torus is complex in its own way."

To be precise, a 'n-dimensional real torus' is a real manifold of the form V/Λ where V is an n-dimensional real vector space and Λ ⊆ V is a lattice of rank n in this vector space. They are all isomorphic.

An 'n-dimensional complex torus' is a complex manifold of the form V/Λ where V is an n-dimensional complex vector space and Λ ⊆ V is a lattice of rank 2n in this vector space. These are not all isomorphic, because there are different ways the lattice can get along with multiplication by i. For example we might have iΛ = Λ or we might not.

And so, it's possible to write a whole book - and indeed a fascinating one - on complex tori. For example a 1-dimensional complex torus is an elliptic curve, and there are whole books just about those.

johncarlosbaez, 17 days ago to random

Since 2020 California has installed more batteries than anywhere in the world except China. Peak power usage comes between 7 and 10 pm, so solar power must be stored to handle this demand. It's starting to happen!

This year battery storage is expected to nearly double in the US, with the biggest growth in Texas, California and Arizona. 😎

From:

• Brad Plumer and Nadja Popovich, Giant batteries are transforming the way the U.S. uses electricity, Washington Post, May 7, 2024, https://archive.is/Vj2py

johncarlosbaez, 18 days ago (edited 18 days ago) to random

If I tell you the radii of the spheres 𝑎 and 𝑏 in this picture, can you figure out the radii 𝑟₁,...,𝑟₆ of the six spheres that touch them and snugly fit inside the big sphere? Can you at least do it if you know 𝑟₁?

Irisawa Shintarō Hiroatsu did it in 1822! He was a merchant who sold tea, textiles and ingredients for traditional Chinese medicine - and he had a hobby of solving math puzzles.

In 1932 his technique was rediscovered by a Nobel-prize-winning chemist, so it's often called Soddy’s Hexlet Theorem. But Hiroatsu did it earlier as part of a Japanese mathematical tradition called 𝑤𝑎𝑠𝑎𝑛 - and as part of this tradition, he donated a plaque containing this result to a shrine!

He wasn't the only one who did this sort of thing. This kind of plaque is called a 𝑠𝑎𝑛𝑔𝑎𝑘𝑢. These plaques were used to commemorate newly discovered solutions to hard math problems during the Edo Period from 1603 to 1868. There's a lot of interesting math in these 𝑠𝑎𝑛𝑔𝑎𝑘𝑢, and you can see some of them here:

• Abe Haruki, Japan’s “𝑊𝑎𝑠𝑎𝑛” mathematical tradition: surprising discoveries in an age of seclusion, https://www.nippon.com/en/japan-topics/c12801/

You can also learn more about the solution to the puzzle I gave! The most surprising thing is that the reciprocals of the opposite pairs of spheres in the "hexlet" of 6 spheres add up to the same number:

1/𝑟₁+1/𝑟₄ = 1/𝑟₂+1/𝑟₅ = 1/𝑟₃+1/𝑟₆

See also:

• Wikipedia, Soddy's hexlet, https://en.wikipedia.org/wiki/Soddy%27s_hexlet

This math is secretly all about conformal transformations, which map spheres to spheres... or planes!

Thanks to @highergeometer for pointing this out!

johncarlosbaez, 21 days ago (edited 20 days ago) to random

Yay! The science fiction author @gregeganSF helped me figure out this picture! It's called the hexagonal tiling honeycomb. A lot is known about it, but we described it in a new way.

An 'Eisenstein integer' is a complex number of the form a+bω where ω=exp(2πi/3). We proved that the center of each hexagon in this picture corresponds to a 2×2 self-adjoint matrix of Eisenstein integers with determinant 1 and positive trace.

We did it here on Mathstodon, and then @ai joined in and gave a different proof. It's a nice example of how people can team up spontaneously in the Fediverse to solve problems. For details, check out the n-Category Café:

https://golem.ph.utexas.edu/category/2024/04/line_bundles_on_complex_tori_p_2.html

In the process, a lot of nice math was revealed. For why this result is important, read my earlier article:

https://golem.ph.utexas.edu/category/2024/04/post_2.html

Briefly, the hexagon centers correspond to principal polarizations of the abelian variety ℂ²/𝔼² where 𝔼 is the lattice of Eisenstein integers. These are concepts that algebraic geometers know and love. Next I want to interpret the other features in this picture using related ideas from algebraic geometry.

johncarlosbaez, 23 days ago to random

For the next 6 weeks I'm meeting with Kris Brown, Nate Osgood, Xiaoyan Li, William Waites and Evan Patterson to design software for epidemiological modeling. And we're meeting in Maxwell's childhood home in Edinburgh! It's been made into a little museum, and this plaque is inside.

It sets a high standard for what counts as doing good science. And the coffee packets in the kitchen are all the same brand: Maxwell House.

johncarlosbaez, 24 days ago (edited 24 days ago) to random

It's time to have that talk with your kid. About quantum mechanics:

https://www.smbc-comics.com/comic/the-talk-3

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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