The Future of Physics

The 20th century was the century of fundamental physics. While we saw immense progress toward discovering the basic laws governing matter, space, and time, this has slowed to a crawl since 1980, despite an immense amount of work. Luckily, there’s plenty of exciting progress in other branches of physics: for example, using the fundamental physics we already know to design surprising new forms of matter! Like all other sciences in the 21st century, physics must also embrace the challenges of the Anthropocene: the era in which humanity is a dominant influence on the Earth’s climate and biosphere.

Last week I gave a Santa Fe Institute Community Lecture on the future of physics. You can see my slides here, or watch it here:

Since my talk was announced on the marquee of a theater, and I myself misread it as a concert by Joan Baez, I asked the audience how many had been expecting her. About ten, it seems! I planned to sing a bit of a song, but thought better of it at the last second.

Galois’ Fatal Duel

On this day in 1832, Evariste Galois died in a duel. The night before, he summarized his ideas in a letter to his friend Auguste Chevalier. Hermann Weyl later wrote “This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.”

That seems exaggerated, but within mathematics it might be true. On top of that, the backstory is really dramatic! I’d never really looked into it, until today. Let me summarize a bit from Wikipedia.

Galois lived during a time of political turmoil in France. In 1830, Charles X staged a coup d’état, touching off the July Revolution. While students at the Polytechnique were making history in the streets, Galois, at the École Normale, was locked in by the school’s director. Galois was incensed and wrote a blistering letter criticizing the director, which he submitted to the Gazette des Écoles, signing the letter with his full name. Although the Gazette’s editor omitted the signature for publication, Galois was expelled.

Galois joined the staunchly Republican artillery unit of the National Guard. He divided his time between math and politics. On 31 December 1830, his artillery unit was disbanded for fear that they might destabilize the government. 19 officers of this unit were arrested and charged with conspiracy to overthrow the government.

In April 1831 these officers were acquitted of all charges. On 9 May 1831, a banquet was held in their honor, with many famous people present, including Alexandre Dumas. The proceedings grew riotous. At some point, Galois stood and proposed a toast in which he said, “To Louis Philippe,” with a dagger above his cup. The Republicans at the banquet interpreted Galois’s toast as a threat against the king’s life and cheered.

The day after that wild banquet, Galois was arrested. He was imprisoned until 15 June 1831, when he had his trial. The jury acquitted him that same day.

All this time, Galois had also been doing math! Earlier, the famous mathematician Poisson had asked Galois to submit a paper to the Academy, which he did on 17 January 1831. Unfortunately, around 4 July 1831, Poisson wrote a reply declaring Galois’s work “incomprehensible” and saying his “argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor”. But Poisson ended on a positive note: “We would then suggest that the author should publish the whole of his work in order to form a definitive opinion.”

Galois did not immediately receive this letter. He joined a protest on Bastille Day, 14 July 1831, wearing the uniform of the disbanded artillery and heavily armed with several pistols, a loaded rifle, and a dagger. He was again arrested. During his stay in prison, Galois at one point drank alcohol for the first time at the goading of his fellow inmates. One of these inmates recorded in a letter what Galois said while drunk:

“And I tell you, I will die in a duel on the occasion of some coquette de bas étage. Why? Because she will invite me to avenge her honor which another has compromised. Do you know what I lack, my friend? I can confide it only to you: it is someone whom I can love and love only in spirit. I’ve lost my father and no one has ever replaced him, do you hear me…?”

In his drunken delirium Galois attempted suicide, and would have succeeded if his fellow inmates hadn’t forcibly stopped him.

Remember Poisson’s letter? While Poisson wrote it before Galois’s arrest, it took until October for this letter to reach Galois in prison. When he read it, Galois reacted violently. He decided to give up trying to publish papers through the Academy and instead publish them privately through his friend Auguste Chevalier.

Later he was released from prison. But then he was sentenced to six more months in prison for illegally wearing a uniform. This time he continued to develop his mathematical ideas and organize his papers. He was released on 29 April 1832.

Galois’s fatal duel took place on 30 May. The true motives behind the duel are obscure. There has been much speculation about them. What is known is that, five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair.

Some archival investigation on the original letters suggests that the woman of romantic interest was Stéphanie-Félicie Poterin du Motel, the daughter of the physician at the hostel where Galois stayed during the last months of his life.

Whom did Galois fight in his fatal duel? Alexandre Dumas named Pescheux d’Herbinville, who was actually one of the 19 artillery officers whose acquittal was celebrated at the banquet that led to Galois’s first arrest. On the other hand, newspaper clippings from only a few days after the duel may suggest that Galois’ opponent was Ernest Duchatelet, who was imprisoned with Galois on the same charges. The truth seems to be lost to history.

Whatever the reasons behind his fatal duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament: his famous letter to Auguste Chevalier outlining his ideas, and three attached papers. But the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated. The papers were already mostly written.

Early in the morning of 30 May 1832, Galois was shot in the abdomen. He was abandoned by his opponents and his own seconds, and found by a passing farmer. He died the following morning at ten o’clock in the Hôpital Cochin after refusing the offices of a priest. Evariste Galois’s younger brother Alfred was present at his death. His last words to Alfred were:

“Ne pleure pas, Alfred! J’ai besoin de tout mon courage pour mourir à vingt ans!”

(Don’t weep, Alfred! I need all my courage to die at twenty!)

On 2 June, Galois was buried in a common grave in the Montparnasse Cemetery. Its exact location is apparently unknown.

Eleven years later, in 1843, the famous mathematician Liouville reviewed one of Galois’ papers and declared it sound. Talk about slow referee’s reports! It was finally published in 1846.

In this paper, Galois showed that there is no general formula for solving a polynomial equation of degree 5 or more using only familiar functions like roots. But the really important thing is the method he used to show this: group theory, and the application of group theory now called Galois theory.

And for something amazing in his actual letter, read this:

• Bertram Kostant, The graph of the truncated icosahedron and the last letter of Galois, Notices of the AMS 42 (September 1995), 959–968.

Symmetric Spaces and the Tenfold Way

I gave a talk about symmetric spaces and the tenfold way in Nicohl Furey’s series Algebra, Particles and Quantum Theory on Monday May 15, 2023. This talk was a followup to an earlier talk, also about the tenfold way.

You can see the slides of my new talk here. They have material on category theory that I didn’t get around to talking about.

Symmetric spaces and the tenfold way

Abstract: The tenfold way has many manifestations. It began as a tenfold classification of states of matter based on their behavior under time reversal and charge conjugation. Mathematically, it relies on the fact that there are ten super division algebras and ten kinds of Clifford algebras, where two Clifford algebras are of the same kind if they have equivalent super-categories of super-representations. But Cartan also showed that there are ten infinite families of compact symmetric spaces! After explaining symmetric spaces, we show how they arise naturally from forgetful functors between categories of representations of Clifford algebras.

The final upshot is this:

Let the Clifford algebra be the free real or complex algebra on n anticommuting square roots of -1. Let

be the forgetful functor between representation categories. Then the essential fibers of this functor are disjoint unions of compact symmetric spaces. Moreover we get all the compact symmetric spaces in the 10 infinite families this way!

(There are also finitely many exceptional ones, which all arise from the octonions.)

Cold-Resistant Trees

The Appalachians are an old, worn-down mountain chain that runs down the eastern side of North America. The ecology of the Appalachians is fascinating. For example:

Ecologists have tested many species of Appalachian trees to see how much cold they can survive. As you’d expect, for many trees the killing temperature is just a bit colder than the lowest temperatures at the northern end of their range. That makes sense: presumably they’ve spread as far north—and as far up the mountains—as they can.

But some other trees can survive temperatures much lower than that! For example white and black spruce, aspen and balsam poplar can survive temperatures of -60° C, which is -80° F. Why is that?

One guess is that this extra hardiness is left over from the last glacial cycle, which peaked 20,000 years ago—or even previous glacial cycles. It got a lot colder then!

So, maybe these trees are native to the northern Appalachians—while others, even those occupying the same regions, have only spread there since it warmed up around 10,000 years ago. Ancient pollen shows that trees have been moving north and south with every glacial cycle.

I learned about this issue here:

• Scott Weidensaul, Mountains of the Heart: a Natural History of the Appalachians, Fulcrum Publishing, 2016.

I bought this book before a drive through the Appalachians.

To add some extra complexity to the story, David C. writes:

I’d love to understand more and reconcile that with the fact that none of these trees do well above around 4500 ft in the northern Appalachians (New Hampshire).

and Brian Hawthorne writes:

Don’t forget that all the tree species had to move back into the areas that were under the last glacier.

Categories for Epidemiology

Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood, Eric Redekopp and I have been creating software for modeling the spread of disease… with the help of category theory!

Lots of epidemiologists use “stock-flow diagrams” to describe ordinary differential equation (ODE) models of disease dynamics. We’ve created two tools to help them.

The first, called StockFlow.jl, is based on category theory and written in AlgebraicJulia, a framework for programming with categories that many people at or associated with Topos have been developing. The second, called ModelCollab, runs on web browsers and serves as a graphical user interface for StockFlow.jl.

Using ModelCollab requires no knowledge of Julia or category theory! This feature should be useful in “participatory modeling”, an approach where models are built with the help of diverse stakeholders. However, as we keep introducing new features in StockFlow.jl, it takes time to implement them in ModelCollab.

But what’s a stock-flow diagram, and what does our software let you do with them?

The picture here shows an example: a simple disease model where Susceptible people become Infective, then Recovered, then Susceptible again.

The boxes are “stocks” and the double-edged arrows are “flows”. There are also blue “links” from stocks to “variables”, and from stocks and variables to flows. This picture doesn’t show the formulas that say exactly how the variables depend on stocks, and how the flows depend on stocks and variables. So, this picture doesn’t show the whole thing. It’s really just what they call a “system structure diagram”: a stock-flow diagram missing the quantitative information that you need to get a system of ODEs from it. A stock-flow diagram, on the other hand, uniquely specifies a system of first-order ODEs.

Modelers often regard diagrams as an informal step toward a mathematically rigorous formulation of a model in terms of ODEs. However, we’ve shown that stock-flow diagrams have a precise mathematical syntax! They are objects in a category while “open” stock-flow diagrams, where things can flow in and out of the whole system, are horizontal 1-cells in a double category If you know category theory you can read a paper we wrote with Evan Patterson where we explain this:

• John C. Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams. To appear in Proceedings of Applied Category Theory 2022.

If you don’t, we have a gentler paper for you:

• John C. Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Evan Redekopp, A categorical framework for modeling with stock and flow diagrams, to appear in Mathematics for Public Health, Springer, Berlin.

Why does it help to formalize the syntax of stock-flow diagrams using category theory? There are many reasons, but here are three:

1. Functorial Semantics

Our software lets modelers separate the syntax of stock and flow diagrams from their semantics: that is, the various uses to which these diagrams are put. Different choices of semantics are described via different functors. This idea, called “functorial semantics”, goes back to Lawvere and is popular in certain realms of theoretical computer science.

Besides the ODE semantics, we have implemented functors that turn stock-flow diagrams into other widely used diagrams: “system structure diagrams”, which I already explained, and “causal loop diagrams”. It doesn’t really matter much here, but a causal loop diagram ignores the distinction between stocks, flows and variables, lumps them all together, and has arrows saying what affects what:

These other forms of semantics capture purely qualitative features of stock and flow models. In the future, people can implement still more forms of semantics, like stochastic differential equation models!

So, instead of a single monolithic model, we have something much more flexible.

2. Composition

ModelCollab provides a structured way to build complex stock-flow diagrams from small reusable pieces. These pieces are open stock-flow diagrams, and sticking together amounts to composing them.

ModelCollab lets users save these diagrams and retrieve them for reuse as parts of various larger models. Since ModelCollab can run on multiple web browsers, it lets members of a modeling team compose models collaboratively. This is a big advance on current systems, which are not optimized for collaborative work.

This picture shows two small stock-flow diagrams being composed in ModelCollab:

Some of the underlying math here was developed in earlier work using categories and epidemiological modeling, which was also done by people at Topos and their collaborators:

• Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson and James P. Fairbanks, An algebraic framework for structured epidemic modelling, Philosophical Transactions of the Royal Society A 380 (2022), 20210309.

3. Stratification

Our software also allows users to “stratify” models: that is, refine them by subdividing a single population (stock) into several smaller populations with distinct features. For example, you might take a disease model and break each stock into different age groups.

In contrast to the global changes commonly required to stratify stock-flow diagrams, our software lets users build a stratified diagram as a “pullback” of simpler diagrams, which can be saved for reuse. Pullbacks are a concept from category theory, and here we are using pullbacks in the category whose objects are system structure diagrams. Remember, these are like stock and flow diagrams, but lacking the quantitative information describing the rates of flows. After a system structure diagram has been constructed, this information can be added to obtain a stock and flow diagram.

This picture shows two different models stratified in two different ways, creating four larger models. I won’t try to really explain this here. But at least you can get a tiny glimpse of how complicated these models get. They get a lot bigger! That’s why we need software based on good math to deal with them efficiently.

References

[AJ] AlgebraicJulia: Bringing compositionality to technical computing.

[B1] John C. Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams. To appear in Proceedings of Applied Category Theory 2022.

[B2] John C. Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams, to appear in Mathematics for Public Health, Springer, Berlin.

[H] P. S. Hovmand, Community Based System Dynamics, Springer, Berlin, 2014.

[L] Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson and James P. Fairbanks, An algebraic framework for structured epidemic modelling, Philosophical Transactions of the Royal Society A 380 (2022), 20210309.

[MC] ModelCollab: A web-based application for collaborating on simulation models in real-time using Firebase.

[SF] Stockflow.jl.

The Bebop Major Scale

Though people say ‘octave’, there are only 7 different notes in the major scale. This is annoying if you’re trying to play scales in melodies with, say, 8 beats per measure. The scale keeps drifting out of synch.

One solution is to add an extra note to your scale! In this video, jazz cat Adam Maness explains the virtues of the ‘bebop major scale’, where you add a minor 6th to the major scale:

1 2 3 4 5 ♭6 6 7

Just playing this scale up and down, 8 beats per measure, already suggests some melodies. Even more so if you play it with both hands in ‘contrary motion’—up with one hand, down with the other. Listen to the video and you’ll see what I mean! This is just the start of the interesting things you can do with the bebop major scale.

Why do they call this the ‘major sixth diminished scale’? The jazz pianist and educator Barry Harris introduced this term: he said this scale is derived from a major 6th chord (1 3 5 6) and a diminished 7th chord starting at the 2 (that is, 2 4 ♭6 7).

You can get other bebop scales by putting the extra note somewhere else:

• Wikipedia, Bebop scale.

Sometimes the extra note in the bebop major scale is described as a ♯5 rather than a ♭6:

Bosons, Fermions and Clifford Algebras

A minus sign can make a huge difference. Einstein discovered that the difference between space and time is all due to a minus sign.

Another amazing fact is that the difference between ‘matter particles’ (or more precisely fermions, like electrons, quarks, etc.) and ‘force particles’ (bosons, like photons, gluons, etc.) is mainly due to the fact that when you switch two fermions their quantum state gets multiplied by -1, while when you switch two bosons it get multiplied by 1.

This was discovered by Pauli, who realized that there must be some reason why the electrons in atoms go into ‘shells’ – why all the electrons in a big atom like iron don’t all fall into the same lowest-energy state. The reason is that if two electrons were in the same state, switching them would do nothing but also multiply that state by -1: a contradiction. This rule, that fermions can’t be in the same state, is called the Pauli exclusion principle.

Bose and Einstein realized that on the contrary, bosons actually like to be in the same lowest energy state at low temperatures! This is called Bose-Einstein condensation. Similarly, a laser beam has many photons in the same state.

Later people realized that if we replace vector spaces (like the Hilbert space of quantum states of some system) by ‘super vector spaces’, where every vector is a sum of a bosonic and fermionic part, we can impose a rule saying that switching two fermionic vectors should always introduce an extra minus sign.

It turns out that this rule is not arbitrary—it’s mathematically very natural and it’s lurking around all over in mathematics, even in contexts that superficially have nothing to do with bosons or fermions!

A nice example of how ‘super’ thinking can clarify things is in the study of Clifford algebras and their representations.

In the Clifford algebra Cliffₙ we start with the real numbers and then throw in n anticommuting square roots of -1. For example:

Cliff₀ = ℝ, Cliff₁ = ℂ, Cliff₂ = ℍ (the quaternions), etc.

With the quaternions, once you throw in i and j with i² = j² = -1 and make them anticommute (ij = -ji) you get k = ij for free.

Each Clifford algebra has ‘representations’: roughly, real vector spaces where elements of the Clifford algebra act as linear operators. The most famous representations of Cliffₙ are the ‘pinor’ representations, which describe spin-1/2 particles in n-dimensional space along with how reflections act on these particles. You get all the other representations by taking direct sums of pinor representations.

For example, in 2d space Cliff₂ = ℍ has a representation on itself, and this is the only pinor representation in 2d space. All other representations of Cliff₂ are direct sums of this one – so its category of representations is the category of quaternionic vector spaces!

This chart shows the categories of representations of the Clifford algebras up to dimension 7. After that they repeat.

The symbol ≃ means that two algebras have equivalent categories of representations. For example, Cliff₆ is the category of 8×8 real matrices! So it’s not isomorphic to ℝ, but you can show they have equivalent categories of representations.

One last thing: Cliff₃ is isomorphic to ℍ⊕ℍ, meaning an element is a pair of quaternions. So a representation is a pair of quaternionic vector spaces – or equivalently, a quaternionic vector space that’s been ‘split’ as the direct sum of two pieces. That’s what I mean by ‘split’ here.

Do you see the surprising pattern in the above chart? It has bilateral symmetry across the diagonal line from Cliff₃ to Cliff₇!

Why such a weird diagonal line? It’s because we’re doing things a bit wrong! The Clifford algebras aren’t just algebras: they are ‘superalgebras’, meaning that every element is a sum of two parts, which we can jazzily call the bosonic and the fermionic part, and multiplication obeys these rules:

bosonic × bosonic = bosonic
bosonic × fermionic = fermionic
fermionic × bosonic = fermonic
fermionic × fermionic = bosonic

These rules are motivated by pure math and also what happens in nature when you combine bosons and fermions into bigger particles.

How do we make the Clifford algebras into superalgebras? We just decree that the square roots of -1 we throw in are fermionic. In Cliff₂ this means that i and j are fermionic and k is bosonic. That may seem weird, but that’s because we’re getting the quaternions from studying 2-dimensional space, which is also a bit weird. (In 3d space it turns out that the quaternions are the bosonic part of Cliff₃, and this is closer to Hamilton’s original thoughts.)

Believe it or not, working with superalgebras and their super-representations takes our chart and rotates it a bit, so the weird diagonal line becomes a vertical line!

It’s super time!

A ‘super vector space’ is a vector space V that’s the direct sum of two subspaces:

V = V₀ ⊕ V₁

We call vectors in V₀ bosonic or ‘even’ and vectors in V₁ fermionic or ‘odd’. And a superalgebra can have a super-representation on a super vector space. It’s just like an ordinary representation except that when we let guys in our superalgebra act on guys in our superalgebra, we require these by now familiar rules:

bosonic × bosonic = bosonic
bosonic × fermionic = fermionic
fermionic × bosonic = fermonic
fermionic × fermionic = bosonic

The chart above shows the categories of super-representations of the Clifford algebras. Amazingly, it’s just like the previous chart rotated an eighth of a turn clockwise! Now the axis of symmetry is the vertical line!

This doesn’t explain yet why the symmetry exists in the first place, but it’s a step in the right direction.

As a quick sanity check, think about super-representations of Cliff₀. The chart claims these are ‘split real vector spaces’. Why is that true? Cliff₀ is just ℝ, with every element bosonic. So a super-representation of Cliff₀ is just a super-vector space where you can multiply by real numbers. But this is just a super-vector space. But a super-vector space is just the same as a split real vector space: it’s a vector space V split as a direct sum of two parts:

V = V₀ ⊕ V₁

which we call fermionic and bosonic. So yes, the category of super-representations of Cliff₀ is just the category of split real vector spaces!

So at least that case checks out.

The Lydian Dominant Scale

I got excited when I learned about the Lydian dominant scale:

1 2 3 #4 5 6 ♭7

and could sing it without much effort on the first try. I seem to be getting better at this sort of thing!

More importantly, it’s a really cool scale. Rob van Hal’s video explains what it sounds like and what you can do with it:

You can think of it as a blend of the Lydian mode, with its mystical raised 4th:

1 2 3 #4 5 6 7

and the Mixolydian, with its lowered 7th:

1 2 3 4 5 6 ♭7

But why does this image from van Hal’s video call the Mixolydian mode ‘dominant’? Probably because it contains the dominant 7th chord, where we take a major triad and extend it by playing a minor 7th on top:

1 3 5 ♭7

Indeed, you can get all the notes of the Lydian dominant scale from a chord where you stack the triad 2 #4 6 on top of a dominant 7th. Since we’re playing them an octave higher, we add seven to 2, #4, and 6 and get this:

1 3 5 ♭7 9 #11 13

This is a nice jazzy chord called a ‘dominant 13th sharp 11’. And if you play it in C, musicians might call it a ‘C13#11 chord’ because the dominant 7th is so prevalent that it’s taken for granted in this notation!

But if you play this chord you might leave out the 3 and 5, which are so taken for granted that actually playing them just muddies the chord with unnecessary notes. So you might just play this:

1 ♭7 9 #11 13

There are lots of other ways to play it.

Another way to think about the Lydian dominant scale is that it’s the 4th mode of melodic minor ascending. Melodic minor is a scale you play like this when you’re going up:

1 2 ♭3 4 5 6 7

If you start it on the 4 you get a Lydian dominant scale. Think about it!

Another name for the Lydian dominant scale is the ‘acoustic scale’. This is because you can get approximately get the notes of this scale by playing all the overtones of a single tone… and quitting at the right point. Read this for the details:

• Wikipedia, Acoustic scale.

To see how people apply the Lydian dominant scale in music, start with van Hal’s video. Then take it to the next level by watching Peter Martin and Adam Maness explain how to dominate the Lydian dominant:

For me, watching these guys talk about music is like watching Ed Witten talk about physics. I get bits here and there. It all goes by too fast. Still, it’s tremendously exciting!

They describe and illustrate several things you can do in jazz with the Lydian dominant scale. Adam Maness tries to go through topics systematically, from simple to complicated. But Peter Martin—the pianist for Dianne Reeves, and a very cool cat—interrupts and talks about fancier topics, so the end result is a bit disorganized.

Nonetheless I really benefit from hearing actual jazz theory experts talk about this stuff. At the very least, it gives me a useful sense of how crappy I am at music theory, and playing the piano. But not in a depressing way. It makes me want to improve!

If you want do jam in Lydian dominant, you can use this backing track:

If you don’t, listening to it is still a good way to absorb the vibe of Lydian dominant. There are tons of backing tracks like this on YouTube.

Kaktovik Numerals

 

I’m fascinated by the Inuit languages in Alaska, Canada and Greenland. There are many of these languages: they ring much of the Arctic Ocean. I just learned that they use a base 20 system for numbers, with a ‘sub-base’ of 5. That is, quantities are counted in scores (twenties) with intermediate numerals for 5, 10, and 15. This makes a lot of sense if you look at your fingers and toes.

But the Inuit didn’t have a written form of their number system until the early 1990s, when high school students in the town of Kaktovik, Alaska invented one! There were just nine students at this small school, and they all joined in.

They used 5 principles:

• Visual simplicity: The symbols should be easy to remember.

• Iconicity: There should be a clear relationship between the symbols and their meanings.

• Efficiency: It should be easy to write the symbols without lifting the pencil from the paper.

• Distinctiveness: There should be no confusion between this system and Arabic numerals.

• Aesthetics: They should be pleasing to look at.

They decided that the symbol for zero should look like crossed arms, meaning that nothing was being counted. So here’s what they came up with:

The students built base-20 abacuses. These were initially intended to help the conversion from decimal to base 20 and vice versa, but soon the students started using them to do arithmetic in base 20.

The upper section of their abacus has 3 beads in each column for the values of the sub-base of 5, while the lower section has 4 beads in each column for the remaining units.

The students discovered their new system made arithmetic easier than it was with Arabic numerals. Adding two digits together often gives a result that looks like the combination of the two digits!

The students also found that long division was more fun with Kaktovik numerals! They noticed visually interesting patterns. They discovered that they could keep track of intermediate steps with colored pencils.

And then something interesting happened: after the students of Kaktovik invented their new numerals, their scores on standardized math tests improved dramatically! Before, their average score was down in the 20th percentile. Afterwards, their scores shot up to above the national average.

Some argue that being able to work in both base 10 and base 20 was helpful—much like being bilingual. Another explanation is that having a written system of numbers that matched the local language was helpful.

But I suspect that even more important was the sheer process of developing their own system of numerals! Getting engaged in mathematics is so much better than learning it passively.

But this was just the start of the story. For more, read this:

• Amory Tillinghast-Raby, A number system invented by Inuit schoolchildren will make its Silicon Valley debut, Scientific American, April 10, 2023.

I’ll quote a bit:

At first students would convert their assigned math problems into Kaktovik numerals to do calculations, but middle school math classes in Kaktovik began teaching the numerals in equal measure with their Hindu-Arabic counterparts in 1997. Bartley reports that after a year of the students working fluently in both systems, scores on standardized math exams jumped from below the 20th percentile to “significantly above” the national average. And in the meantime, the board of education in the North Slope Borough’s district seat, Utqiagvik, passed a resolution that spread the numerals almost 500 miles along the Arctic coast. The system was even endorsed by the Inuit Circumpolar Council, which represents 180,000 Inuit across Alaska, Canada, Greenland and Russia.

But under the federal No Child Left Behind Act, from 2002 to 2015, schools faced severe sanctions—or even closure—for not meeting state standards, provoking a “scare” that some local educators say squeezed the Kaktovik numerals into a marginal role despite the system’s demonstrated educational impact. “Today the only place they’re really being used is in the Iñupiaq language classrooms,” says Chrisann Justice, the North Slope Borough’s Iñupiaq education department specialist. “We’re just blowing on the coal.”

Why is Scientific American talking about Kaktovik numerals just now? It’s because some linguists working with the Script Encoding Initiative at U.C. Berkeley recently got them added to Unicode! See here:

• Wikipedia, Kaktovik numerals (Unicode block).

For more on the cool mathematical properties of Kaktovik numerals, try this:

• Wikipedia, Kaktovik numerals.

On Mathstodon David Nash wrote:

I read the Wikipedia article about this numeral system and came across this long division example:

My nearly exact thought process while reading the caption:

Article “The divisor goes into the first two digits of the dividend one time, for a one in the quotient.”

Me “OK, sure, got it.”

Article “It fits into the next two digits (red) once if rotated…”

Me: “What do you mean, “rotated”?”

…

Me “holy *SHIT*, that’s totally genius”

I suspect there’s more to be said. Maybe someone has worked out more details somewhere?

The Kaktovik numerals were invented by high school students who spoke Iñupiaq:

• Wikipedia, Iñupiaq language.

As you can see, this endangered language is spoken in northern Alaska:

This map also shows other Inuit languages:

• Wikipedia, Inuit languages.

These are part of a larger group called Inuit-Yupik-Unangan languages:

• Wikipedia, Inuit-Yupik-Unangan languages.

Euler

 

It’s Leonhard Euler’s birthday today! He was born in Basel on April 15, 1707.

Euler was a relentlessly energetic mathematician, physicist, astronomer, geographer, logician and engineer who founded the subjects of graph theory and topology and made pioneering discoveries in analysis, number theory, mechanics, fluid dynamics, optics, and even music theory. When he lost sight in his right eye, he remarked

“Now I will have fewer distractions.”

and even when he became almost blind in his other eye he averaged one paper a week.

I love how he seized the day and explored all the possibilities available to him. Harnessing the tools of calculus, he came up with dozens of amazing formulas, like this:

But how do you find a formula like this???

It’s not magic: it uses a method called ‘Euler’s continued fraction formula’ which can produce many similar formulas. This method is not easy to guess, but you can check it using high-school algebra, especially if you don’t worry about convergence—which Euler didn’t, not much anyway. Then, to apply this method and get Euler’s formula for 4/π, you need to know some calculus.

Let’s see how it works!

Here is Euler’s continued fraction formula:

It’s a way to rewrite a sum

as a fraction. To check it, just try some small choices of n like 1 and 2 and 3. Use the rules of algebra to simplify the fraction at right. You’ll get the desired sum! With some thought, you should be able to see the pattern of why it works for any n.

There are probably better things to say about this, but I don’t know them. So, I’ll just show you how to use this trick to get Euler’s formula for 4/π.

There are many ways to apply Euler’s continued fraction formula. For example, we can start by remembering that

Then we apply the geometric series

to write

Integrating this from 0 to x, we get a nice infinite series for arctan(x).

To apply Euler’s continued fraction formula, we need to massage this series until it’s of the form

Luckily this is not hard to do:

Next, hit this formula for arctan(x) with Euler’s continued fraction formula!

The result looks pretty complicated, but we can simplify it a bit by multiplying the top and bottom of one fraction by 3, another by 5, and so on:

Next, to get something impressive and cover our tracks, let’s choose some very simple number x whose arctangent is also nice. Like x = 1. The arctangent of 1 is π/4, since a line of slope 1 has an angle of 45° from the horizontal. So taking x = 1, we’ll get a cool formula for π/4:

This was not a rigorous proof by today’s standards, since I didn’t investigate whether any of the series converge. But neither did Euler! He probably thought about it, but he didn’t let it slow him down.

You have to imagine Euler spending all day, every day, trying stuff like this.

For more fun examples of what you can do with Euler’s continued fraction formula, go here:

• Wikipedia, Euler’s continued fraction formula.

I think the moral of the story is that to achieve greatness, it helps to take what you’re good at and run wild with it: try everything you can, and let people know about the good stuff.

Happy Euler’s day!