@gregeganSF @duetosymmetry @_onionesque Luckily the names of the guys e_1, ... , e_7 are completely irrelevant (unless you're trying to talk to someone else), so you just need to remember the picture. It also doesn't matter if the arrows are circulating around clockwise or counterclockwise, as long as... (1/n)

@gregeganSF @duetosymmetry @_onionesque ...they all go around the same way. You might think you need to remember if the other arrows point from the triangle's corners to the midpoints of the opposite sides or vice versa. But you can get that mixed up too as long as you're consistent: it doesn't really matter! (2/n)

@gregeganSF @duetosymmetry @_onionesque So, if you're just trying to get an algebra isomorphic to the octonions instead of the "official standard octonions", you can forget a lot of stuff and still do fine. This is useful if you get stuck on a long bus ride with nothing to do. (3/n, n = 3)

What's "free energy"? I don't mean energy you get for free. I mean the concept from physics: roughly, energy that you can use to do work.

More precisely, free energy is energy that you can use to do work at constant temperature. But why the fine print?

(1/n)

More precisely, free energy is energy that you can use to do work at constant temperature. But why the fine print?

(1/n)

A red-hot rock has a lot of energy due to the random motion of its molecules. You can't do anything with this energy if the rock is in an equally red-hot furnace. You can if you put it in contact with something colder: you can boil water, make steam and drive a piston.

(2/n)

(2/n)

The thermal energy in a red-hot rock can't do work in an environment at the same temperature. So this energy is not "free energy".

But if the rock is moving, it has "free energy". You can do work with this energy - even in an environment at the same temperature!

(3/n)

But if the rock is moving, it has "free energy". You can do work with this energy - even in an environment at the same temperature!

(3/n)

A function from a finite set to itself consists of cycles with trees attached.

For a random function from a huge n-element set to itself, there are log(n)/2 cycles on average. sqrt(πn/2) points lie on cycles, on average. n/e points lie on leaves of trees, on average. (1/n)

For a random function from a huge n-element set to itself, there are log(n)/2 cycles on average. sqrt(πn/2) points lie on cycles, on average. n/e points lie on leaves of trees, on average. (1/n)

These days I'm wanting to know what a typical map from a large set to itself looks like.

Such a simple question! But it's actually many questions, and answering them uses a mix of category theory, combinatorics and complex analysis. (2/n)

Such a simple question! But it's actually many questions, and answering them uses a mix of category theory, combinatorics and complex analysis. (2/n)

For a great introduction to this exciting little corner of mathematics, try this:

Philippe Flajolet and Andrew M. Odlyzko, Random mapping statistics, hal.inria.fr/inria-00075445…

Guaranteed hours of pleasure.

(3/n)

Philippe Flajolet and Andrew M. Odlyzko, Random mapping statistics, hal.inria.fr/inria-00075445…

Guaranteed hours of pleasure.

(3/n)

I'm falling in love with random permutations.

The average length of the longest cycle in a random permutation of a huge n-element set approaches the "Golomb-Dickman constant" times n.

(1/n)

The average length of the longest cycle in a random permutation of a huge n-element set approaches the "Golomb-Dickman constant" times n.

(1/n)

The Golomb-Dickman constant also shows up in number theory... in a very similar way!

If you randomly choose a huge n digit integer, the average number of digits of its largest prime factor is asymptotically equal to the Golomb-Dickman constant times n.

(2/n)

If you randomly choose a huge n digit integer, the average number of digits of its largest prime factor is asymptotically equal to the Golomb-Dickman constant times n.

(2/n)

So, there's a connection between prime factorizations and random permutations!

You can read more about this in Jeffrey Lagarias' paper about Euler's constant:

arxiv.org/abs/1303.1856

The Golomb-Dickson constant seems to be a relative of Euler's constant.

(3/n)

You can read more about this in Jeffrey Lagarias' paper about Euler's constant:

arxiv.org/abs/1303.1856

The Golomb-Dickson constant seems to be a relative of Euler's constant.

(3/n)

Pluto is a spooky planet so it makes sense that this huge dark plain is named after Cthulhu, the malevolent entity hibernating in an underwater city in the Pacific in Lovecraft's fiction.

There's also a dark region called Balrog, after the monster in Lord of the Rings.

(1/n)

There's also a dark region called Balrog, after the monster in Lord of the Rings.

(1/n)

Lovecraft wrote about Pluto:

"Yuggoth... is a strange dark orb at the very rim of our solar system... There are mighty cities on Yuggoth—great tiers of terraced towers built of black stone... The sun shines there no brighter than a star, but the beings need no light."

(2/n)

"Yuggoth... is a strange dark orb at the very rim of our solar system... There are mighty cities on Yuggoth—great tiers of terraced towers built of black stone... The sun shines there no brighter than a star, but the beings need no light."

(2/n)

The famous mathematician James Sylvester, born in 1814, got in lots of trouble. He entered University College London at age 14. But after just five months, he was accused of threatening a fellow student with a knife in the dining hall!

(1/n)

(1/n)

His parents took him out of college and waited for him to grow up a bit more. He began studies in Cambridge at 17. Despite being ill for 2 years, he came in second in the big math exam called the tripos. But he couldn't get a degree... because he was Jewish.

(2/n)

(2/n)

At age 24 he became a professor at University College London. At 27 he got his BA and MA in mathematics. In the same year he moved to the United States, to become a professor of mathematics at the University of Virginia. But his troubles weren't over.

(3/n)

(3/n)

Take the real numbers. Throw in some square roots of +1 and some square roots of -1, all anticommuting. You get a Clifford algebra.

Clifford algebras are important in geometry and physics - we need them to understand spin! They also display some amazing patterns.

(1/n)

Clifford algebras are important in geometry and physics - we need them to understand spin! They also display some amazing patterns.

(1/n)

How many patterns can you see in this table?

The table stops here because if you throw 8 more square roots of +1 into a Clifford algebra, you get 16x16 matrices with entries in that Clifford algebra. This is also true if throw in 8 more square roots of -1!

(2/n)

The table stops here because if you throw 8 more square roots of +1 into a Clifford algebra, you get 16x16 matrices with entries in that Clifford algebra. This is also true if throw in 8 more square roots of -1!

(2/n)

To learn more about Clifford algebras, go here:

en.wikipedia.org/wiki/Clifford_…

To see how they're connected to normed division algebras, go here:

math.ucr.edu/home/baez/octo…

Here all that matters is those generated by square roots of -1.

(3/n, n = 3)

en.wikipedia.org/wiki/Clifford_…

To see how they're connected to normed division algebras, go here:

math.ucr.edu/home/baez/octo…

Here all that matters is those generated by square roots of -1.

(3/n, n = 3)

"Screw theory" was invented by a guy named Ball. There should be a joke in there somewhere. But what it is it?

For starters, any rigid motion of 3d Euclidean space has a "screw axis" - a line mapped to itself. We translate along this axis, and rotate about it. (1/n)

For starters, any rigid motion of 3d Euclidean space has a "screw axis" - a line mapped to itself. We translate along this axis, and rotate about it. (1/n)

Screw theory is about the *Euclidean group*: the group of rigid motions of Euclidean space. A *screw* is an element of the Lie algebra of this group. It's a 6d vector built from a pair of 3d vectors, an infinitesimal translation and an infinitesimal rotation. (2/n)

An object moving through space and rotating has a velocity and an angular velocity. These combine to form a screw.

When you push on this object you exert a force and a torque on it. These also combine to form a screw. (3/n)

When you push on this object you exert a force and a torque on it. These also combine to form a screw. (3/n)

To get the complex numbers, you take the real numbers and throw in a new number i that squares to -1. But other alternatives are also interesting! Different choices are connected to geometry in different ways.

(1/n)

(1/n)

A hydrogen molecule does *not* look like this! I only just now realized how misleading this picture is.

The electron wavefunctions do *not* form two separate blobs. In fact a hydrogen molecule is almost round! It looks like this....

(1/n)

The electron wavefunctions do *not* form two separate blobs. In fact a hydrogen molecule is almost round! It looks like this....

(1/n)

Here's a better picture of a hydrogen molecule. It's almost round!

The color and brightness shows how likely you are to find an electron per unit volume. For details on what the colors mean, go here:

phelafel.technion.ac.il/~orcohen/h2.ht…

Why does this matter?

(2/n)

The color and brightness shows how likely you are to find an electron per unit volume. For details on what the colors mean, go here:

phelafel.technion.ac.il/~orcohen/h2.ht…

Why does this matter?

(2/n)

Here's *one* reason it matters that hydrogen molecules are almost round. When you freeze hydrogen at low pressure it forms a low-density crystal. Since they're almost round and far apart, each molecule can rotate independently!

(3/n)

(3/n)

It's the strongest known acid - so acidic it can't be prepared in liquid form. It was one of the first compounds formed in the Universe. It's also the lightest ion made of two different kinds of atoms.

Helium hydride! It was first seen in outer space on April 2019.

(1/n)

Helium hydride! It was first seen in outer space on April 2019.

(1/n)

Helium hydride looks like a laugh with a plus sign: HeH⁺.

The cool part, to me, is that extra helium atoms can attach to HeH⁺ to form larger clusters such as He₂H⁺, He₃H⁺, He₄H⁺, He₅H⁺ and He₆H⁺. Hexahelium hydride is especially stable.

(2/n)

The cool part, to me, is that extra helium atoms can attach to HeH⁺ to form larger clusters such as He₂H⁺, He₃H⁺, He₄H⁺, He₅H⁺ and He₆H⁺. Hexahelium hydride is especially stable.

(2/n)

A narrow speciality: chemistry with just hydrogen and helium! Dihelium hydride can be formed from a dihelium ion and a hydrogen molecule:

He⁺₂ + H₂ → He₂H⁺ + H

People have even studied He₂H⁺⁺ and He₂H⁺⁺⁺.

Fun stuff:

en.wikipedia.org/wiki/Helium_hy…

(3/n)

He⁺₂ + H₂ → He₂H⁺ + H

People have even studied He₂H⁺⁺ and He₂H⁺⁺⁺.

Fun stuff:

en.wikipedia.org/wiki/Helium_hy…

(3/n)

A Petri net describes how various kinds of things (a,b,c,d,e here) can turn into others via "transitions" (shown in blue). I've been thinking about them a lot.

But I just learned something new: a Petri net gives a quantale! "Quantales" were invented in quantum logic.

(1/n)

But I just learned something new: a Petri net gives a quantale! "Quantales" were invented in quantum logic.

(1/n)

Any Petri net gives a commutative monoidal category, where the objects are "markings" of the Petri net - that is, bunches of dots saying how many things of each kind you have. Morphisms say how a bunch of things can turn into another bunch of things.

(2/n)

(2/n)

If we don't care *how* a bunch of things can turn into another bunch, just *whether*, we can take our category and turn it into a poset. So, any Petri net gives a commutative monoidal poset.

But from this we can get a commutative quantale!

(3/n)

johncarlosbaez.wordpress.com/2019/10/06/qua…

But from this we can get a commutative quantale!

(3/n)

johncarlosbaez.wordpress.com/2019/10/06/qua…

@robinhouston "The current that flows into each node must equal the current that flows out, which corresponds to the fact that the squares above and below a shared horizontal segment have the same total width."

But what corresponds to them being *squares*? (1/n)

But what corresponds to them being *squares*? (1/n)

@robinhouston Any way of dissecting a big rectangle into rectangles gives a solution of Kirchhoff's current law: think of rectangle widths as currents; drawing any horizontal line through the big rectangle gives a bunch of widths ("currents") that sum to the same total current. (2/n)

@robinhouston So, given any dissection of a rectangle into rectangles I think I can get a graph with edges labelled by "currents" that obeys Kirchhoff's current law. So far the *heights* of the rectangles plays no role! I want to see how that enters the game. Kirchoff's voltage law? (3/n)

When it first really sank in that I would die, I started crying. My mom consoled me, saying it would be just like going to sleep.

Mostly I don't fear death too much. Since age 40 my finite time span is always on my mind. But I just try to do lots of good stuff.

(1/n)

Mostly I don't fear death too much. Since age 40 my finite time span is always on my mind. But I just try to do lots of good stuff.

(1/n)

When my wife had some medical problems a few years ago, I started worrying about "managing the decline" - losing our good health, losing our mental acuity. That will clearly be the biggest challenge to come... sometimes terrifying.

But that's different than death.

(2/n)

But that's different than death.

(2/n)

Sometimes when I'm on an airplane and it plunges a bit due to turbulence, I get really scared. My body floods with adrenaline and I start sweating.

This is some sort of animal reaction to a threat. It seems different from the metaphysical fear of no longer existing.

(3/n)

This is some sort of animal reaction to a threat. It seems different from the metaphysical fear of no longer existing.

(3/n)

1/3 of meteorites hitting Earth today come from one source: the "L-chondrite parent body", an asteroid ~100 kilometers across that was smashed in an impact 468 million years ago. This was biggest smash-up in the last 3 billion years!

How did it affect life on Earth?

(1/n)

How did it affect life on Earth?

(1/n)

Back in the 1800s, physicists noticed that an electron should get extra mass, and extra energy, from its electric field. Some of the very best - Heaviside, Thomson and Lorentz - calculated the relation between this energy and this mass, and they got E = (3/4)mc².

(1/n)

(1/n)

What were they doing wrong? It's an interesting mistake.

First, they assumed the electron was a little sphere of charge. Why? In their calculations, the electron were a point, the energy in its electric field would be infinite.

(2/n)

First, they assumed the electron was a little sphere of charge. Why? In their calculations, the electron were a point, the energy in its electric field would be infinite.

(2/n)

Assuming the electron was a tiny sphere of charge, they could compute the total energy in its electric field. They could also work out how much extra force it takes to accelerate an electron due to this electric field. That gives it an extra mass using F = ma.

(3/n)

(3/n)

A group is a set with a way to "add" elements that obeys (x+y)+z = x+(y+z), with an element 0 obeying 0+x=x=x+0, where every element x has an element -x with x + -x = -x + x = 0.

Classifying finite groups is really hard. And there are some surprises!

(1/n)

Classifying finite groups is really hard. And there are some surprises!

(1/n)

The complex numbers together with infinity form a sphere called the "Riemann sphere".

The 6 simplest numbers on this sphere lie at the north pole, the south pole, the east pole, the west pole, the front pole and the back pole. 🙃 They're the corners of an octahedron!

(1/n)

The 6 simplest numbers on this sphere lie at the north pole, the south pole, the east pole, the west pole, the front pole and the back pole. 🙃 They're the corners of an octahedron!

(1/n)

On the Earth, let's say the "front pole" is where the prime meridian meets the equator at 0°N 0°E. It's called Null Island, but there's just a buoy there. You can see it here:

en.wikipedia.org/wiki/Null_Isla…

Where's the back pole, the east pole and the west pole?

(2/n)

en.wikipedia.org/wiki/Null_Isla…

Where's the back pole, the east pole and the west pole?

(2/n)

The Antikythera mechanism, found undersea in the Mediterranean, dates to roughly 60-200 BC. It's a full-fledged analogue computer! It had at least 30 gears and could predict eclipses, even modelling changes in the Moon's speed as it orbits the Earth.

(1/n)

(1/n)

The Riemann Hypothesis is one of the most important unsolved math problems. @rperezmarco explains how *not* to prove the Riemann Hypothesis:

1) Don't expect simple proofs to ever work. It would be very naive to think otherwise.

(cont.)

1) Don't expect simple proofs to ever work. It would be very naive to think otherwise.

(cont.)

@rperezmarco 2) Don't work on it unless you have very novel and powerful ideas. Many of the best mathematicians of all times have failed. Something more than existing techniques and tools is needed. You need a good new idea. Most of what you believe is a good new idea is not.

(cont.)

(cont.)

@rperezmarco 3) Don't work on it without a clear goal. You must first decide if you believe the conjecture or not. There is no point in trying to prove the conjecture one day and trying to disprove it the next day. A clear goal is a source of strength that is needed.

(cont.)

(cont.)

Riemann came up with a formula that counts the primes < x. It's a sum of waves, one for each nontrivial zero of the Riemann zeta function. Here's the sum of the first k waves. There are steps at the primes, but note the "glitches" at prime powers: 4, 8, 9, etc. (1/n)

π(x) is the number of primes < x. Li(x) is a good estimate: it's the integral from 2 to x of 1/log(t), where log means the natural logarithm here.

Notice that π(x) < Li(x) in this picture. But that stops being true around x = 10^316. They cross infinitely often! (2/n)

Notice that π(x) < Li(x) in this picture. But that stops being true around x = 10^316. They cross infinitely often! (2/n)

The Prime Number Theorem says the ratio π(x)/Li(x) approaches 1 as x → ∞.

Riemann's formula for π(x) starts with Li(x) and adds corrections. All but two of these corrections wiggle up and down (but are not sine waves). (3/n)

Riemann's formula for π(x) starts with Li(x) and adds corrections. All but two of these corrections wiggle up and down (but are not sine waves). (3/n)

Suppose we take a set X and freely start multiplying its elements in a commutative and associative way. For example if X = {x,y} we get things like

x

xx

xy = yx

yy

xxx

xxy = xyx = yxx

xyy = yxy = yyx

yyy

and so on. (1/n)

x

xx

xy = yx

yy

xxx

xxy = xyx = yxx

xyy = yxy = yyx

yyy

and so on. (1/n)

Let's include an identity for multiplication, 1. Then we get the "free commutative monoid" on X. A "monoid" is a set with an associative multiplication and identity 1.

Example: the free commutative monoid on the set of prime numbers is the set of positive integers!

(2/n)

Example: the free commutative monoid on the set of prime numbers is the set of positive integers!

(2/n)

An element of the free commutative monoid on X is the same as an unordered n-tuple of elements of X, where n = 0,1,2,3...

For example xxy = xyx = yxx is the unordered triple [x,x,y]. This is not a set, since we count x twice! Sometimes it's called a "multiset".

(3/n)

For example xxy = xyx = yxx is the unordered triple [x,x,y]. This is not a set, since we count x twice! Sometimes it's called a "multiset".

(3/n)

Is the limit of a sequence of continuous functions again continuous?

As @panlepan shows, the functions xⁿ are all continuous for x in the interval [0,1]. For each x in that interval, xⁿ converges as n → ∞. But the limit is a function that's not continuous!

(cont)

As @panlepan shows, the functions xⁿ are all continuous for x in the interval [0,1]. For each x in that interval, xⁿ converges as n → ∞. But the limit is a function that's not continuous!

(cont)

@panlepan The problem is that while the functions xⁿ converge "pointwise", they don't converge "uniformly". That is, they take longer and longer to converge to zero as x gets closer and closer to 1 but still stays < 1.

@panlepan My advisor's advisor's advisor's advisor's advisor's advisor, Karl Weierstrass, invented the concept of uniform convergence around 1841, so he could prove this theorem:

The limit of a uniformly convergent sequence of continuous functions is continuous.

en.wikipedia.org/wiki/Uniform_c…

The limit of a uniformly convergent sequence of continuous functions is continuous.

en.wikipedia.org/wiki/Uniform_c…

Why are Penrose tiling so fascinating? One reason is that no pattern with 5-fold symmetry in the plane can repeat in a periodic way.

But why is this? It's because no lattice in the plane can contain a regular pentagon.

And why is that?

(cont)

But why is this? It's because no lattice in the plane can contain a regular pentagon.

And why is that?

(cont)

@JDHamkins gave a nice proof that no *square* lattice can contain a regular pentagon. If it did, find a smallest possible example. Rotate its edges 90 degrees as shown. You get new edges going between lattice points... and they form a smaller regular pentagon!

(cont)

(cont)

@JDHamkins But why can *no* lattice have vertices forming a regular pentagon?

A "lattice" is a discrete set of vectors in the plane such that if two vectors are in the set, so are their sum and difference. So if 3 vertices of a parallelogram are in a lattice, so is the 4th.

(cont)

A "lattice" is a discrete set of vectors in the plane such that if two vectors are in the set, so are their sum and difference. So if 3 vertices of a parallelogram are in a lattice, so is the 4th.

(cont)

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