GerardWestendorp

johncarlosbaez, 7 days ago (edited 7 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.

GerardWestendorp, 1 month ago to random

A rigidly foldable tessellation, based on the "square twist". Square twists are alternately connected to their mirror image.

Foldable tesselation

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

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

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

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

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

t² − x² − y² − z² = 1 and t > 0

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

a + bω

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

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

(1/n)

GerardWestendorp, 2 months ago to random

I was at the G4G (maths, puzzles, gadgets etc) conference this week. This was one of the “exchange gifts”. I don’t know the creator, if someone knows, please let me know.
It is a torus tiled by 5 squares. I was surprised this is possible, so I figured out a fundamental polygon. This also turns out to be a square, which is possible because ( 5 = 2^2 + 1^2) .

johncarlosbaez, 3 months ago to random

Today I got some questions about logic, like:

what do provability, decidability, consistency, and completeness mean?
how do we work with equality in set theory?

I don't know articles that explain topics like this to non-mathematicians, clearly and crisply, without becoming overlong or heavy with notation. Do you?

I looked around and found this "Introduction to First-Order Logic":

https://builds.openlogicproject.org/content/first-order-logic/introduction/introduction.pdf

but the very first sentence is

"You are probably familiar with first-order logic from your first introduction to formal logic."

which is basically a way of saying "fuck you - if you don't know this stuff already I won't explain it to you".

As a student I liked Boolos and Jeffrey's book "Computability and Logic":

http://alcom.ee.ntu.edu.tw/system/privatezone/uploads/Logic/20090928151927_George_S._Boolos,_John_P._Burgess,_Richard_C.

but that's more like a course than what I'm thinking of here: a collection of essays that explain different topics in plain English.

I also liked Hofstadter's "Gödel, Escher, Bach", but that's a massive quirky elaborate tale, not a simple clear explanation.

Wikipedia articles are packed with information but they aren't self-contained, clearly written essays. Articles in the Stanford Encyclopedia of Philosophy are better in some ways, but they often "show off" by including more advanced material.

Sigh....

johncarlosbaez, 3 months ago to random

So much of what we're suffering in the US - and elsewhere - is due to the capture of income by the rich. To quote:

"How big is this elephant? A staggering $50 trillion. That is how much the upward redistribution of income has cost American workers over the past several decades.

This is not some back-of-the-napkin approximation. According to a groundbreaking new working paper by Carter C. Price and Kathryn Edwards of the RAND Corporation, had the more equitable income distributions of the three decades following World War II (1945 through 1974) merely held steady, the aggregate annual income of Americans earning below the 90th percentile would have been $2.5 trillion higher in the year 2018 alone. That is an amount equal to nearly 12 percent of GDP—enough to more than double median income—enough to pay every single working American in the bottom nine deciles an additional $1,144 a month. Every month. Every single year.

Price and Edwards calculate that the cumulative tab for our four-decade-long experiment in radical inequality had grown to over $47 trillion from 1975 through 2018. At a recent pace of about $2.5 trillion a year, that number we estimate crossed the $50 trillion mark by early 2020. That’s $50 trillion that would have gone into the paychecks of working Americans had inequality held constant—$50 trillion that would have built a far larger and more prosperous economy—$50 trillion that would have enabled the vast majority of Americans to enter this pandemic far more healthy, resilient, and financially secure."

https://time.com/5888024/50-trillion-income-inequality-america/

johncarlosbaez, 4 months ago to random

Sometimes when Lisa and I make yoghurt, something goes wrong and the result resembles ricotta: it's bland rather than tart, and it's more grainy in texture. What makes this happen? Is some bacterium other than 𝐿𝑎𝑐𝑡𝑜𝑏𝑎𝑐𝑖𝑙𝑙𝑢𝑠 𝑑𝑒𝑙𝑏𝑟𝑢𝑒𝑐𝑘𝑖𝑖 𝑏𝑢𝑙𝑔𝑎𝑟𝑖𝑐𝑢𝑠 taking over? If so, which one(s)?

And more importantly, why? It can happen even if we add enough of the correct culture. My current hypothesis is that it happens when we keep the pot too warm. I suppose I should do experiments.... but maybe someone knows already!

This ricotta stuff is not bad tasting, but it's not like yoghurt so right now Lisa is using it to make pancakes.

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

Yesterday I asked: if you made an "atom" out of two neutrons bound together by gravity, how big would it be?

HUGE.

@lilacperegrine worked it out and got an orbital radius of 5.608 ⋅10²⁴ meters, or about 1/100th the radius of the observable universe. Their calculations are below.

Let me check. The Bohr radius of a hydrogen atom is r = ℏ²/me² where m is the mass of the electron, e is its charge and ℏ is Planck's constant. We can use the same formula with the force of gravity replacing the electromagnetic force, and the neutron mass replacing the electron mass.

Making the force weaker makes the atom larger in inverse proportion. Making the mass of the orbiting particle larger makes the atom smaller in inverse proportion.

The Bohr radius of a hydrogen atom is roughly 10^{-10} meters.

The force of gravity between two electrons is about 10⁴⁰ times weaker than the electromagnetic force between them. But the neutron mass is about 1000 times the electron mass, so the force of gravity between two neutrons about 1000 × 1000 times bigger than between two electrons. Thus, the force of gravity between two neutrons is roughly 10³⁴ times the electromagnetic force between a proton and electron.

But the mass of the orbiting particle is about 1000 times larger for our gravitational atom.

So, the radius of our gravitational atom is roughly 10^{-10} meters times 10³⁴ divided by 1000, which is 10²¹ meters.

This doesn't match @lilacperegrine's calculation. Now I can have fun figuring out why I'm off by a factor of 1000. [EDIT: it looks like my 10⁴⁰ should have been 10⁴³.]

But we agree that gravitational "atoms" made of neutrons would be huge. In reality they'd be much too fragile to exist.

johncarlosbaez, 8 months ago to random

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

Recently Quanta came out with an article explaining modular forms:

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

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

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

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

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

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

christianp, 9 months ago to random

Me, earlier today: "weird, this image was 1.6MB but when I made each dimension a quarter as big, it's only 0.1MB. That's smaller than I was expecting!"

:euclid: : :qed:

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

People already knew that a changing magnetic field makes the electric field curl around. But in 1861, Maxwell proposed that also a changing electric field makes the magnetic field curl around. This means that electric and magnetic fields can form waves, with each one changing the other. Maxwell computed the speed of these waves and got an answer close to the speed of light! The rest is history...

... in a way. Maxwell reached his conclusions using a strange mechanical model involving "molecular vortices". In this model magnetic fields are spinning vortices - the black hexagons here - while electric fields are the displacement of little particles between these vortices - the green circles.

It took him a few years to drop this model and simply write equations governing fields. And it took longer for Heaviside and others to take Maxwell's 20 equations, throw some out, and boil the rest down to 4 vector equations - the ones we now learn in school.

I want to understand Maxwell's molecular vortices because they're the key to how he guessed that a changing electric field makes the magnetic field curl. People already knew a lot about the electric and magnetic fields from experiments. But Maxwell's extra effect completed our theory of electromagnetism!

There are modern ways to see that this extra effect is necessary. Few people go back and try to understand Maxwell's original thinking. Here's one great exception:

https://royalsocietypublishing.org/doi/10.1098/rsta.2014.0473#d3e647

I also want to read Daniel M. Siegel's book "Innovation in Maxwell’s Electromagnetic Theory: Molecular Vortices, Displacement Current, and Light."

By the way, Maxwell's model correctly says E is a vector field while B is a pseudovector field!

johncarlosbaez, 1 year ago to random

Don't tell the kids, but sometimes mathematicians consider infinity as a number on the same footing as all the rest. This lets us curl up the complex plane into a sphere with ∞ as its north pole. 0 is at the south pole. Around the equator we have 1, -1, i and -i. These 6 points are the corners of an octahedron!

Now we can can create cool stuff like the 'Bolza curve'. Make up a polynomial that vanishes at 0, 1, -1, i and -i:

x(x-1)(x+1)(x-i)(x+i) = x(x² - 1)(x² + 1) = x(x⁴ -1) = x⁵−x

The equation y² = x⁵−x defines the Bolza curve. You can draw its real solutions but the fun starts when you consider complex solutions, even allowing infinity as a number.

Say you try to solve it for y. To do this, you just take the square root of both sides. Since most numbers have two square roots, you usually get two solutions. The only numbers that don't have two square roots are 0 and ∞. So you get just one solution when x⁵−x = 0, and this happens at five points:

x = 0, 1, -1, i, -i

But you also get just one solution when x⁵−x = ∞, and this happens when x = ∞. (Honest.)

So, if I were better at drawing, I could draw all the solutions (x,y) of y² = x⁵−x, and you'd see a surface with two sheets sitting over each point x on the sphere - except at the six corners of the octahedron, where the two merge into one!

This is the Bolza curve in all its glory. Alas, so far it seems to be visible only by the 'inner eye' of mathematicians - I don't see a picture online anywhere. But I wrote more about it here:

https://math.ucr.edu/home/baez/diary/january_2022.html

The moral is: it's important to learn the rules in math, follow them carefully... and then very carefully BREAK them to discover new things.