I recently learnt from John Baez on Mastodon that there’s a beautiful honeycomb in hyperbolic 3-space that’s related to the Eisenstein integers: complex numbers of the form a + b ω, where a, b are integers and ω is a complex cube root of 1.

To see how this works, start with the hyperboloid model of hyperbolic 3-space: the set of all points (t,x,y,z) in space-time that lie on the hyperboloid:

–t^2 + x^2 + y^2 + z^2 = –1.

But it’s often convenient to repackage these vectors as 2×2 Hermitian matrices:

t+z x-iy
x+iy t-z

with determinant 1.

If we take

[1/15] What does quantum mechanics say about two particles that are “entangled”? In this thread, I’ll sketch an example, describing two entangled photons.

First, a quick summary of how QM describes the polarisation of a single photon. If a photon is travelling in some direction [2/15] x, pick any two mutually orthogonal directions to x, and call them y and z. If we prepare a photon by requiring it to have passed through a polarising filter aligned in the y or z direction, we describe its quantum state with a vector in a 2-dimnsional complex vector space

[1/10] Here’s an example of how to calculate the difference between quantum and classical correlations — of the kind that led to the 2022 Nobel Prize — with a few lines of high school mathematics.

Let’s start with a simple inequality:

|x + y + z + ... | ≤ |x| + |y| + |z| + ... [2/10] This just says that if we add up some numbers, then take the absolute value, the result will never be greater than adding up the absolute values of the individual numbers.

But we calculate averages by adding up numbers, and then dividing by how many numbers there are.

[1/16] You might have heard it claimed that the temperature of a system is an “emergent” property that only makes sense if the system has a large number of degrees of freedom. This isn’t true!

There are certainly things that can be said about the temperature of a system that are [2/16] only approximately true, in general, but become *vastly* better approximations for systems with many degrees of freedom. But temperature itself is perfectly well defined, regardless.

In classical mechanics, suppose we have a system with f degrees of freedom whose total

[1/5] You and your friend separately shuffle two decks of 52 cards, then you go through them together, with both of you revealing successive cards in the shuffled order.

On average, how many times would you expect both of you to reveal exactly the same card? [2/5] To be honest, I had no intuition about this, but I recently learned the answer from @Caleb_Speak.

The average number of matches is exactly 1, independent of the number of cards in the deck!

Here’s a proof by induction.

Clearly for n=1, you must match for that one card.

I thought I'd look at a numerical solution (related to) this fun problem by @gravity_levity.

If we set g=1, I think the relevant PDE, where tension in the string is proportional to z measured from the bottom, is

∂_z(z ∂_z u(z,t)) - ∂_{t,t} u(z,t) = 0

https://twitter.com/gravity_levity/status/1534022485418643457

If we make the initial conditions a pulse near the bottom of the string, with the string instantaneously at rest, and we have the bottom of the string free and the top fixed, the solution looks like this (at regularly spaced time intervals):

[1/5] The LRL vector, which points along the axis of the orbit of a particle subject to an inverse-square force, can be shown to be constant by many different methods.

In this page, I show a very easy way …. and also how to do it with Noether’s Theorem:

gregegan.net/SCIENCE/LRL/LR… [2/5] To use Noether’s Theorem, you start with a formula for three vectors you can use to displace points from their original trajectory:

X_s = ½ (2 x_s v – v_s x – (x · v) e_s)
s=1,2,3

If you shift the trajectory this way, the total energy of the particle is unchanged, while

[1/9] Me: To see how gravity/electrostatics works in multiply connected space, pretend there are layers of charges and dipoles creating discontinuities.

Sommerfeld: No, let’s complexify one coordinate of the source and replace it with a contour integral!

https://twitter.com/gregeganSF/status/1435021713071243264

[2/9] Me: Well … you did get 84 Nobel nominations, help found quantum mechanics, and mentor an entire generation of the Golden Age of 20th century physics, so I should probably listen to your advice.

Sommerfeld came up with this very cool technique when he was only about 29!

[1/8] Take N copies of Euclidean space and glue them together along a disk, so that passing through the disk from one side in region j takes you to region j+1, but passing through it from the other side takes you to region j–1 (mod N).

How does *gravity* pass through the disk? [2/8] If we have a point mass in one region, what is its gravitational field across *all* regions?

In the Newtonian approximation, gravity is just like electrostatics. So, we can ask the same question for a point charge.

One way to answer this is to think about the sudden jump

[1/4] Take a de Bruijn pentagrid: 5 sets of uniformly spaced parallel lines, whose directions form a 10-pointed star.

Treat the intersections as vertices in a graph and the lines between them as the graph’s edges.

This infinite graph is easy to 3-colour: pick a point P not [2/4] equidistant from any pair of vertices, then choose colours for the vertices in order of their increasing distance from P.

Every vertex has 4 neighbours on two grid lines, but at most 2 of its neighbours will lie closer to P, so you can always assign it a different colour.

[1/2] Suppose you want to find the shortest path through the curved space near a black hole.

You have 2 towers protruding from a shell that encloses the hole. Light is bent by the curved spacetime … so can you just trace the light [red] that shines from one tower to the other? [2/2] No! The shortest path between the towers is the spatial geodesic shown in blue.

Light follows a geodesic through 4D spacetime, not 3D space. And the geodesics through curved space are *not*, in general, the paths that light follows.

[1/4] In Newtonian physics, a squad of aerobatic planes could, in principle, stay locked in a fixed formation while following any flight plan: they just all need to move the same way.

In special relativity, apart from staying still, there are only 4 special classes of … [2/4] “Born rigid motion” where all the spaceships in a flotilla measure the same constant distance between each other.

These flight plans can be found by applying “exponential curves” through the Poincaré group to an initial set of events, to sweep out the ships’ world lines.

[1/3] Any proper Euclidean transformation of R^3 (one that doesn’t involve reflections) can be thought of as:

• a rotation by an angle θ around an axis through the origin, parallel to some unit vector n

followed by

• translation by a vector v.

But in 1763, Giulio Mozzi … [2/3] … figured out that if the axis of rotation is *not* required to pass through the origin, you can always rewrite this as a “screw rotation”, where the translation occurs entirely along the axis of rotation, like the motion of a screw as you turn it.

en.wikipedia.org/wiki/Chasles%2…

[1/4] “Frame dragging” by a rotating black hole is a very cool phenomenon — and less arcane than it’s often portrayed to be.

Suppose you woke up in some kind of space habitat, with a dome that showed you a sky full of stars that remained perfectly still. You feel 1 gee’s weight. [2/4] You toss a ball straight up. As it rises, it veers sideways, and it lands over to your left — as if the habitat were spinning along an axis that passed through the floor, and the Coriolis force sent the ball askew.

But … the stars aren’t moving!

The only explanation is

[1/4] Sometimes you do need to go backwards to go forwards.

Suppose you want to match position *and* velocity with a constantly accelerating car (the blue curve), ASAP, in a car that can accelerate (or decelerate) up to twice as fast, and starts from rest either ahead or behind. [2/4] I looked at the more general problem in two dimensions in another thread:

https://twitter.com/gregeganSF/status/1218322842963460096

But the 1D case is easier, and helps show why, if you have a head start, it makes sense to go *backwards* initially.

If you start by assuming that the quickest interception

[1/8] In the 1978 rock-opera of Faust, by Richard O’Brien and Tom Stoppard, there’s a wonderful duet between Faust and Mephistopheles where they’re negotiating the contract for Faust’s soul.

The Devil wants Faust to toss a 6-sided die every hour, and if the last 6 numbers seen [2/8] are 6 5 4 3 2 1, his soul must descend to Hell.

Faust says, sure, but then if the numbers:

1 2 3 4 5 6

ever come up, his soul should ascend to Heaven.

Mephisto is having none of that, so they haggle. Faust argues that the infernal sequence should be:

6 6 6 6 6 6

What these wondrous numbers giveth with one hand, they taketh with the other.

— Leonardo Bigollo Pisano, Liber Abaci (1202) Consider the sets K_n of coin tosses that end with HH, but contain no other pair of consecutive heads.

We can modify any such sequence of tosses by:

• (blah)HH → (blah)tHH

or

• (blah)HH → (blah)HtHH

This recursively gives us all of them, and shows why:

|K_n| = F_{n-1}

[1/2] Newton famously proved that, throughout the interior of a perfectly spherical shell of uniform density, you would feel no gravitational force at all.

But what would you experience inside a hollow *prolate ellipsoid* (one with two short equal axes and one longer axis)? [2/2] There is no net force at the exact centre — but the equilibrium there is unstable, so given a small disturbance you will fall “upwards” towards the smaller, circular “equator” of the shell.

The images show the potential for two slices through the interior.

[1/5] Here’s a fun relativistic thought experiment.

In your coordinates, a particle of rest mass 1 has velocity v_x=0.5, v_y=0.5, in units where c=1.

Its mass-energy is 1/√(1–0.5^2–0.5^2) =√2.

You bounce a laser pulse off it, aimed along the y-axis, increasing v_y to 0.6. [2/5] You’ve contributed a change in momentum solely in the y direction, and you’ve caused the particle to speed up along the y-axis.

But by increasing its total speed, haven’t you increased its mass-energy—and so increased its momentum in the x direction?

How is it conserved?

A Lithuanian nun is searching a junkyard in Alberta for spare parts for her soul.

#MagicRealismNonBot A 17th century caliph covets a clockwork bird that only eats parables.

#MagicRealismNonBot

[1/4] You’re in interstellar space, and your radio is stuck on one band.

Can you head for your destination (yellow +) without missing a transmission from home (white dot) due to Doppler shift?

Yes — if the initial angle, θ, between home and your destination is greater than 90°. [2/4] If θ>90°, you can always find a velocity such that the signal from home is unshifted (then turn the antenna to keep it aimed at home).

But if θ>90°, why does the yellow + appear in the same hemisphere?

Because relativistic aberration shifts it from its initial bearing.