The physical motion of the billiards is two-dimensional, but wouldn’t the phase space be four-dimensional (since it tracks both position and momentum)?

I think the reason chaos needs at least three-dimensional phase space for continuous-time systems is that the orbits in phase space can’t intersect (which the paths of the billiards in real space obviously do).

Some further reading has led me to hyperchaos:

A hyperchaotic system is a dynamical system with a bounded attractor set, on which there are at least two positive Lyapunov exponents. Since on an attractor, the sum of Lyapunov exponents is non-positive, there must be at least one negative Lyapunov exponent. If the system has continuous time, then along the trajectory, the Lyapunov exponent is zero, and so the minimal number of dimensions in which continuous-time hyperchaos can occur is 4.

I edited/clarified my comment before I saw your reply—apparently the restriction to three dimensions only apples to continuous-time systems, which is what I had in mind when I posted the question.

Hmm… the book I was reading (Complex and Adaptive Dynamical Systems by Claudius Gros) actually said strange attractors only occur in three dimensions or higher, but I changed that to “chaos” in the title to fit the character limit—I thought they were essentially synonymous.

Here’s the quote:

Strange attractors can only occur in dynamical system of dimension three and higher, in one dimension fixpoints are the only possible attracting states and one needs at least two dimensions for limit cycles.

Edit: I see that Gros clarifies a few paragraphs later:

Chaos may arise in one dimensional maps … but continuous-time dynamical systems need to be at least three dimensional in order to show chaotic behavior.

I guess I was mentally thinking of continuous-time systems, but there was no room to fit that in the title.

Imagine you were making a ladder by sticking together a bunch of identical blocks, each of which was half a rung and the attached bit of rail. But imagine the rails aren’t perfectly straight—each one is a little bit skewed, so when you join the two halves of each rung the connected rails skew in opposite directions. When you add the next rung the skew of the rails will continue, but the rotational symmetry of the rung will keep it centered over the previous rung. The net result will be a skewed helix.

It might help to look at the actual structure of the molecule—you can see how the sides of the “ladder” are made of a lumpy sugar bit that has no reason to stack directly on top of itself. Also notice that the halves of the rungs vary in two ways: there are long and short pieces, and there are pieces that connect with two or three connections. These two variables give a total of four possible half-rung types, and each can joint with only one other type (long to short, short to long, two to two, and three to three).

Companies that are leading the market are already profiting from existing science, and new discoveries threaten to disrupt the market—it’s often in their interest to prevent new advances from coming to market even if they’d be the ones selling them. (Say, a cheap vaccine that prevents a disease they sell highly profitable drugs to treat.)

If they make these potentially disruptive advances privately, they can patent them to preemptively keep anyone else from introducing them. But if the advances were made using public funding, they have to make the case that whatever they do with it is in the public interest (which can conflict with their own).

The more important factor isn’t the funding or lack thereof, it’s the ability to control the results.

Because the US government is aligned with companies that exploit science for profit, and publicly-funded science is harder to exploit.

why is DNA in a helix?

The nucleotides that make up DNA have a natural angle that they attach together at. So they will stack in a helix unless that angle is exactly 90 degrees, and why would it be?

how come we can’t gene edit?

Most traits are affected by many genes governed by complex gene regulatory networks; and likewise, most genes affect many traits at once. That makes it difficult or impossible to change an isolated trait by making a few point edits to the genome (the exceptions are traits that arose recently due to novel mutations that haven’t yet found their way into deeper GRNs).

We naturally move through spacetime on geodesics, which could be defined as “the trajectory along which you don’t feel the effects of gravity or acceleration”.

I think the real difference is that fantasy (and science fantasy) has parallel sets of rules (natural vs supernatural), neither of which reduces to the other. It’s the irreducibility that’s critical, whether or not the rules are strictly spelled out.

I can’t see the content (except for a thumbnail).

Time follows geodesics on a curved spacetime manifold, so it’s linear locally but not necessarily globally. In theory a manifold with the right curvature could have geodesics that form loops (which would imply time travel), but there’s no practical way to cause such curvature.

Not just travel, but faster-than-light travel. Either one implies the other.

• The rate at which pathogens, predators, etc. evolve to exploit a species is dependent on the size of the species’s population—and once a pathogen anywhere in the world evolves to spread through the species, it can take out the whole population at once.

• The more different species fill the same niche around the world, the more likely it is that some of those species will successfully adapt to changing conditions. The local ecosystems with species that don’t adapt might collapse temporarily, but the successful alternatives can replace them and re-diversify.

So reduced biodiverity makes the remaining species and ecosystems more fragile—and species like humans that are dependent on all the remaining globally-distributed monospecies are as vulnerable as all of them combined.

Currently reading Hiron Ennes’ The Works of Vermin—it’s very reminiscent of Miéville, especially Perdido Street Station.

Yeah—Kuhn describes reading Aristotle’s Physics and being shocked at how nonsensical it was. Then he spent months going through it term by term, until everything finally clicked: Aristotle’s world has a fundamentally different ontology and causal structure that’s perfectly internally consistent, and in that world post-Galilean physics is wrong—as nonsensical as Aristotle’s physics initially seemed to Kuhn. And there’s no way to get from that world to this one without abandoning the former in toto.

In Kuhn’s view, paradigms are like parallel, mutually-incompatible worlds that happen to share the same surface-level phenomena.

the whole reason it was possible to experiment cheaply and come across this serendipity was because 9 months ago, faced with the choice to either do the bad easy thing or the good nothing, I chose to do the bad easy thing. The SQLite database worked! I understood how it worked, behind the scenes with its B-trees and its Full Text Search extension.

I always wonder what the tradeoffs are for using SQLite for personal projects… being a stepping-stone to rolling your own non-sql alternatives is one I hadn’t thought of.

RSS feeds seem like the obvious replacement—how is your vibe-coded solution better?

Consider the Lorentz boost to convert events from Alice’s reference frame to Jane’s. The time of an event (such as Bob’s current point in Alice’s frame) will differ in Jane’s frame by a factor of vx/c² (times the Lorentz factor). Since v (their velocity relative to each other) and x (the distance to Bob) are both nonzero, the times of events in Bob’s history will differ between Alice’s frame and Jane’s by a nonzero amount. And the time shift is proportional to x, which is why the distance to Bob matters.

Instead of Alice (aka Jane) turning around, say Alice and Jane are two non-accelerating observers moving toward and away from Bob, who pass each other four light years out. Would you agree that, at the moment they pass each other, Alice and Jane would see different points in Bob’s history as being simultaneous in their respective reference frames? It would be measurable even though no acceleration is occurring.

And it’s related to distance because the further they are from Bob, the greater the discrepancy in their calculations of Bob’s relative time will be.

You’re right that this isn’t how Alice/Jane would think of it in practice, because everyone would convert their own calculations into a stationary reference frame that would take their own time dilation into account. But that doesn’t illustrate OP’s question about the reference frame symmetry as well.