Yeah, that’s my understanding.

I guess one would not have 2D flat strange attractors like the Lorentz butterfly, but the chaotic region must then be at least 3D (two expanding directions + the time direction)

That’s how I understand it: the volume element surrounding any given point has to be contracting, but it can expand in some direction(s) if it has at least one other direction to contract in for compensation.

As for qualitative differences, I'm not really picturing what the difference would be—but trying to intuit the effect of extra parametric dimensions might be like trying to explain trichromancy to a color-blind person.

Would that “just” mean that along any given orbit there are two expanding directions? What sort of qualitative difference would that entail?

You are correct in the identification of the phase space for billiards.

Sensitivity is only one of three requests, there is also mixing and dense periodic orbits.

It is yet unknown how this effect is brought about. Here is a recent article about the topic that is interesting:

https://www.nytimes.com/2026/07/14/magazine/terminal-lucidity.html

https://archive.ph/20260717083237/https://www.nytimes.com/2026/07/14/magazine/terminal-lucidity.html

What you say about war is true. You can also attribute jet engines and rockets to Nazi Germany though, they made the most progress there during the war.

Thanks for referencing the [obligatory XKCD] (https://xkcd.com/1814)

Exactly. Physically, inductance and capacitance affect how quickly a circuit responds to a change in voltage.

A capacitor lets through current at first but pushes back as it fills up, with its energy stored in an electrical field.

An inductor doesn't like to suddenly change current, so it acts as if it has inertia where it won't let through current immediately upon a voltage being applied to it, and it won't immediately stop current flowing through it even if the voltage is taken off, because the energy being fed into the inductor gets stored as a magnetic field.

In a DC circuit, you can just map the changes to current in response to voltage changes, over time.

But in an AC circuit where the voltage doesn't stay still, the inductance and capacitance end up acting to shift the timing of the sinusoidal wave where the current and the voltage aren't timed in sync with each other.

So mathematically, because sinusoidal waves work well with phase angles and the math of imaginary numbers, it's easier to just think of the "resistance" as being a complex number with both a real and an imaginary component, and applying that math of complex numbers to the circuit behavior. And then we need new words to separate the real component (resistance), the imaginary component (reactance) and then both put together (impedance).

It's a mathematical model that very elegantly fits the physical reality of the time delay of how capacitance and inductance cause current to respond to changes in voltage. But it takes a step to explain why imaginary numbers are necessary to explain physical phenomena.

Electric current moving through a wire creates a magnetic field. Creating that magnetic field is work and takes energy. If it has to do a lot of work, that energy all gets stored in the magnetic field instead of in the load. Then if the electric current stops, the magnetic field collapses and dumps its energy back into the wire. That's impedance. It behaves like the electricity has been delayed and slowed down but it cannot be explained and understood as electricity alone like resistance or capacitance can, it is an electromagnetic phenomenon and can only roughly and imperfectly fit into an electricity-only model. The devil's in the details, and reactance is the devil's name.

The way I remember it from E&M (Phys 102) is that adding an impedance coil to a circuit causes it to resist changes in current flow. This is because the magnetic fields of the coil loops line up in such a way that they basically reinforce one another (due to spatial geometry). More or tighter loops means greater impedance.

It's easiest to think about a DC circuit of course. There, adding an impedance coil gives an effective "inertia" for current changes, i.e. if there's no current flowing at first then there will be a temporary resistance to starting it. Or vice versa if current is already flowing, then the coil will "induce" it to continue for a short time even after the EMF (e.g. battery) source is cut off.

I'm not very familiar with reactance but Wikipedia says it's an element of impedance (alongside reactance as you noted) that arises as opposition to alternating current. As I understand it affects how much current a given voltage can push (i.e. more reactance means a given voltage will yield less current). It's kind of like resistance but instead of converting the energy to heat, it converts it to a phase shift.

I'm far from an expert on any of this and may be way of base from what you're asking so I'm sure someone more knowledgeable could correct me or elaborate.

https://en.wikipedia.org/wiki/Electrical_impedance

First, genetics are only a blueprint for the structure of the body. It's not the final structuee. It's like asking why a building collapses when the blueprints look sound. The blueprints might have physically impossible information for the location it's being built on, or something could have interfered with the building process, or there might not be good quality materials to do the construction properly, etc. Identical twins often are not identical right out of the womb.

But let's assume you could accurately create the exact structure out of the womb. The initial physical body might have some effect on your ability to develop intelligence or not (e.g. blood flow to the brain, etc.), ability to generate and use certain hormones that affect mood, etc. This might make the two copies have the same disorders like ADHD, propensity towards violence, or other things, but is every person with ADHD exactly the same? So, no even that isnt enough to create the same personality.

Then there are environmental factors, nutrition, radiation, pollution, etc., all of which could alter the brain in different ways over the person's lifetime.

But personality has a lot to do with the neural connections and memories you create throughout your lifetime than just the brain you were born with and the physical environment you grew up in. Just look at twins raised in the same home. They can have significantly different personalities based on different experiences, different memories that were impactful, etc.

And personalities change over time because of all of this. So someone at age 30 and the same person at age 50 is often going to have somewhat different personalities based on all of this.

What might be possible, though it would take technology we may never have, would be to individually copy every single atom in a person's brain with something like a star trek style replicator, and implant that in a cloned body that was as close as possible to their body at the age you copied the brain. This would include every single neuron and neural pathway, every single drop of hormones, etc. And somehow do this instantly or at absolute zero temperatures and then instantly bring them to normal temperature to prevent any damage to the cells. Of course that would be absurdly difficult and is only science fiction, of likely to ever be reality. Too many variables, too many ridiculous technological dependencies, etc.

Other pollinators include flies, moths, butterflies, birds, bats, and rodents to name a few.

Mosquitos.

The question is about species going extinct. Despite all our laws, environmental devastation is still driving countless species to extinction, it’s just happening in other countries.

Algae. It makes more than half of our breathable oxygen, some studies say up to 70ish%.

Most important to humans?

Humans.

Really if all protections were removed, the soil, water, and air would be so badly polluted with heavy metals, radiation, etc., from large corporations dumping that we wouldn't have the ability to grow enough food, probably within a generation or so. Long before climate change dealt its deadly blow. And the wealthy who survive that won't have enough servants to keep them alive for too long.

Most important to the Earth?

That would be more interesting and it would be really difficult to pick just one. Probably something like Prochlorococcus which is one of a few ocean plankton that produce the most oxygen on Earth. That will die off much sooner than Humans from climate change as the ocean temperatures and acidity are rising more quickly than the air.

As a side note, one thing I think a lot of people miss from the movie Soylent Green,...

I had plural in mind, but single is good too if you have one.

The most important single species? I don't think there is one. Pretty much any one species disappearing would have its niche filled by another. There are a number of fragile ecosystems where that wouldn't happen and there would be a local collapse, but those would not have large-scale impacts.

If all environmental protections were removed, we'd see broad destruction of the environment across many species. Think broad deforestation, mining, waterway contamination, air pollution. It wouldn't be species, it'd be whole ecosystems killed.

It depends on how you define "environmental protections". If you're talking about actual laws (in the US, because that's what I have real world experience with) that establish management of hazardous waste (RCRA); force polluters to either remediate or pay to have their own pollution remediated (CERCLA); mandate clean air (CAA) and clean water (CWA); and ban manufacture, import, and production of toxic substances (TSCA), then those protections going away would be devastating. People as a whole forget how awful environmental conditions were 50 years ago before agencies were established and laws enacted. Obviously, these things are not universal, but the laws are sound and effective. The recent gutting of enforcement of those laws and prevention of implementing new protections will have lasting effects for generations.

Plankton

Primary producers (photo-synthesizers and chemo synthesizers). All other life gets energy from them.

Edited in response to OP's edit.

Better move fast, there’s not much time left.