More then 3 planets.

Yeah. The thing that made me “get” quaternions was thinking about clocks. The hands move around in a 2d plane. You can represent the tips position with just x,y. However the axis that they rotate around is the z axis.

To do a n dimensional rotation you need a n+1 dimensional axis. So to do a 3D rotation you need a 4D axis. This is bassicly a quat.

You can use trig to get there in parts but it requires you to be careful to keep your planes distinct. If your planes get parallel you get gimbal lock. This never happens when working with quats.

No. It’s more what the previous poster said about encoding rotation. It’s just not a xyz axes. It’s current, charge, flux as axes. The trig is how you collapse the 3d system into a 2d or 1d projection. You lose some information but it’s more useful from a spefic reference.

Without complex numbers you can’t properly represent the information.

Yeah a good friend of mine has a PHD in math. I asked him how he would describe math. His answer was a set of rules for dealing with numbers. What’s important is how useful the rules are.

Yup. When you have a circuit that is not purely resistive the inductive or capacitive load causes the voltage and current to not be in phase. It looks like ohms law is being violated. However the missing part of the energy is in the imaginary component to be returned latter.

Fair point.

I get the joke but Auschwitz was in Poland. They were careful to keep all the concentration camps out of Germany.

Tesla is not private. He borrowed a huge amount against his Tesla shares to buy twitter. That’s one of the reasons he wanted that giant bonus. He needs the money to keep paying off the loans.

Which is odd that some people care so much about this. Games Workshop used to make female space marine models but stopped because they didn’t sell well.

My head canon is that with all the hormones to boost mucles, organs changed out, bones merged with ceramics, extra heart, special space marine reproductive organ in the chest, black carapace, acid spit, memory eating organ, etc. Doesn’t matter if you start with Judy Garland or Arnold Schwarzenegger you get Lou Ferrigno at the end.

That’s my thought. I think his plauges spread suffering and despair. Tempting you to join him as a follower making you no longer mind all of the suffering.

Until the show was banned from flying the helicopter and had to use recycled footage.

The thing is countries can change. Russia was ill equipped to fight a war against a prepared equipped country. Supplies were missing because people sold off supplies they thought were never gong to be needed. Now they know they need that equipment and the countries economy is on a war footing.

Look at how much a difference being prepared made for Ukraine the recent invasion compared to the earlier invasion of Crimea.

Yeah should have that. Mornings are hard.

Electrolytic capacitors use the chemistry to make a very high dielectric allowing the plates to get very close and increase the capacitance and decrease the size.

A cell in a battery is a capacitor then converts the charge on the plates into chemical energy and vice versa allowing much more energy storage and a flat operating range as the plates charge is replenished by the chemical reaction.

This article doesn’t go into details but it sounds like the breakthrough is a much better dialectic then storing energy in a chemical reaction.

It’s a hot mess. Or it was 10 years when I was last forced to use it.

I remember there was one setting I needed to adjust from time to time. However the menu tree it was in had been removed leaving it a orphaned tree. So you couldn’t get to it through the ui. The only way to get to the setting was to go to a unrelated page in the documentation and click on the goto link in it. That would launch the menu that otherwise didn’t exist. From there you could back up and get into the menu to adjust the setting.

There is a difference between race and culture even if we tend to use the same words for both. I have a friends who is racially Chinese but was born and raised in the UK so very British culturaly.