Two-dimensional space

Which, you know, brings up a profoundly interesting paradox.

If virtual lighting is just this collection of clever mathematical hacks and like 2D textures pretending to be 3D, how does it consistently trigger such deep, authentic emotional responses from us?

I mean, a fake digital sunset, meet a pixels on a flat monitor, can actually make a human being feel biological calm.

It's wild.

When designers need acute, localized control, they drop down to the second tier, fog cards.

Fog cards are the most elegant cheat in the entire industry.

It is literally just a flat 2D rectangular plane, like a pane of glass, with a translucent, cloudy noise texture painted onto it.

That's it.

it comes from your phone now when you're living your world inside this phone you're living in that tron 2d reality right and that flat earth information is literally devolving people.

It's the devolution into the lower dimensions, which is the exact opposite way of the ascension.

And if you liken it to a video game, like I like to do, a 2D video game that we remember playing as kids, the side scrolling games, like let's just say there's a moon on the screen.

That moon is a flat disc on the 2D video game.

And just through the evolution of the graphics, the games became 3D.

And what happened? That flat disk became a sphere in the 3D video game.

The same thing happens in our reality.

If you go from a 2D reality to a 3D reality, just by the nature of physics alone, it's a spherical planet.

Rigorously justifying this would go beyond the scope of an already long lesson, but it's not too hard a fact to believe.

What we're basically saying is that if you have some function that you know satisfies whatever equations light must obey, namely the object wave O, and you have another function describing a steady state wave where you know that O shows up as a component along a 2D boundary like this, then in the free space beyond that boundary, the wave must also include as a component O in its entirety.

This actually has a nice connection to the key mystery from the very beginning of the lesson of how a three-dimensional scene could possibly be stored on 2D film.

Part of what's required for this to be possible is that light obeys laws regular enough that the state of a light wave in 3D space is sufficiently constrained by its value on a 2D boundary.

Finding a doctor is hard enough.

Finding the right doctor, even harder.

Even if you did see it, but it was a while ago, I'd recommend taking another look just to make sure those ideas are fresh in your mind.

For the 2D cross product, v cross w, what you do is you write the coordinates of v as the first column of a matrix, and you take the coordinates of w and make them the second column, then you just compute the determinant.

This is because a matrix whose columns represent v and w corresponds with a linear transformation that moves the basis vectors i hat and j hat to v and

w.

So composing sliding actions is another way to think about what adding complex numbers actually means.

This collection of all sliding actions on the 2D complex plane goes by the name the additive group of complex numbers.

Again, the upshot here is that numbers, even complex numbers, are just one example of a group, and the idea of addition can be thought of in terms of successively applying actions.

But numbers, schizophrenic as they are, also lead a completely different life as a completely different kind of group.