Gabriel's Horn

Take an asymptotic function then revolve it in 3rd dimension around x. When considering at what point infinity is defined, then you can clearly see how with infinite x dimensions, it is an unending object. Typically we define it as I think y = 1/x and rotate it around the x axis.

Easily enough, you can calculate it's volume, it's area however, shit outta luck. One of them Paradoxes you really can't get away from

How we figure into this.

Imagine you are one line, a plane that intersects this 3d object and then slices it once. You are one revolution. There are so many lines of 1/x on this fucking thing. One might even say an infinite amount! see the Banach-Tarski Paradox! You could always find yet another one too! because this is just infinitely large! That's an oversimplification, but you own one projection, if you can own a 2d projection, in the whole larger 3 dimensional object.

Now, one interesting thing about this is, this infinity means that we have an infinite amount of potential unique lines of unique 1/x s. If we consider those 1/x s as representative Maybe they don't even end up being 1/x s and they look all wavy and shit, all really shitty, probably none of our loss functions look smooth lol. You've seen graphed loss functions. Though if we take the general look and collection of those, and revolve them around an X axis. We get a mesh that looks quite fucking similar to Gabriel's Horn But still eventually look horn like. One interesting part of this is understanding that our understanding of the world, our Schema will produce as a projection, something similar or akin to a Loss Functions. THATS WHATS FUCKING INSANE.

One thing to note, and also similar to GEB

We will never completely understand or satisfy the understanding the full unfolding of the manifold that is the reality manifold. We must be okay with that. We must imagine Sisyphus as Happy. We will never fully comprehend the x axis, as it is both without dimension, and all encompassing in every dimension. Thus, again, We must imagine Sisyphus as Happy.

Credit where Credit is due

I had a great time discussing these and other thoughts with Karim. It's kinda fun how it's called Gabriel's Horn

Other thoughts

If you are seeing a projection, a backprojection, just like in CryoEM we can see all these different projections and eventually maybe, we can even just piece the reality manifold back from those! We could use those loss functions, which we do know, because we have us, into a huge cool shit!

I doubt we could, but it's an interesting thought, We definitely do not have the fucking compute for that though, and I'm sure we'll just get an approximation, just like any other single particle piecing.