Geometry from Another Universe • Hacker News Board

Re: Geometry from Another Universe

#11

Post by Jeff_Brown » Thu, Jan 20, 2022, 4:08 PM UTC

melissalobos wrote: post #7

mark_l_watson wrote: post #3

I had a dream a year ago in which I was traveling at very high speed towards an edge of our universe. In the dream I entered a null space before entering the next universe in which all star systems obeyed solid state physics type laws: stars formed an outer lattice and planets in small lattices around stars. After traveling through the second universe in my dream, I then entered a third universe that was more compact…

> everything was happening in lower dimensional manifolds curving through that space where “all the action was.” I feel like this would have to be true in a very high dimensional space, since an infinite dimensional orange is all skin. > systems obeyed solid state physics type laws: stars formed an outer lattice and planets in small lattices around stars. This sounds really fun, could you try explaining a bit more ab…

> an infinite dimensional orange is all skin

Huh?

Re: Geometry from Another Universe

#12

Post by Jeff_Brown » Thu, Jan 20, 2022, 4:09 PM UTC

ericbarrett wrote: post #2

Would love to see this where you can toggle between a three-sphere, flat, and 3-hyperbolic space.

And arbitraily gluable diffrrentiable manifolds!

Re: Geometry from Another Universe

#13

Post by echelon » Thu, Jan 20, 2022, 4:15 PM UTC

pavel_lishin wrote: post #10

If you're interested in this, you may be interested in Hyperbolica, which is a whole game about this concept: https://store.steampowered.com/app/1256230/Hyperbolica/ https://www.youtube.com/watch?v=EMKLeS-Uq_8

These are great!

(There's also miegakure: https://www.youtube.com/watch?v=uWsBnVtl8tA )

I can't wait for these games to start being built for VR. Maybe it'll be possible to develop a sense for higher dimensionality intuitively using our senses.

Maybe young mathematicians and physicists can explore higher dimensions in VR to get accustomed to it, which might help with their theorizing and explorations.

Re: Geometry from Another Universe

#14

Post by ryandrake » Thu, Jan 20, 2022, 4:17 PM UTC

mark_l_watson wrote: post #3

I had a dream a year ago in which I was traveling at very high speed towards an edge of our universe. In the dream I entered a null space before entering the next universe in which all star systems obeyed solid state physics type laws: stars formed an outer lattice and planets in small lattices around stars. After traveling through the second universe in my dream, I then entered a third universe that was more compact…

Wow, if you’re having these dreams without the aid of substances, I envy you. The most intellectual dreams I seem to be able to come up with “unenhanced” are falling off buildings and being late to college exams while having no pants on.

Re: Geometry from Another Universe

#15

Post by jlpom » Thu, Jan 20, 2022, 4:24 PM UTC

pavel_lishin wrote: post #10

If you're interested in this, you may be interested in Hyperbolica, which is a whole game about this concept: https://store.steampowered.com/app/1256230/Hyperbolica/ https://www.youtube.com/watch?v=EMKLeS-Uq_8

Antichamber too https://store.steampowered.com/app/219890/Antichamber/

Re: Geometry from Another Universe

#16

Post by hwers » Thu, Jan 20, 2022, 4:32 PM UTC

No css plaintext blogs is such a statement and cool signalling, love it.

Re: Geometry from Another Universe

#17

Post by all2 » Thu, Jan 20, 2022, 4:44 PM UTC

ryandrake wrote: post #14

mark_l_watson wrote: post #3

I had a dream a year ago in which I was traveling at very high speed towards an edge of our universe. In the dream I entered a null space before entering the next universe in which all star systems obeyed solid state physics type laws: stars formed an outer lattice and planets in small lattices around stars. After traveling through the second universe in my dream, I then entered a third universe that was more compact…

Wow, if you’re having these dreams without the aid of substances, I envy you. The most intellectual dreams I seem to be able to come up with “unenhanced” are falling off buildings and being late to college exams while having no pants on.

My favorite are the dreams I have before a semester starts: it is the end of the semester, and I've just discovered I've forgotten about an entire class for most of the semester. Then I wake up.

Re: Geometry from Another Universe

#18

Post by all2 » Thu, Jan 20, 2022, 4:48 PM UTC

Jach wrote: post #9

I thought this might have been something about a Greg Egan book ( https://www.gregegan.net/DICHRONAUTS/DICHRONAUTS.html is a recent one with space-time composed of two dimensions of space and two of time), but these demos are really fun too. Especially like the connectedness of everything that's not in some sense cheated with portals.

Being able to set off a chain reaction in time 1 and then inspect that chain reaction at various moments in time 2 would be fascinating. Can you imagine a bullet hell in two dimensions of time? Or even a physics puzzle game that operates in two dimensions of time?

Re: Geometry from Another Universe

#19

Post by zackmorris » Thu, Jan 20, 2022, 4:53 PM UTC

Does anyone know of a good visualization of a hypercube? Specifically, I'd like to be able to rotate it around various axes to see what its shadow looks like in 3D. On that note, a VR rendering might help since a 2D projection overcomplicates the shadow.

It's easy to construct, just build a point, extrude the point out to make a line, extrude the line out to make a square, extrude the square out to make a cube, then extrude the cube out to make a hypercube. It looks like two cubes with the corners connected by lines through a 4th dimension that we can't construct in 3D.

One way to visualize it is to use time, so if you translate a cube from point A to point B, the "extrusion" would be along time and you can kind of visualize that extra "dimension". But what would scaling, rotation, etc along that invisible axis look like? To fully grok it, we'd need to be able to sculpt the hypercube's shadow in 3D as easily as drawing a cube on a piece of paper.

Asking in the hopes of building a mental bridge to 4D and then possibly 5D in order to generalize to higher dimensions.

Re: Geometry from Another Universe

#20

Post by ogogmad » Thu, Jan 20, 2022, 4:54 PM UTC

Very cool, but I have a question.

> This is done using WebGL, an API for drawing 3D graphics in a browser. You might think it's only designed to draw ordinary, Euclidean things, but it turns out it's perfectly suited to rendering this world as well. WebGL (and OpenGL) works with four-dimensional coordinates (x,y,z,w). Normally, you'll be advised to just set that w coordinate to 1 all the time. If you do that, the x, y and z coordinates will behave like ordinary 3-D geometry. I just ignored that rule and used the coordinates to represent points on the 3-sphere instead. It all works out. (You have to be a bit careful if you want to put textures on things without seeing seams between the triangles, but it can be done. The key is to make sure every flat surface is still a flat surface in 4-D.)

Are the 4D coordinates treated as ratios, so that (w,x,y,z) = (tw, tx, ty, tz)? Because then that would express RP^3 instead of S^3, i.e. projective 3-space, instead of the 3-sphere. Equivalently, this produces a model of 3D elliptic geometry, instead of 3D spherical geometry. My doubts are because 3D computer graphics is known to use homogeneous coordinates, which results in a model of projective 3-space or elliptic 3-space.

S^3 can be coordinatised using elements of R^4 of unit length, which might have been done here.