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Geometry from Another Universe

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Re: Geometry from Another Universe

#41

Post by mnadkvlb »

ck45 wrote: post #40
all2 wrote: post #17

Earlier quoted context omitted.

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.

This is extremely common, I googled a lot about it some years ago. I have it in two versions, the first one is that I have to repeat a maths course in school, being aware about the age difference with the other students, the other one is missing a course in university and not having graduated. The latter one feels so real that in the beginning I had to remind myself about having been at the graduation ceremony :)

holy crap, i had the ‘missing a course’ nightmare roughly every 2 months for 2 years after graduating.

Re: Geometry from Another Universe

#42

Post by Asooka »

ogogmad wrote: post #20

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,…

The w coordinate is there to aid the perspective transform and translations. Because in homogeneous coordinates everything becomes a matrix multiplication. At the end, before rasterisation, the coordinates are normalised by dividing by w. Yes, you can use it for whatever else you want, GPUs are just vector processors, as long as you keep in mind how the rasterisation step will happen.

Re: Geometry from Another Universe

#43

Post by ogogmad »

Asooka wrote: post #42
ogogmad wrote: post #20

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,…

The w coordinate is there to aid the perspective transform and translations. Because in homogeneous coordinates everything becomes a matrix multiplication. At the end, before rasterisation, the coordinates are normalised by dividing by w. Yes, you can use it for whatever else you want, GPUs are just vector processors, as long as you keep in mind how the rasterisation step will happen.

Thanks. I'm aware of all those uses. But that strongly suggests that he's constructed a simulation of projective / elliptic 3-space, not spherical 3-space. The image on the screen is a perspective projection thereof.

Additionally:

To clarify my understanding of projective vs elliptic geometry, because I keep conflating them: Projective space is a topological space, while elliptic space is projective space endowed with a certain metric, turning it into a metric space. Additionally, every elliptic isometry is a projective symmetry, while the other way round isn't true. On a purely topological level, there's no difference between elliptic geometry and projective geometry.

Re: Geometry from Another Universe

#44

Post by kroltan »

echelon wrote: post #13
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.

Hyperbolica is to have VR support too, the creator discussed [0] how he had to use a specific technique to ensure the rendering was compatible with VR optimizations.

[0]: https://youtu.be/rBr-0bHQfxc?t=357

Re: Geometry from Another Universe

#45

Post by throw10920 »

ck45 wrote: post #40
all2 wrote: post #17

Earlier quoted context omitted.

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.

This is extremely common, I googled a lot about it some years ago. I have it in two versions, the first one is that I have to repeat a maths course in school, being aware about the age difference with the other students, the other one is missing a course in university and not having graduated. The latter one feels so real that in the beginning I had to remind myself about having been at the graduation ceremony :)

I wonder why it's common? Easiest explanation would be extreme stress/mild trauma from the stress of college...

Re: Geometry from Another Universe

#46

Post by geodel »

jerf wrote: post #36
all2 wrote: post #17

Earlier quoted context omitted.

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.

I'm in my 40s. For a good long while, my recurring school dream was that my high school discovered that I didn't actually finish correctly, and because that invalidates my college degrees (in dream logic land), I can't do my job until I go back and finish high school properly. For some reason in the last six months to a year it has shifted to the idea that I signed up for a full semester's worth of courses, while sti…

Wow, seeing so many comments high school/college exam dreams do seem universal. I have had these dreams like forever. To think about high school was almost 25 years ago.

Another theme for me is not able to board on plane. More than missing flight it is about being stuck to places where I do not want to be.

Re: Geometry from Another Universe

#47

Post by advantager »

mnadkvlb wrote: post #41
ck45 wrote: post #40

Earlier quoted context omitted.

This is extremely common, I googled a lot about it some years ago. I have it in two versions, the first one is that I have to repeat a maths course in school, being aware about the age difference with the other students, the other one is missing a course in university and not having graduated. The latter one feels so real that in the beginning I had to remind myself about having been at the graduation ceremony :)

holy crap, i had the ‘missing a course’ nightmare roughly every 2 months for 2 years after graduating.

I've had variations on this dream several times. I was talking to my father about it, and he still occasionally dreams about this. As a 67 year old, 5 years retired, 40 years after graduating...

Re: Geometry from Another Universe

#48

Post by JoeDaDude »

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

'Nother 4Dgame. An old classic, just get the ball out of the (hyper)cube.

http://harmen.vanderwal.eu/hypercube/

Re: Geometry from Another Universe

#49

Post by JoeDaDude »

zackmorris wrote: post #19

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…

I created a Hypercube screen saver back in the day from instructions provided in a Scientific American article, which if memory serves was written by Martin Gardner. A casual search leads me to believe the article is in his book Mathematical Carnival, though I could be wrong.

https://www.amazon.com/Mathematical-Carnival-Martin-Gardner/...

While you program it up, you can play this little Hypercube game. Do it in 3D, then turn on the 4D to get an intuitive feel for the 4D world.

http://harmen.vanderwal.eu/hypercube/

Re: Geometry from Another Universe

#50

Post by unsui »

all2 wrote: post #17
ryandrake wrote: post #14

Earlier quoted context omitted.

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.

I'm genuinely curious why this dream pattern seems to be quite distributed across many people.

I still regularly have the dream where I realized after graduation that I had missed an entire college class or test (often varies depending on the specific dream), but this then nullifies my graduation and all my subsequent work.

Also over 20 years ago, but still a regular visitor to my dreamscape.

Taking very liberal inspiration from the idea of something like Jungian archetypes, more along the lines of there being attractors in the informational dream-space that create narrative clusters linked to common emotions (such as anxiety), I can imagine of set of emotion-to-dream-narrative mappings that would lead to some of the regularities often seen in distributed recurring dreams:

- losing one's teeth - riding naked in the subway - being late to a class or missing a class/test entirely etc.

Just wondering if there has been any research on this that doesn't try to fall on the old standbys of Jungian archetypes, collective unconscious, and other similar handwaving

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