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May 9th, 2017, 08:13 AM   #1
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Math Focus: Abstract Simulations
Building an intuition for infinite-dimensional points, lines, surface etc.

I come from a javascript background. My goal is to build simulations of "dream" stuff. Not in javascript

Yesterday my mind was blown when I was listening to a lecture on youtube and the professor brought up "a point with infinite dimensions". In that moment I asked myself "how could you do that?" and then I figured that you could just define the value on every axis with a function, if you wanted to know the 8th dimension's value then just plug in 8 to the function.
The interesting thing about this is that a simple point that is 1 at every axis is infinitely distant from the center. I'm sensing an interesting computational shift when one works with purely infinite dimensions.
I don't know, but it would almost seem like everything is either infinitely distant from each other, or the distance between two of these points is always divergent.
I feel like I am absorbing some stuff just by brute-force reading the threads and playing youtube lectures like talk radio/podcasts. I also feel that because I'm trying to apply all features of math to this simulation I'm getting a very real hands on experience & it can really go wherever my imagination wants to. The interesting thing is, the more I discover about math, it's properties are useful for what I'm trying to do, and I can't quite explain it, but my efforts are converging on something.

So back to the concepts. If I can have a point with infinite dimensions, what does my axis value defining function mean at 1.5? Because I'm sure that there is some value between two natural numbers that may indicate that one point is within a certain distance or distance definition to some other point & I feel like when we have infinite dimensions the algebra/calculous is already going to look at all real numbers in-between natural numbers. Can I have partial dimensions? What happens to the distance formula? Can only the tail end dimension be partial? or can any dimension be partial? If so, then are dimensions independent; so my axis-value defining function really hasn't covered all dimensions because dimensions are defined independently of a sequence or without name? And if there is some type of proof to figure out what the distance to something is, even if it is divergent then what about negative dimensions? Negative dimensions feel like they're gonna be big if I can find a way to figure out distance with points that have infinite dimensions.

What can I call any given point, line, surface, manifolds? Is there a name to group all three? Substances?
OH yeah. and I heard about infinitely dense curves, and these is great for representing distortions in space (topology) ...
There's so many properties that naturally come from pure geometry.

There's more I want to dive into.
-Continuous vector definitions throughout a "substance".

And finally. None of these really means much of anything unless I can find out how to fluidly (by plugging in T) find the x & y coordinates of two particles pulling on each other for any length of time.
I think I've asked this question here before, but I didn't understand the answer or maybe the answer wasn't enough. I think they showed me some langrangian or hamiltonian stuff. What confused me was that the value being returned was only the distance between 2 points, when my real issue is when I have more than 2 points and they are all acting on each other in continuous space and time. If there was a way to get the actual x y and z coordinates of 2 or more particles I don't think you could plug in T and get infinite values returned with a single algebraic trick. I'm guessing it would involve calculating it to a some kind of limit and then running the function again at that limit "key-frame".

I COMPLETELY forgot calculous could define the area under a curve, which I believe is a similar challenge to what I'm trying to do. If what I'm theorizing is correct, then ... yeah, let's do this.
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May 9th, 2017, 05:51 PM   #2
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Relax... Keep working on your sim, but make yourself some clear objectives of what you want it to do, predict the results, and then go from there..
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May 10th, 2017, 05:22 AM   #3
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The things I'm reaching for can only be 100% implemented in a 2.0 version. Exploring higher concepts has helped me solidify what needs to be done with a simpler version.

I do have code, it works. I promise you I'm not entirely crazy.

Am I just wasting my time? Maybe.
The fact that I know the 4th dimension is a weird place for spheres, tells me something I should look out for when casting shadows down to the 3rd dimension. I think it's valuable to have these nuggets of information.

This is the projection involving a 4-dimensional object called a dodecaplex. Image crated by Paul Nylander

Last edited by InkSprite; May 10th, 2017 at 06:08 AM.
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May 10th, 2017, 07:03 AM   #4
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Quote:
Originally Posted by InkSprite View Post
The fact that I know the 4th dimension is a weird place for spheres, tells me something I should look out for when casting shadows down to the 3rd dimension.
you might enjoy a short story by Greg Bear called "Tangents".

It's about a kid that can see and to some extent interact with the 4th dimension.
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May 11th, 2017, 06:56 AM   #5
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Infinite dimensional point:
$\displaystyle p=\lim_{n\rightarrow \infty}(x_{1},x_{2},x_{3},...,x_{n})$

Infinite dimensional line:
$\displaystyle p=p_{1}+k(p_{2}-p_{1})$

Infinite dimensional plane:
$\displaystyle \lim_{n\rightarrow \infty}a_{0}(p-p_{0})+a_{1}(p-p_{1})+a_{2}(p-p_{2})+...+a_{n}(p-p_{n})=0
$

Infinite dimensional circle:
$\displaystyle \lim_{n\rightarrow \infty}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2},...,x_{n}^{2 })=r^{2}
$
There is nothing mysterious about a 4-dimensional circle. It is simply a definition:
$\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=r^{2}$
If $\displaystyle (x_{1}, x_{2}, x_{3})$ are space coordinates and $\displaystyle x_{4}$ is time t, then it describes a spherical shell contracting from r.

$\displaystyle x_{i}, k, r, a_{i}$, real
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May 12th, 2017, 04:48 AM   #6
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Quote:
Originally Posted by zylo View Post
There is nothing mysterious about a 4-dimensional circle. It is simply a
$\displaystyle x_{i}, k, r, a_{i}$, real
That's not what I gathered from this link
https://plus.maths.org/content/richard-elwes

They say that they don't know how many "spheres" the 4th dimension has. It just has a question mark.
15 dimensions has 16256 spheres. 7 dimensions, 28. There's something weird about the 4th dimension I don't quite understand.
Quote:
The realm which remains the most mysterious, even today, is 4-dimensional space. No exotic spheres have yet been found here. At the same time no-one has managed to prove that none can exist. The assertion that there are no exotic spheres in four dimensions is known as the smooth Poincaré conjecture. In case anyone has got this far and is still not sure, let me make this clear: the smooth Poincaré conjecture is not the same thing as the Poincaré conjecture! Among other differences, the Poincaré conjecture has been proved, but the smooth Poincaré conjecture remains stubbornly open today.
oh... "exotic" spheres. not regular ones.

Last edited by InkSprite; May 12th, 2017 at 04:57 AM.
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