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Author Topic: Folding boxes in higher dimensions?  (Read 3503 times)
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KRAFTWERK
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« on: May 05, 2010, 09:08:49 AM »

Hello geniouses!

I have a weak mathematical mind, but one of my brain cells had a spark on the way to work today.

I thought about Tglads mandelbox, and the discussion about it having no “volume”.

If I understand things correctly, the “mandelbox” or Tbox maybe wink is a folding of 3D space in the fourth dimension, my thought was: What if you could fold 3D space in the fifth or sixth dimension?

Somebody probably thought about this before, but I just wanted to hear what you have to say about the idea...  undecided

J
« Last Edit: May 05, 2010, 02:47:12 PM by KRAFTWERK » Logged

reesej2
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« Reply #1 on: May 05, 2010, 04:54:01 PM »

I can easily imagine how to fold a 4D box in 5 dimensions, but a 3D box in 5 is tricky. It'd be like trying to fold a line using all three dimensions. If it's possible, and someone with more brainpower than I at this time of day (it's 8 am where I am and I haven't had my coffee yet tongue stuck out) thinks about it a bit, I'm sure it'll produce some interesting results. I've found that, as a general rule, conflict between higher and lower dimensions produces fascinating fractal results.
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hobold
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« Reply #2 on: May 05, 2010, 06:17:34 PM »

It'd be like trying to fold a line using all three dimensions.
I think you are talking about knots. Yes, those are tricky.

Funny trivia: three dimensions is the only space in which there are "interesting" knots. In fewer dimensions, you cannot pull ends of the string through loops of the same string. In higher dimensions, you can basically untangle every knot just by pulling on both ends - there is too much space.


I am not aware of generalizations of this, i.e. how many dimensions are needed to make interesting hyper-knots from a two dimensional hyper-string.
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reesej2
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« Reply #3 on: May 05, 2010, 08:11:23 PM »

In the case of the Mandelbox, our "string" is 3D--a box. So maybe some sort of... nine-dimensional knot could be tied? How would you describe that mathematically? Sounds pretty awesome though.
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KRAFTWERK
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« Reply #4 on: May 07, 2010, 02:47:21 PM »

I can easily imagine how to fold a 4D box in 5 dimensions, but a 3D box in 5 is tricky. It'd be like trying to fold a line using all three dimensions. If it's possible, and someone with more brainpower than I at this time of day (it's 8 am where I am and I haven't had my coffee yet tongue stuck out) thinks about it a bit, I'm sure it'll produce some interesting results. I've found that, as a general rule, conflict between higher and lower dimensions produces fascinating fractal results.

I am glad to get some answers to my idea, and when you say it would be easier to fold a 4D box in 5D I got this question:
When doing the folding for the mandelbox, the box must be in 4D space before going back to 3D?
Would it be possible in that step to fold the 4D box in 5D?

Hope you get the idea...  afro alien evil hurt
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reesej2
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« Reply #5 on: May 07, 2010, 11:53:17 PM »

I get your meaning, but the problem I see is that the 4D version of the 3D box isn't a box anymore. It's more like a plane. Of course, we're talking about 4-dimensional analogues here.
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KRAFTWERK
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« Reply #6 on: May 10, 2010, 01:35:40 PM »

Of course..  afro

Hard enough to imagine 4D folding.

I will probably never be able to even comprehend my own idea...  embarrass
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trafassel
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trafassel
« Reply #7 on: October 30, 2010, 11:20:41 PM »

// I  added another entry q (for the 4th dimension) and used the same foldings as (x,y,z)
// The result feels very 4 dimensional :-)

//              ; fold box onto itself 
              if (x > 1)
                x = 2.0 - x;
              else if (x < -1)
                x = -2.0 - x;
             
              if (y > 1)
                y = 2.0 - y;
              else if (y < -1)
                y = -2.0 - y;
             
              if (z > 1)
                z = 2.0 - z;
              else if (z < -1)
                z = -2.0 - z;

   if (q > 1)
                q = 2.0 - q;
              else if (q < -1)
                q = -2.0 - q;
              if (q > 1)
                q = 2.0 - q;
              else if (q < -1)
                q = -2.0 - q;
             
            //  ; fold sphere onto itself
              double fixedRadius = 1;
              double minRadius = 0.00001;
 //     double length=   r = Math.Pow(x*x + y*y + z*z,aj);
   double length=   r = x*x + y*y + z*z+q*q;

if(r!=0) {
additionalPointInfo.red+=x*x/r;
additionalPointInfo.green+=y*y/r;
additionalPointInfo.blue+=z*z/r;

}
            if (length > gr) {
              tw = n; break;
            }

   double mult=1;

    if (length < minRadius) {
      //  mult=Math.Sqrt(fixedRadius)/(minRadius);
    }   else
 if (length < fixedRadius) {
        mult=Math.Sqrt(fixedRadius)/(length);
  }
   mult*=scale;
   x*=mult;
   y*=mult;
   z*=mult;
q*=mult;
   x+=br;
   y+=bi;
   z+=bj;
q+=bq;
          }


* Data495pic10200a.jpg (232.85 KB, 1200x1200 - viewed 337 times.)
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KRAFTWERK
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« Reply #8 on: November 05, 2010, 03:23:08 PM »

Cool Trafassel, but this is not a 5D-folding (?)
Nice image though!!!
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trafassel
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trafassel
« Reply #9 on: November 08, 2010, 09:02:22 PM »

No, just 4D folding. Like this:

<a href="http://www.youtube.com/v/YHKaBY7kNDk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/YHKaBY7kNDk&rel=1&fs=1&hd=1</a>
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Tglad
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« Reply #10 on: November 08, 2010, 11:30:53 PM »

Kraftwerk "When doing the folding for the mandelbox, the box must be in 4D space before going back to 3D?"
Take the example of folding the 2d box (folding a plane), you need a 3rd dimension to visualise the plane folding over itself, but the end result is just a reflection, which doesn't need the 3rd dimension to exist.
Equally, we can imagine the 3d volume folding around the 4th dimension, or just think of it as a reflection about a plane.

"Would it be possible in that step to fold the 4D box in 5D?"
Yes, or think of it as reflecting the 4d hypervolume about a volume. This is what Trafassel did, e.g.

Fold around the volume where x=1:
if x>1
 x = 2-x
Fold around the volume where y=1:
if y>1
 y = 2-y
Fold around the volume where z=1:
if z>1
 z = 2-z
Fold around the volume where w=1:
if w>1
 w = 2-w

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KRAFTWERK
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« Reply #11 on: November 09, 2010, 08:55:11 AM »

Wooohaaa, thanks Trafassel & Tglad, so Trafassels animation is actually a vision of the idea I had?
I love it, it is like ice-chrystals forming, thank you very much Trafassel, and thank you Tglad for explaining!

Jesse? wink
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trafassel
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trafassel
« Reply #12 on: December 14, 2010, 09:12:22 AM »

The 4D variant does not give new (... but only until now) types of shapes, but I like the flow in animations very much.

<a href="http://www.youtube.com/v/ovjo9JwKBqs&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ovjo9JwKBqs&rel=1&fs=1&hd=1</a>
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KRAFTWERK
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« Reply #13 on: December 14, 2010, 09:23:33 AM »

Wow! It is like rings on the water... or some strange radio transmission...
I like it too T!
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