Re: 4D Visualisierung

Op 15-9-2024 om 21:28 schreef Chris M. Thomasson:

On 9/15/2024 2:11 AM, guido wugi wrote:

Op 15-9-2024 om 02:17 schreef Chris M. Thomasson:

On 9/14/2024 5:10 PM, Chris M. Thomasson wrote:

On 8/28/2024 12:30 PM, guido wugi wrote:

Hallo,

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This is your artificial 4d axis, right?
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https://i.ibb.co/rMqqp9k/image.png

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Exactly. I called them (x,y,z,v) here, but (x,y,u,v) for complex functions. The positions are initiated by the six angle controls for coordinate plane rotations (or four angle controls for "spherical" coordinate rotations).

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Okay. I see. Thanks.
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To be quite honest, 4d kind of freaks me out a little bit... If 3d is comprised of infinite 2d planes, then 4d is comprised of infinite 3d planes...

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Yes, 3D-manifolds aren't much indicated for visualisation of course.
It's all about *surfaces and edges*:
Pure surfaces and their parameter curves as for complex functions.
Or border edges of border surfaces, of (border) volumes of 4D-volumes, as for the tesseract.
If you want 3D-volumes in 4D, that's another pair of sleeves (as we say in Dutch:-).

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Yeah. That's an interesting one for sure. So, a 3d volume would be one 3d plane out of the infinity of them in the 4'th dimension? Humm...

 Yes and no. 4D-space may indeed be generated by piling up 3D spaces along a 4th-dimension axis.
But 3D volumes/manifolds may also evolve in 4D space, just like 2D surfaces and 1D curves may evolve in (x,y,z) space. I was rather referring to such 3D objects ("volumes", manifolds, whatever you call them) existing in 4D. Those are beyond my 4D visualisation scope. But if they are contained within lowerdimensional limits, say, border surfaces and edges, then, like any surface and curve, those are the things one can visualise (example: the tesseract).
 

Btw, I have created a lot of 3d volumes. Even in DICOM format. They are all good candidates for holograms... :^)
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Check these out if you can get to the link:
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https://www.facebook.com/share/p/n2nMhW5G2PhRzyfx

 Sorry but half of your links are unavailable.

 

They can all be 3d printed. Humm... Sometimes I think that a 3d "observer" would only be able to see 2d. As in a 3d scene projected onto a 2d plane with lights and shadows, ect... However, a 4d observer would be able to see in pure 3d. Make any sense? Thanks.

 Exactly. We 3D observers see only outer layers = border surfaces of 3D objects. OK, we can see through transparent media like air and water and glas, but as soon as opaque things are to be observed, it is their outer surface we see.
And a 4D observer would indeed see us in full 3D, our entire internal body, organs etc. included. As for us, we can see the inner parts of a Flatlander picture on a flat, transparent sheet.
(Another nicety: if we turn the Flatlander sheet upside down, our Flatlander image will have swapped its left/right sides! So, a 4D observer can look at us "from one side" and agree with us about our left and right sides, and then look "from the other side" and see us with swapped left/right sides.)