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From: guido wugi <wugi@brol.invalid>
Newsgroups: sci.math
Subject: Re: 4D Visualisierung
Date: Sun, 15 Sep 2024 22:21:49 +0200
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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,
>>>> [...]
>>>>
>>>> This is your artificial 4d axis, right?
>>>>
>>>> https://i.ibb.co/rMqqp9k/image.png
>>
>> 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).
>
> Okay. I see. Thanks.
>
>
>>> 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...
>>
>> 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:-).
>
> 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... :^)
>
> Check these out if you can get to the link:
>
> 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.)

-- 
guido wugi