Hallo,

Für alle, die sich für „echte 4D“-Renderings (natürlich in Projektion) von Sachen wie komplexen Funktionen w=f(z), 3-Sphären, Clifford-Torus
und anderen Tesserakten interessieren, habe ich diesen kleinen
4D-Grapher mit interaktiven controls in Desmos3D erstellt.

NB1.) Die ersten Versionen verwendeten willkürige Achsenprojektionen und rotierten nicht richtig, d. h. sphärisch. Nach ein paar Wochen des Ausprobierens und Kopfzerbrechens haben sich die Dinge geklärt, und
alles passte gut kalibriert an seinen Platz.

NB2.) Ich kann nicht verstehen, warum professionelle und gängige Mathematiksoftware diese 4D-Rendering-Methoden hartnäckig ignoriert.
Seit Jahren verwende ich den unprätentiösen Graphing Calculator 4.0 von Pacific Tech, der 4D vollständig integriert hat.

Wie auch immer, willkommen zum Ausprobieren hier:

https://www.desmos.com/3d/x7w6jdpxgx?lang=nl : Methode und Beispiele https://www.desmos.com/3d/krq32ylqjd?lang=nl : 3-Sphäre https://www.desmos.com/3d/3ci8qmdzaf?lang=nl : Clifford-Torus https://www.desmos.com/3d/dwujqpjry3?lang=nl : Tesserakt

Mehr hier:
https://www.wugi.be/qbinterac.html https://www.youtube.com/@wugionyoutube/playlists (suche nach "4D" und
"Complex Functions")

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 28-8-2024 om 21:49 schreef Chris M. Thomasson:

On 8/28/2024 12:38 PM, Chris M. Thomasson wrote:

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

Hallo,

[...]

Actually, it's impossible to visualize a true tesseract in 3d space?

A question I have is where do I plot a 4d point, say:

(0, 0, 0, 1)

in a 3d space? Humm...

I've been doing that for a few decades by now ;o)

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 28-8-2024 om 21:49 schreef Chris M. Thomasson:

On 8/28/2024 12:38 PM, Chris M. Thomasson wrote:

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

Hallo,

[...]

Actually, it's impossible to visualize a true tesseract in 3d space?

A question I have is where do I plot a 4d point, say:

(0, 0, 0, 1)

in a 3d space? Humm...

How do you plot a photo of a 3D scene?

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 29-8-2024 om 00:31 schreef FromTheRafters:

guido wugi explained :

Op 28-8-2024 om 21:49 schreef Chris M. Thomasson:

On 8/28/2024 12:38 PM, Chris M. Thomasson wrote:

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

Hallo,

[...]

Actually, it's impossible to visualize a true tesseract in 3d space?

A question I have is where do I plot a 4d point, say:

(0, 0, 0, 1)

in a 3d space? Humm...

How do you plot a photo of a 3D scene?

Oh, now you're projecting. :)

Sorry, couldn't help myself. In another group they all think that they
are psychologists.

Most "3D" renderings of math objects are done in 2D, whether on paper or
on screen.
As for surfaces and curves, which is what we do, there is no difference
in rendering 3D or 4D ones. The main problem is having a coherent
coordinate projection base (conserving spherical rotation symmetry).
Which I've had to resolve the last couple of weeks :)

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 29-8-2024 om 20:47 schreef Chris M. Thomasson:

On 8/29/2024 7:56 AM, guido wugi wrote:

Op 29-8-2024 om 00:31 schreef FromTheRafters:

guido wugi explained :

Op 28-8-2024 om 21:49 schreef Chris M. Thomasson:

On 8/28/2024 12:38 PM, Chris M. Thomasson wrote:

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

Hallo,

[...]

Actually, it's impossible to visualize a true tesseract in 3d space? >>>>>>

A question I have is where do I plot a 4d point, say:

(0, 0, 0, 1)

in a 3d space? Humm...

How do you plot a photo of a 3D scene?

Oh, now you're projecting. :)

Sorry, couldn't help myself. In another group they all think that
they are psychologists.

Most "3D" renderings of math objects are done in 2D, whether on paper
or on screen.
As for surfaces and curves, which is what we do, there is no
difference in rendering 3D or 4D ones. The main problem is having a
coherent coordinate projection base (conserving spherical rotation
symmetry). Which I've had to resolve the last couple of weeks :)

I don't think you can truly project a _true_ 4d object into a 3d
space. We can get some insights, but the projection does not really
represent the 100% true 4d object... It does not capture all of the information? Actually, this kid did an interesting explanation, well
at least to me: :^)

https://youtu.be/eGguwYPC32I

What do you think?

I find it obvious that we can project from 4D space into 3D space in the
same way that we can, and do (everytime you look at a photograph;),
project 3D into 2D. What we can't do really, is project
3D-volumes/manifolds. But projecting surfaces and curves works just fine.

Of course the projected image isn't the "real [4D] thing". But then a photograph isn't the real 3D world it depicts either. Still we like
looking at and interpreting photographs/pictures and find them
interesting. So how for heaven's sake could one not find 4D-to-3D
projected images equally interesting, I ask you???

So then, my renderings aren't "true 4D" objects alright, but they are
"true 4D" projections.
Just as the ubiquitous pictures of the Tesseract are already.
But contrary to the usual 3D extractions of complex functions, like
Re(w), Im(w) etc, which are effectively cutting off a 4th dimension.

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 11-9-2024 om 10:15 schreef Chris M. Thomasson:

On 9/11/2024 1:12 AM, Chris M. Thomasson wrote:

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

Check this out:

https://youtu.be/IVR5I5mnrsg

;^)

Also, iirc, this experiment of mine has a vector with a non-zero 4d component...

https://youtu.be/KRkKZj9s3wk

I don't understand it, but they're beautiful graphics alright! But 4D?

Meanwhile I've put a Desmos4D graph of Clifford tori, Dupin cyclides
(also shown together!) and Hopf fibration.
bolnorm4D.CT-DC-HF | Desmos <https://www.desmos.com/3d/rwj9vo31yc?lang=nl> https://www.youtube.com/watch?v=1y6qrsJff-g&list=PL5xDSSE1qfb6c7UHcURl6wXh0pH4ARB75&index=21


--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 13-9-2024 om 23:58 schreef Chris M. Thomasson:

On 9/11/2024 1:22 PM, guido wugi wrote:

Op 11-9-2024 om 10:15 schreef Chris M. Thomasson:

On 9/11/2024 1:12 AM, Chris M. Thomasson wrote:

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

Check this out:

https://youtu.be/IVR5I5mnrsg

;^)

Also, iirc, this experiment of mine has a vector with a non-zero 4d
component...

https://youtu.be/KRkKZj9s3wk

I don't understand it, but they're beautiful graphics alright! But 4D?

Meanwhile I've put a Desmos4D graph of Clifford tori, Dupin cyclides
(also shown together!) and Hopf fibration.
bolnorm4D.CT-DC-HF | Desmos
<https://www.desmos.com/3d/rwj9vo31yc?lang=nl>
https://www.youtube.com/watch?v=1y6qrsJff-g&list=PL5xDSSE1qfb6c7UHcURl6wXh0pH4ARB75&index=21

Here is an example of a 4d vector ping ponging through -1...1 wrt its
w component:

https://www.facebook.com/share/v/PC17LfU94uUjW6DY

So, the single attractor is at point (0, 0, 0, w) for the animation.
There is a major effect on the field. Here is another simulation that
shows the attractor at a fixed (0, 0, 0, 0) for the entire duration:

https://www.facebook.com/share/v/DXKhRoGZmpB9fX5Y/

I don't see really the difference, sorry.
And: where is the w component *in* the graph? If it isn't *in* the
graph, it's just some external parameter upon a 3D-graph, isn't it?

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 14-9-2024 om 21:20 schreef Chris M. Thomasson:

On 9/14/2024 2:08 AM, guido wugi wrote:

I don't see really the difference, sorry.

There is a massive difference. Humm... The animation is rather fast.
Try it in slow motion.

The thing is, it doesn't expose dim 4. It looks and feels like just an
external parameter acting on the output.

And: where is the w component *in* the graph? If it isn't *in* the
graph, it's just some external parameter upon a 3D-graph, isn't it?

Well, the vector field algorithm is working on 4d vectors. However, I
don't know where to plot a vector like (0, 0, 0, 1) unless I define
some other axis in 3d. This does not seem quite "kosher" to me.
Anyway, I can only see what the non-zero w components do to a field
that has all zero w's. The 4d definitely casts an influence on the 3d components (x, y, z).

Humm... I need to work on another animation that shows this off more, clearly...

It's precisely what my thread is about.
*Graphing 4 dimensions in 3D, spherical-symmetry-true.*
You might try it out with your app ;-)
Mainstream math apps ought to try it out. But they prefer going on
happily ignoring 4D possibilities. Except for tesseracts, for some reason.

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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).

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:-).

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 15-9-2024 om 22:26 schreef Chris M. Thomasson:

Sorry but half of your links are unavailable.

Damn! Try this one, a screenshot of the link above:

https://i.ibb.co/n7FFKvq/image.png

That works. I can't interpret it of course. But they're fine volumes
alright. Or rather, volume border surfaces ;-)

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

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

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 16-9-2024 om 00:07 schreef Chris M. Thomasson:

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

Hallo,

Für alle, die sich für „echte 4D“-Renderings (natürlich in
Projektion) von Sachen wie komplexen Funktionen w=f(z), 3-Sphären,
Clifford-Torus und anderen Tesserakten interessieren, habe ich diesen
kleinen 4D-Grapher mit interaktiven controls in Desmos3D erstellt.

NB1.) Die ersten Versionen verwendeten willkürige Achsenprojektionen
und rotierten nicht richtig, d. h. sphärisch. Nach ein paar Wochen
des Ausprobierens und Kopfzerbrechens haben sich die Dinge geklärt,
und alles passte gut kalibriert an seinen Platz.

NB2.) Ich kann nicht verstehen, warum professionelle und gängige
Mathematiksoftware diese 4D-Rendering-Methoden hartnäckig ignoriert.
Seit Jahren verwende ich den unprätentiösen Graphing Calculator 4.0
von Pacific Tech, der 4D vollständig integriert hat.

Wie auch immer, willkommen zum Ausprobieren hier:

https://www.desmos.com/3d/x7w6jdpxgx?lang=nl : Methode und Beispiele
https://www.desmos.com/3d/krq32ylqjd?lang=nl : 3-Sphäre
https://www.desmos.com/3d/3ci8qmdzaf?lang=nl : Clifford-Torus
https://www.desmos.com/3d/dwujqpjry3?lang=nl : Tesserakt

Mehr hier:
https://www.wugi.be/qbinterac.html
https://www.youtube.com/@wugionyoutube/playlists (suche nach "4D" und
"Complex Functions")

Fwiw, is another test. I call it Spiralina... ;^)

https://i.ibb.co/Khg4TKK/image.png

Beauty! Trajectory bundles: now these, being curves, can be done in 4D
as well...

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 15-9-2024 om 23:06 schreef Chris M. Thomasson:

On 9/15/2024 1:54 PM, Chris M. Thomasson wrote:
[...]

Actually, here is some of my test code for one of my experimental
stacked mandelbulbs. The code generates ppm's images as its final
image stack for any volumetric renderer to get a hold of them. Can
you run the code?

https://pastebin.com/raw/07TWQQYF

https://groups.google.com/g/comp.lang.c/c/ve7UtNFAYH0

Iirc, it should create something akin to the following volumetric
result of mine:

https://i.ibb.co/zrHBdcz/image.png

An alien baby chick?

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 17-9-2024 om 08:45 schreef Chris M. Thomasson:

On 9/16/2024 3:31 AM, guido wugi wrote:

Op 15-9-2024 om 23:06 schreef Chris M. Thomasson:

On 9/15/2024 1:54 PM, Chris M. Thomasson wrote:
[...]

Actually, here is some of my test code for one of my experimental
stacked mandelbulbs. The code generates ppm's images as its final
image stack for any volumetric renderer to get a hold of them. Can
you run the code?

https://pastebin.com/raw/07TWQQYF

https://groups.google.com/g/comp.lang.c/c/ve7UtNFAYH0

Iirc, it should create something akin to the following volumetric
result of mine:

https://i.ibb.co/zrHBdcz/image.png

An alien baby chick?

Not sure! Some sort of stone idol or something? Actually, this has an
alien face in it. Very insect like:

https://youtu.be/k9qpHcfiDho

A look at the face:

https://i.ibb.co/rw6NxH0/image.png

Can you see it?

Rather some scanning result :)

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 16-9-2024 om 21:49 schreef Chris M. Thomasson:

Trajectory bundles: now these, being curves, can be done in 4D as
well...

I need to study existing your work to see where I should/could plot
all of my vectors that have non-zero 4d w's as in (x, y, z, w). That
would be interesting. I just need to find some time to give it a go,
been really busy lately. Shit... Well... Now, when I do it, I will
start small and create 4 axes in the 3d plane. Ask you a lot of
questions... ;^) It would be a learning experience for me.

Also, I think it might help a bit if I colored any vector with a
non-zero w with a special color spectrum... Humm... Keep in mind that
I am only plotting the (x, y, z) parts of the vectors that my field
algorithm generates. So, I can see how non-zero w's cast an influence
upon the field wrt the (x, y, z) parts of an n-ary vector.

I can do the coloring thing in my current work. If any vector has a
non-zero w, make its color _unique_ among all colors used in the field render. Humm...

I propose you try this example file.
bolnorm4D. Parabola | Desmos <https://www.desmos.com/3d/igi6shir3e?lang=nl>

A graph of the complex Parabola w=z^2.

The axes can modified/put to rotation with one of two angle control sets
(or both;-) :
1. "initial axis position controls", a 'spherical coordinate'-like set
of angles α,β,γ,δ; and
2. "axis plane rotation controls", a set of angles for the six possible axis-plane rotations: ζ1,η1,ζ2,η2,ζ3,η3.
The resulting projected axis points are called X,Y,Z,V, defined by 3D coordinates.

A 4D coordinate (a,b,c,d) is graphed as a point
E(a,b,c,d)=aX+bY+cZ+dV.
The graph w=f(z) or u+iv=f(x+iy) is produced by the 4D points
E(x,y,u,v)

The function definitions are stated apart, eg,
Fre(x,y)=xx-yy, Fim(x,y)=2xy
(Desmos lacks yet complex function handling)

A surface is defined with variables u,v (not to be confused with
variables u+iv=w!!).
A curve is defined with variable t. Parameter curves are obtained using
a parm list L=[a,b...c]

The parabola is rendered by
E(u,v,Fre(u,v),Fim(u,v))
In polar coordinates we'd have
E(u cos v, u sin v, Gre(u,v),Gim(u,v))

You can try out 4D rendering right away with this file!
If you have a function definition with parms u and v, or t and L,
or x,y,z,w, making z a 3D-function z=f(x,y) and w a list or a slider parm),
all you need to render is
E(u,v,F1(u,v),F2(u,v)) or
E(t,L,F1(t,L),F2(t,L)) and another by swapping t and L, or
E(u,v,f(u,v),w)
...

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 17-9-2024 om 21:52 schreef Chris M. Thomasson:

On 9/17/2024 12:46 PM, guido wugi wrote:

Op 16-9-2024 om 21:49 schreef Chris M. Thomasson:

Trajectory bundles: now these, being curves, can be done in 4D as
well...

I need to study existing your work to see where I should/could plot
all of my vectors that have non-zero 4d w's as in (x, y, z, w). That
would be interesting. I just need to find some time to give it a go,
been really busy lately. Shit... Well... Now, when I do it, I will
start small and create 4 axes in the 3d plane. Ask you a lot of
questions... ;^) It would be a learning experience for me.

Also, I think it might help a bit if I colored any vector with a
non-zero w with a special color spectrum... Humm... Keep in mind
that I am only plotting the (x, y, z) parts of the vectors that my
field algorithm generates. So, I can see how non-zero w's cast an
influence upon the field wrt the (x, y, z) parts of an n-ary vector.

I can do the coloring thing in my current work. If any vector has a
non-zero w, make its color _unique_ among all colors used in the
field render. Humm...

I propose you try this example file.
bolnorm4D. Parabola | Desmos
<https://www.desmos.com/3d/igi6shir3e?lang=nl>

[...]

This moves the object along the 4d axis right:

https://i.ibb.co/k1XR3FT/image.png

13: s_p

right?

Exactly, an example to 'move along v' (your w).

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 19/09/2024 om 21:39 schreef Chris M. Thomasson:

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

Hallo,

[...]

Fwiw, I finally ported my work to a realm where I have real time. My experimental modern opengl thing... I can fly around my simulations. Can
you get to the following link to an animation of a simulation?

https://www.facebook.com/chris.thomasson.31/videos/1217820042822507

Seems like a little ball game between anemonies ;-)

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Op 19/09/2024 om 23:19 schreef Chris M. Thomasson:

On 9/19/2024 2:04 PM, wugi wrote:

Op 19/09/2024 om 21:39 schreef Chris M. Thomasson:

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

Hallo,

[...]

Fwiw, I finally ported my work to a realm where I have real time. My
experimental modern opengl thing... I can fly around my simulations.
Can you get to the following link to an animation of a simulation?

https://www.facebook.com/chris.thomasson.31/videos/1217820042822507

Seems like a little ball game between anemonies ;-)

Indeed it does! Humm... Actually, here is an older one I made. This has
some of my personal midi music to go along with it:

https://youtu.be/HwIkk9zENcg

:^)

What fields look like octopuses or anemonies?

Here's a 4D Clifford torus rotating with my
"fraktet", a Tue-Morse-like tune with three "fractal" voices. https://www.youtube.com/watch?v=R96uu4Il9Jo&list=PL5xDSSE1qfb6c7UHcURl6wXh0pH4ARB75&index=6
the score: https://www.youtube.com/watch?v=zGZ2aG_yviU&list=PL5xDSSE1qfb6ybEuZ5XWxpKUIFKdO9rK7&index=13

--
guido wugi

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)