Discussion:
4D Visualisierung
(too old to reply)
guido wugi
2024-08-28 19:30:51 UTC
Permalink
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
Chris M. Thomasson
2024-08-28 19:38:05 UTC
Permalink
Post by guido wugi
Hallo,
[...]

Actually, it's impossible to visualize a true tesseract in 3d space?
Chris M. Thomasson
2024-08-28 19:49:07 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
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...
guido wugi
2024-08-28 19:55:43 UTC
Permalink
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(0, 0, 0, 1)
in a 3d space? Humm...
I've been doing that for a few decades by now ;o)
--
guido wugi
Chris M. Thomasson
2024-08-29 18:39:54 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(0, 0, 0, 1)
in a 3d space? Humm...
I've been doing that for a few decades by now ;o)
You can't just create another axis in 3d space and say its 4d because
this axis exists in 3d space. Now, with some of my n-ary field work I
can add a 4d point (a non-zero 4d component of a vector) to the field
and see how it effects it, but I cannot actually plot a 4d point. I can
just see how the 4d field point mutates the results.

The true 4d axis is not visible... ;^)
FromTheRafters
2024-08-29 20:15:24 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(0, 0, 0, 1)
in a 3d space? Humm...
I've been doing that for a few decades by now ;o)
You can't just create another axis in 3d space and say its 4d because this
axis exists in 3d space. Now, with some of my n-ary field work I can add a 4d
point (a non-zero 4d component of a vector) to the field and see how it
effects it, but I cannot actually plot a 4d point. I can just see how the 4d
field point mutates the results.
The true 4d axis is not visible... ;^)
There is, in a way, a way to imagine it though. We can't really draw a
cube on a piece of paper, but if a cube were backlit in a way which
casts a shadow on the plane of paper - we must move it in time to
ascertain its shape. Sometimes it looks square and sometimes a hexagon.
Consider now your mental image of a cube, and imagine it being a cube
in 4D, just a projected shadow the actual object, that you must see in
motion to ascertain its shape.

That having been said, mathematically it doesn't matter that we can't
get a clear mental 4D image.
Chris M. Thomasson
2024-08-29 20:32:26 UTC
Permalink
Post by FromTheRafters
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(0, 0, 0, 1)
in a 3d space? Humm...
I've been doing that for a few decades by now ;o)
You can't just create another axis in 3d space and say its 4d because
this axis exists in 3d space. Now, with some of my n-ary field work I
can add a 4d point (a non-zero 4d component of a vector) to the field
and see how it effects it, but I cannot actually plot a 4d point. I
can just see how the 4d field point mutates the results.
The true 4d axis is not visible... ;^)
There is, in a way, a way to imagine it though. We can't really draw a
cube on a piece of paper, but if a cube were backlit in a way which
casts a shadow on the plane of paper - we must move it in time to
ascertain its shape. Sometimes it looks square and sometimes a hexagon.
Consider now your mental image of a cube, and imagine it being a cube in
4D, just a projected shadow the actual object, that you must see in
motion to ascertain its shape.
That having been said, mathematically it doesn't matter that we can't
get a clear mental 4D image.
Fair enough. We can try to get some ideas about it, but even the shadow
implies a light source. We will never get to see its full 4d beauty.
Actually, sometimes I think of an alpha channel wrt transparency as
being kind of like the 4'th dimension. We can see through things...
Think about being able to walk in the middle of things. Say a big
boulder. Well, if your 4'd component was non-zero, then you could walk
into the middle of the boulder because the boulder is in the 3d and you
with your non-zero 4d component is off axis. So, you can walk through
walls and shit. ;^) What that too far off the rails? Sorry.

Say if your standing on Earth and went into the off axis, the 4d with a
non-zero 4d component. You would be able to look down through the Earth
and out into space on the other side?
guido wugi
2024-08-28 21:08:06 UTC
Permalink
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(0, 0, 0, 1)
in a 3d space? Humm...
How do you plot a photo of a 3D scene?
--
guido wugi
FromTheRafters
2024-08-28 22:31:35 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(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.
guido wugi
2024-08-29 14:56:00 UTC
Permalink
Post by FromTheRafters
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(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
Chris M. Thomasson
2024-08-29 18:47:34 UTC
Permalink
Post by guido wugi
Post by FromTheRafters
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(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: :^)



What do you think?
guido wugi
2024-08-29 22:01:21 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Post by FromTheRafters
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(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: :^)
http://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
Chris M. Thomasson
2024-08-29 22:08:54 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
Post by FromTheRafters
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(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: :^)
http://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.
Fair enough. It just seems that to create another axis in 3d space and
say its 4d is not quite right. But, then again wrt my fields, I can have
attractors in 4d space with non-zero 4d components, and I can only see
what they do to the field as its rendered.
Post by guido wugi
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.
Are you familiar with the triplex numbers? How about using quaternions
to get 4d projections from Julia sets? This is a nice one:



I need to find my anaglyphic glasses! :^)
Chris M. Thomasson
2024-08-29 22:10:12 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
Post by FromTheRafters
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(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: :^)
http://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".
Agreed. Now, I think we can convert any volumetric image into a
hologram, right?
Post by guido wugi
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.
Chris M. Thomasson
2024-09-15 07:21:34 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
Post by FromTheRafters
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(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: :^)
http://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".
Agreed. Now, I think we can convert any volumetric image into a
hologram, right?
[...]

Have you ever messed around with DICOM?
Chris M. Thomasson
2024-08-28 23:40:20 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(0, 0, 0, 1)
in a 3d space? Humm...
How do you plot a photo of a 3D scene?
Are you referring to equirectangular projections for 360 plots?
Chris M. Thomasson
2024-08-28 23:44:33 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
Actually, it's impossible to visualize a true tesseract in 3d space?
(0, 0, 0, 1)
in a 3d space? Humm...
How do you plot a photo of a 3D scene?
You can take slices of it as in a volumetric. I have some pretty
interesting volumetrics such that each frame is where a 3d printer head
should be. Here is an simple example:

https://www.facebook.com/share/p/9NVZ8qwmQjQ6bsJe/
Post by Chris M. Thomasson
Are you referring to equirectangular projections for 360 plots?
Chris M. Thomasson
2024-08-29 20:43:02 UTC
Permalink
On 8/28/2024 12:30 PM, guido wugi wrote:
[...]

Fwiw, here is a complex 3d field from one of my experimental n-ary field
algorithms. This is in a 4d space, however every point plotted in here
has a zero 4d component. I should have some time to experiment with it
wrt adding 4d points that have non-zero 4d components:

Loading Image...
Chris M. Thomasson
2024-08-29 20:55:34 UTC
Permalink
Post by Chris M. Thomasson
[...]
Fwiw, here is a complex 3d field from one of my experimental n-ary field
algorithms. This is in a 4d space, however every point plotted in here
has a zero 4d component. I should have some time to experiment with it
https://i.ibb.co/hstKmyX/ct-p30.png
Here is what it looks like by adding in a single 4d point (0, 0, 0, 2)
and using it as an attractor:

Loading Image...
Chris M. Thomasson
2024-09-11 08:12:17 UTC
Permalink
On 8/28/2024 12:30 PM, guido wugi wrote:
[...]

Check this out:



;^)
Chris M. Thomasson
2024-09-11 08:15:47 UTC
Permalink
Post by Chris M. Thomasson
[...]
http://youtu.be/IVR5I5mnrsg
;^)
Also, iirc, this experiment of mine has a vector with a non-zero 4d
component...


guido wugi
2024-09-11 20:22:33 UTC
Permalink
Post by Chris M. Thomasson
Post by Chris M. Thomasson
[...]
http://youtu.be/IVR5I5mnrsg
;^)
Also, iirc, this experiment of mine has a vector with a non-zero 4d
component...
http://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>

--
guido wugi
Chris M. Thomasson
2024-09-11 20:40:25 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
[...]
http://youtu.be/IVR5I5mnrsg
;^)
Also, iirc, this experiment of mine has a vector with a non-zero 4d
component...
http://youtu.be/KRkKZj9s3wk
I don't understand it, but they're beautiful graphics alright! But 4D?
Thanks. Iirc, it had a single vector in the field that had a w component
that was .0001 non-zero.

Here are some other fields, can you get to them? The FB is infested with
javascript... ;^o

https://www.facebook.com/share/p/Urg2ehsEBLq91NNt/

https://www.facebook.com/share/p/cFeUgQdE723LpT5N/

Now, the only way I can actually "see" the affects of any non-zero 4d
components (w) in (x, y, z, w) is how they alter the field lines. So, I
plot a field A with all zero w components. Then I plot field B with
non-zero w components. It's pretty interesting. My experimental n-ary
field is basically a source sink vector field.
Post by guido wugi
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>
http://youtu.be/1y6qrsJff-g
Chris M. Thomasson
2024-09-13 21:58:21 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
[...]
http://youtu.be/IVR5I5mnrsg
;^)
Also, iirc, this experiment of mine has a vector with a non-zero 4d
component...
http://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>
http://youtu.be/1y6qrsJff-g
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/
guido wugi
2024-09-14 09:08:52 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
[...]
http://youtu.be/IVR5I5mnrsg
;^)
Also, iirc, this experiment of mine has a vector with a non-zero 4d
component...
http://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>
http://youtu.be/1y6qrsJff-g
Here is an example of a 4d vector ping ponging through -1...1 wrt its
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
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
Chris M. Thomasson
2024-09-14 19:20:12 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
[...]
http://youtu.be/IVR5I5mnrsg
;^)
Also, iirc, this experiment of mine has a vector with a non-zero 4d
component...
http://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>
http://youtu.be/1y6qrsJff-g
Here is an example of a 4d vector ping ponging through -1...1 wrt its
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
https://www.facebook.com/share/v/DXKhRoGZmpB9fX5Y/
I don't see really the difference, sorry.
There is a massive difference. Humm... The animation is rather fast. Try
it in slow motion.
Post by guido wugi
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...
guido wugi
2024-09-14 21:41:00 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
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.
Post by Chris M. Thomasson
Post by guido wugi
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
Chris M. Thomasson
2024-09-14 23:46:50 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
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.
No. Its an integral part of the vector field. If I did create an
artificial axis in 3d and say okay, that's the 4d axis... Then I would
be able to plot points like (-1, .1, -.4, 2). Right now I am only
plotting the 3d components. The 4d field makes some very strange field
lines. Here is one where a strong attractor is in the 4'th dimension.
The field lines want to go off into a spiral like formation. Strange to me:

https://www.facebook.com/share/p/RqFqkaeLxQhN7UkS

Another one using different colors so its a little bit easier to see in
a sense:

https://www.facebook.com/share/v/H6jeFAV1TprPyyaR

Putting an attractor in the 4'th dimension causes some very strange
effects on the field. It almost makes the field lines wants to spiral
into a tube and go off in another direction. Keep in mind that I can
only see what the 4d does to the 3d lines. I can't actually plot a 4d point.

Humm... Perhaps I will experiment with creating an "artificial axis" in
3d to place 4d points on. So, (0, 0, 0, -1) and (0, 0, 0, 1) would both
be on that artificial axis. Humm... interesting.
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
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 ;-)
Well, humm... You know, I just might give it a go. However, I am a bit
busy right now with a pretty neat collision avoidance thing using my
field algorithm. Some examples:

https://www.facebook.com/share/v/rf2AQAFmivdtJxkf/

https://www.facebook.com/share/v/VRv48n97MRoMPxwn/


Well, the fun part is that my 4d fields generates points that are
actually 4d and plotting them on an artificial axis might be
interesting. Humm... Each axis would have a different color, so we can
see whats going on. Humm...
Post by guido wugi
Mainstream math apps ought to try it out. But they prefer going on
happily ignoring 4D possibilities. Except for tesseracts, for some reason.
It's nice work. Thanks for sharing it. :^)

thanks.
Chris M. Thomasson
2024-09-15 00:10:41 UTC
Permalink
Post by guido wugi
Hallo,
[...]

This is your artificial 4d axis, right?

Loading Image...
Chris M. Thomasson
2024-09-15 00:17:20 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
This is your artificial 4d axis, right?
https://i.ibb.co/rMqqp9k/image.png
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...
guido wugi
2024-09-15 09:11:51 UTC
Permalink
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
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).
Post by Chris M. Thomasson
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
Chris M. Thomasson
2024-09-15 19:28:04 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
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.
Post by guido wugi
Post by Chris M. Thomasson
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.
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...

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

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.
guido wugi
2024-09-15 20:21:49 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
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.
Post by guido wugi
Post by Chris M. Thomasson
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.
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).
Post by Chris M. Thomasson
Btw, I have created a lot of 3d volumes. Even in DICOM format. They
are all good candidates for holograms... :^)
https://www.facebook.com/share/p/n2nMhW5G2PhRzyfx
Sorry but half of your links are unavailable.
Post by Chris M. Thomasson
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
Chris M. Thomasson
2024-09-15 20:26:54 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
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.
Post by guido wugi
Post by Chris M. Thomasson
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.
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).
Post by Chris M. Thomasson
Btw, I have created a lot of 3d volumes. Even in DICOM format. They
are all good candidates for holograms... :^)
https://www.facebook.com/share/p/n2nMhW5G2PhRzyfx
Sorry but half of your links are unavailable.
Damn! Try this one, a screenshot of the link above:

Loading Image...
Post by guido wugi
Post by Chris M. Thomasson
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
2024-09-15 20:47:11 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Sorry but half of your links are unavailable.
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
Chris M. Thomasson
2024-09-15 20:58:32 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
Sorry but half of your links are unavailable.
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 ;-)
Indeed. Now, think of viewing any volumetric image with a
transparency... An alpha blend. The alpha channel wrt RGB vs RGBA kinda
sorta seems like a 4'th dimension quality in a strange sense, so to
speak... ;^)

Fair enough? Humm...
Chris M. Thomasson
2024-09-15 20:54:42 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
Post by Chris M. Thomasson
Post by guido wugi
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.
Post by guido wugi
Post by Chris M. Thomasson
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.
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).
Post by Chris M. Thomasson
Btw, I have created a lot of 3d volumes. Even in DICOM format. They
are all good candidates for holograms... :^)
https://www.facebook.com/share/p/n2nMhW5G2PhRzyfx
Sorry but half of your links are unavailable.
Post by Chris M. Thomasson
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.)
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
Chris M. Thomasson
2024-09-15 21:01:46 UTC
Permalink
On 9/15/2024 1:54 PM, Chris M. Thomasson wrote:
[...]
Post by Chris M. Thomasson
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
Here is the most relevant function:


/* The Power Stack Mandelbulb
___________________________________*/
void
ct_iterate_mbulb_pixel(
struct ct_plot* const plot,
struct ct_vec3 const z0,
struct ct_vec3 const c0,
double power,
long x,
long y,
unsigned int n
){
struct ct_vec3 zs = z0;
struct ct_vec3 cs = c0;

for (unsigned long i = 0; i < n; ++i)
{
double r = ct_vev3_length(zs);
double rpower = pow(r, power);

double angle0 = atan2(zs.z, sqrt(zs.x * zs.x + zs.y * zs.y));
double angle1 = atan2(zs.y, zs.x);

struct ct_vec3 znew;
znew.x = rpower * cos(power * angle1) * cos(power * angle0);
znew.y = rpower * sin(power * angle1) * cos(power * angle0);
znew.z = rpower * sin(power * angle0);

zs = ct_vec3_add(znew, cs);

if (r > 2.0)
{
return;
}
}

struct ct_rgb rgb = { 0, 255, 0 };
ct_plot_pixel(plot, x, y, &rgb);
}
Chris M. Thomasson
2024-09-15 21:06:55 UTC
Permalink
On 9/15/2024 1:54 PM, Chris M. Thomasson wrote:
[...]
Post by Chris M. Thomasson
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:

Loading Image...
guido wugi
2024-09-16 10:31:23 UTC
Permalink
Post by Chris M. Thomasson
[...]
Post by Chris M. Thomasson
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
https://i.ibb.co/zrHBdcz/image.png
An alien baby chick?
--
guido wugi
Chris M. Thomasson
2024-09-17 06:45:09 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
[...]
Post by Chris M. Thomasson
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
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:



A look at the face:

Loading Image...

Can you see it?
guido wugi
2024-09-17 16:25:59 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
[...]
Post by Chris M. Thomasson
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
https://i.ibb.co/zrHBdcz/image.png
An alien baby chick?
Not sure! Some sort of stone idol or something? Actually, this has an
http://youtu.be/k9qpHcfiDho
https://i.ibb.co/rw6NxH0/image.png
Can you see it?
Rather some scanning result :)
--
guido wugi
Chris M. Thomasson
2024-09-20 21:42:50 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
Post by Chris M. Thomasson
[...]
Post by Chris M. Thomasson
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
https://i.ibb.co/zrHBdcz/image.png
An alien baby chick?
Not sure! Some sort of stone idol or something? Actually, this has an
http://youtu.be/k9qpHcfiDho
https://i.ibb.co/rw6NxH0/image.png
Can you see it?
Rather some scanning result :)
Exactly. It is a 3d volume. Here is an example rendering:

https://www.facebook.com/photo/?fbid=334035991088739&set=a.110008616824812

A screenshot:

Loading Image...

It's pretty interesting... This can be 3d printed and/or holographically
projected. :^)

Chris M. Thomasson
2024-09-15 06:56:30 UTC
Permalink
Post by guido wugi
Hallo,
[...]

This shows a very strange effect of non-zero w components for attractors:

https://www.facebook.com/photo?fbid=1239041850588144&set=pob.100034470228086

Can you get to it?
Chris M. Thomasson
2024-09-15 06:59:36 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Hallo,
[...]
https://www.facebook.com/photo?fbid=1239041850588144&set=pob.100034470228086
Can you get to it?
Strange that the 4d to me right now is dark in that I cannot plot those
points with non-zero w components. I can plot their x, y, z for sure.
When an attractor goes non-zero w component, it casts strange effects on
the visible field lines, aka (plotting x, y, z). Is sort of dark. Not
WM's dark numbers, because my program generates all of the vectors.
Well, they are not dark. It's just that I don't know where to plot
point, say (-1, -2, 3, 4)? fuck!

;^o
Chris M. Thomasson
2024-09-15 22:07:24 UTC
Permalink
Post by guido wugi
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.
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
https://www.wugi.be/qbinterac.html
"Complex Functions")
Fwiw, is another test. I call it Spiralina... ;^)

Loading Image...
guido wugi
2024-09-16 09:17:11 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
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.
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
https://www.wugi.be/qbinterac.html
"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
Chris M. Thomasson
2024-09-16 19:49:57 UTC
Permalink
Post by guido wugi
Post by Chris M. Thomasson
Post by guido wugi
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.
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
https://www.wugi.be/qbinterac.html
"Complex Functions")
Fwiw, is another test. I call it Spiralina... ;^)
https://i.ibb.co/Khg4TKK/image.png
Beauty!
Thanks. :^) This has non-zero w's in it...
Post by guido wugi
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...
guido wugi
2024-09-17 19:46:33 UTC
Permalink
Post by 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
Chris M. Thomasson
2024-09-17 19:52:53 UTC
Permalink
Post by guido wugi
Post by 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:

Loading Image...

13: s_p

right?
guido wugi
2024-09-17 20:38:05 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
Post by 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>
[...]
https://i.ibb.co/k1XR3FT/image.png
13: s_p
right?
Exactly, an example to 'move along v' (your w).
--
guido wugi
Chris M. Thomasson
2024-09-19 19:39:00 UTC
Permalink
Post by guido wugi
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
wugi
2024-09-19 21:04:11 UTC
Permalink
Post by Chris M. Thomasson
Post by guido wugi
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
Chris M. Thomasson
2024-09-19 21:19:36 UTC
Permalink
Post by wugi
Post by Chris M. Thomasson
Post by guido wugi
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:



:^)
wugi
2024-09-19 22:05:39 UTC
Permalink
Post by Chris M. Thomasson
Post by wugi
Post by Chris M. Thomasson
Post by guido wugi
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
http://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.

the score:

--
guido wugi
Chris M. Thomasson
2024-09-20 19:54:33 UTC
Permalink
Post by wugi
Post by Chris M. Thomasson
Post by wugi
Post by Chris M. Thomasson
Post by guido wugi
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
http://youtu.be/HwIkk9zENcg
:^)
What fields look like octopuses or anemonies?
A vector field.
Post by wugi
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
https://www.youtube.com/watch?
v=zGZ2aG_yviU&list=PL5xDSSE1qfb6ybEuZ5XWxpKUIFKdO9rK7&index=13
I finally ported my code to my real time test using modern OpenGL:

https://www.facebook.com/share/v/7CnYAwg5BK3Y3fna/

https://www.facebook.com/chris.thomasson.31/videos/331769053297125

Now, can you view both links, or just one, or none!? Thanks.
Chris M. Thomasson
2024-09-20 19:57:28 UTC
Permalink
[...]
Post by wugi
What fields look like octopuses or anemonies?
A vector field.[...]
Actually, an n-ary vector field...
Chris M. Thomasson
2024-09-20 03:37:58 UTC
Permalink
On 8/28/2024 12:30 PM, guido wugi wrote:
[...]

Check this one out:


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