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Guido "Wugi" Wuyts @ Dilbeek, Belgium, Europe, World, Solar System, Milky Way, Local Cluster, ...

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... QBasics :  originally a QB application site, now with new Grapher applications

... QBComplex :  Complex Functions  Y = Y(X), now also other higher-dimensional objects

... 4D :  visualisation as 4D surfaces in 2D/3D projections

... *** NEW 2016 ***  Other 4D tools and visuals

... *** NEW 2017 ***   Wugi's Youtube Videos here

... references on the net


Maths

Visualising Complex Functions Y = F(X) as 4D objects

Let X = x1 + ix2, Y = y1 + iy2 be complex variables. A Complex Function Y = Y(X),
or its  real equivalent (y1, y2) = F(x1, x2), will correspond to a 2D object, or surface, in a special-structured 2x2D, or 4D space.

We can't see such objects directly, BUT we can see their 3D or 2D projections, or even "3D in 2D" projections.
The video next explains the method.
So this site shows, using the method described, and with different tools, various functions, mainly typical ones like conics and trigonometrics.

According to their "history", the visuals include snapshots, animated gifs, and Youtube links for video.
By now there is quite some material available on the web, on graphic rendering of complex space. See for instance some examples below and in complex net.pps. However, I still miss the basic approach and picturing you'll find here: if there exist similar pages I'll be glad to receive reference at them !
I 'discovered' this visual description of complex space back in highschool, while unsatisfied there with the treatment of 'complex coordinates' and some absurd theorems about 'isotropic straight lines' (being perpendicular to themselves, and having zero distance between all their points).

My then teacher was amazed with my first tentative descriptions and drawings of 3D parametric lines belonging to such objects. My later university prof, typically for math wizzes, didn't think the topic worth much bothering about. But he did give me Thomas Banchoff as a reference back in 1979 ! (I had the priviledge of exchanging mails with him around the end of 2015, but all was lost due to, guess what, computer problems).
QuickBasic

For examples of my 1978 or so hand-drawn visuals (tentative sketches, and mm-paper drawings with the help of little HP calculator programs:-), and a 1985 abouts Amigabasic screenshot, see Complex past.pps
Then came QBasic, and my "comprehensive" complex.bas and complex.exe program,
see an intro and some menus in Complex intro.pps
Notes

Circle - HyperbolaMost menu items in the program should be clear by themselves. You pick a figure and a rotation, and a pseudoanimation will be shown. However, as this program dates back from Amigabasic (!) where each image would need a painstaking minute or so to build,  the sole "animation" effect,  the default number of steps for a 360 rotation is but 9. You should first increase this in the menu Preferences>More>Number of images. Also, the line scanning option in the menu  Preferences>Parameter curves,  helpful after the slow buildup in the early systems, is now outdated (or should be re-tuned...).
Math Grapher 3D

During 2016, I discovered this nice free smartphone app:
Math Grapher 3D

It produces some nice results, amazing for a smartphone tool.
Plot
                            circ-hypb.pngUsing the projection trick
X = x1 + A y1 + B y2
Y = x2 + C y1 + D y2
Z = E y1 + F y2
with coefficients A...F corresponding with projection angles of y1 and y2 axes onto the XYZ system,
this app is able to render some of the complex functions considered here.

Graphing Calculator 4.0

The smartphone app above having its limitations, eg, with entering long or repetitive formulas, I went looking further. And found a tool that has been there already for a number of years, witness it being host of one of the references on the net down this page, since at least 2008 (!) ...
It's called the Graphing Calculator 4.0, and has proved to be a powerful tool, easy to learn and use.
A free Viewer can be downloaded there. Then, after downloading the Grapher files (they are actually text files name.gcf) and having installed the free Viewer you will be able to open and explore them that way!
graph
                            Circl-Hyp.pngThe Grapher accepts true complex input as w=w(z) or as 4-vectors, and renders it as 3D objects in some rotational movement. It allows saving one rotation as movie (or else I also used AVS4YOU screen capture), see some Youtube videos around this page.
Graphing Calculator 4.0 bis : 2D

Only after rethinking the 3D graphics of above, did I realise how it must also be possible to use the same Graphing Calculator 4.0 to make a 2D representation, similar to the QBasic pictures.

The results are displayed with a flat grey background, and may be  compared with both their 3D and QBasic analogs.

Download 2D grapher files with diverse animated rotations (1 zip file) here.
graph Circl-Hyp.png





Complex Functions visualised


Complex planes : Constant angle theorem

This is about a nice little theorem I discovered back then.

Just like
y = ax + b is the equation of a real Straight line,
Y = AX + B is the equation of a Complex plane ("straight surface").

The theorem states that for any Complex plane, all of its straight lines make a constant angle with the X-plane. So the notion of "angle" exists in complex space. By extension, between any two Complex planes there exists a (consant) angle.
See video next, developing this theorem. 

The Circle-Hyperbola : Y = 1 / X

In complex variables, "Circle" and "Hyperbola" are the same manifold (surface) in a different orientation. Other "orientations":
YY + XX = 1 ; YY - XX = 1
Hence, the presence of both circle and hyperbola curves on this surface. Notice also the two blades extending towards the asymptotic planes, here X=0 and Y=0.


Circle - Hyperbola

Grapher file

Other Youtube link




Circle - Hyperbola     Plot circ-hypb.png

graph Circl-Hyp.png     graph Circl-Hyp.png

The Parabola : Y = X2

A 4D paraboloid of a rotating parabola.

Parabola

Grapher file

Other Youtube link
Parabola     Plot Parab.png

graph Parab.png     graph parab 2D.png

The Exponential : Y = eX

Progressive rotation of an exponential curve, resulting in an asymptotic blade X=0, and a screw-form blade. The function is periodic along the imaginary X-axis with period 2pi.

Exponential

Grapher file

Other Youtube link
Exponential     Plot expon.png

graph Expon.png     graph Circl-Hyp.png

The Cosine : Y = cos X

A combination (sum for the Cosine, difference for the Sine) of the Exponential and its reverse. The asymptote blades ("zero") disappear and give way to a pair of screw blades. Where both meet, the (co)sine curve appears in the real plane, with periodicity 2p. The hyperbolic sine and cosine are also typical curves of this function surface.

Cosine


Grapher file

Other Youtube link
Cosine     Plot Cosin.png

graph Cos.png     graph Circl-Hyp.png

The Cosecant : Y = cosec X

A quarter of a period of the function, with a half "up" cosec curve in the real plane (X between 0 and p/2), and bordered by the asymptotic planes Y=0 and X=0, by cosech curves at one side, and a sech curve at the other.

Cosec

Grapher file

Other Youtube link


Cosec   

graph Cosec.png     graph Circl-Hyp.png

The Tangent : Y = tan X

A half period of the function (X between 0 and p/2), with a half tan curve in the real plane, and bordered by the asymptotic planes Y=0 and X=0, by cotanh curves at one side, and a tanh curve at the other.
(A less 'fluid' gif animation here, as this item caused an overflow bug for some rotation/step choices in the QB-program:-)

animgif/Tan.png

Grapher file

Other Youtube link
animgif/Tan.png   

graph Tan.png     graph Circl-Hyp.png

The Quadratic Hyperbola : Y = 1 / X2

A "square" Circle-Hyperbola, with a double bladed asymptot Y=0, a "squared" hyperbola in the real plane and a minimal closed centre curve somewhat like a doublewinding circle.

Double Hyperbola

Grapher file

Other Youtube link
Double Hyperbola     Plot Dble-Hypb.png

graph Dbl
                  Circ-Hypb.png     graph Circl-Hyp.png

More Complex function visuals
on my Youtube and Tumblr pages


Look at the videos for each explanation...



Youtube Playlists:

wugi's Visualization of Complex Functions

wugi's Complex Function Graphs



On Tumblr:

namelessengineerbel













wugi's Tumblr Files

Complex-function-w-z-z-w-u-iv-and-z-x

Morphing between complex functions Y = (-2)^X and Y = e^X, and back

Morphing-between-complex-functions-y-2-x-and

Complex-function-y-2-x-or-y1-iy2-e

Complex-functions-y-exp-ax-or-y1-iy2-exp









***NEW 2016***

The following came up as an extension of my "Complex Functions in 4D" topic, so they belong here as well.
functions of 4 variables (5D)

An idea I got after reading Quora question

Is-there-anyway-to-intuitively-visualize-a-function-of-4-variables-in-a-computer-as-in-f-R-4-rightarrow-R?

This function, z = z(x,y,u,v), concerns 5 parameters, that means, a 5D graph of a 4D manifold.
So, generally, that would count as not 'vizualizable'.


I proposed, however, juxtaposing two 3D graphs, by keeping  parameters constant pair-wise. That way we'd get
fig 1: u=c, v=d; z = z(x,y,c,d), and
fig 2: x=a, y=b; z = z(a,b,u,v)
Then let the "constants" a,b vary [Slider values !] along the x,y range, and fig 2 evolve with them;
likewise, let c,d, "Slider" along the u,v range, and fig 1 evolve with them.

This is perfectly possible with the Graphing Calculator 4.0.
The Graphing Calculator uses parameters xyz and x'y'z' for twin graphs. So one should keep in mind z' = z; x'=u and y'=v.

*****     *****     *****     *****     *****     *****

Next, I realised that the same method can be applied to two other "pairs of pairs of variables viz. constants":
z = z(x,b,u,d) and z(a,y,c,v), and
z = z(x,b,c,v) and z(a,y,u,d)
Initially I achieved this by creating two more "twin" grapher files, swapping the function variables and constants accordingly in the formulae.
Conclusion: This function is vizualized with three coupled twin graphs!
The drawback of this method is that each "constant" a,b,c,d occurs in the three twin graphs, but cannot be manipulated simultaneously to observe its "triple" effect.

What is needed really, is a set of 6 graphs synchronised by a single set of 4 slider values. Unfortunately Graphing Calculator does not allow multiple graphs beyond 2. So I thought up a trick to at least have a demo of what I would like to see: by off-setting the z function values for the two remaining pairs with a z0 value (z0 = 3 in my demo), so whereas the first pair remains at z = z + 0, the second pair is displayed at z + z0, and the third at z - z0. That way I managed to accommodate three graph pairs in one display pair.
Of course there is always a risk of overlap with this system. So what I would be glad to see is grapher software that can handle multiple graphs, each
independent but capable of being monitored by common parameters such as sliders.
! If you know of one, better yet, if you feel like translating this method into it, please let me know ! ;-)


The grapher files are useable as is.

The only things to change are: redefine the function z(x,y,u,v) and the ranges that one wants vizualized.


Pair of coupled functions and constants.          Grapher file1
In (x,y,z) : function z(x,y,c,d) and constants (a,b,0)
In (x',y',z') : function z(a,b,u,v) and constants (c,d,0)

Similar pairs for           Grapher file2 , Grapher file3
z(x,b,u,d) & (a,c,0), and z(a,y,c,v) & (b,d,0)
and
z(x,b,c,v) & (a,d,0), and z(a,y,u,d) & (b,c,0).

    


*****     *****     *****     *****     *****     *****
3 Pairs of coupled functions and constants.          Grapher file
In (x,y,z) : (offset values z0 = 0 ; +3 ; and -3)
function z(x,y,c,d) and constants (a,b,0)
function z(x,b,u,d) + z0 and constants (a,c,z0)
function z(x,b,c,v) - z0 and constants (a,d,-z0)

In (x',y',z') : (offset values ditto)
function z(a,b,u,v) and constants (c,d,0)
function z(a,y,c,v) + z0 and constants (b,d,z0)
function z(a,y,u,d) - z0 and constants (b,c,-z0)


***NEW 2016***

 
Clifford Torus (...)

The Clifford Torus is often represented by its 3D stereographic projection torus, actually a "skew" torus or Cyclide, see e.g.
wikipedia : Clifford_torus and
https://www.youtube.com/watch?v=8oZhO9asPxI .
Sometimes the "real" 4D torus is shown, see e.g.
https://www.youtube.com/watch?v=oVryLr9rm8E .

The Clifford torus evolves in 4D space: (x,y,u,v).
The Clifford torus' 
coordinates satisfy initially xx+yy=1, and uu+vv=1, so their sum=2. In other words, it resides (also when rotated) within the 3D shell of the hypersphere, or "3-Sphere", with radius r = SQRT(2).

The Clifford Torus being a 4D beast, it is often represented by a 3D projection. This is done, e.g., from the v axis point on the 3-Sphere, i.e. v = SQRT(2), toward the 3D space (x,y,u).
Depending on the torus' rotational position (always within its 3-Sphere), its 3D projection Cyclide however varies from a symmetrical torus onward, to an ever more skew one, to become an infinitely opened surface and, beyond, reverse to a torus but in inversed form... before starting another half such cycle, to return to its initial form.

I wanted to show both objects, the projected and its projection, "in the same picture".  Due to the overall 4D-in-2D rendering, both objects overlap in this animation, whereas in 4D space they wouldn't.

The first grapher file and video show this combination.
The (x,y,u) axes are shown in magenta, the v axis in white, since the latter is used as the projection origin towards 3D space (x,y,u). One may occasionally observe the projective correspondency of the white v-axis' extremity with the Torus' and its projective Cyclide's positions, in their extremities...

The second grapher file and video show the Clifford Torus only, now rotating in its 4D space (x,y,u,v), successively along the 6 planes (u,v) - (x,u) - (x,v) - etc.

The music is "Fraktet" - see under my
muziekte.htm#Instrumentaal.
Though dating from bygone times (when I was young:-) I discovered only recently that it is actuelly a
"Thue-Morse sequence", fractally elongated in 3 voices. The form is indeed a repeated theme AB-BA ... ABBA-BAAB ... ABBABAAB-BAABABBA, etc...
See also wikipedia : Thue-Morse_sequence .


A sequel.

Thinking about the 3-Sphere and the special position of the Clifford Torus in it, I then realised that a whole family of toruses exists, which moreover fills up the 3D space of that 3-Sphere.
These toruses have equations
xx + yy = rr and uu + vv = 2 - rr
The overall sum remains 2, so the embedding in the 3-Sphere is maintained.
The x,y circles vary their radius from 0 to SQRT(2), when the u,v circles do theirs from SQRT(2) to 0. They "meet halfway" with both values = 1, i.e. the Clifford Torus.

I wanted to visualise this feature too, once again with the "4D torus" only, and with 4D torus and 3D Cyclide joint.
So, these are the third and the fourth grapher files and videos...

Final remarks.

I had the impression that I had thought up some unpublished graphic material on the Clifford and other 3-Sphere toruses, but then I stumbled into this:
https://en.wikipedia.org/wiki/Talk:3-sphere
(see the projections... :-o)

Anyway, (...)


Clifford torus Clifford torus

Clifford torus Clifford torus

The 4D Clifford Torus and its 3D projection Cyclide :          Grapher file1

Clifford Torus     Clifford Torus




The Clifford Torus rotating in 4D :         
Grapher file2

Clifford Torus




Visiting the 3-Sphere with 4D Toruses (and their Cyclides) :         
Grapher file3




Visiting the 3-Sphere with 4D Toruses :         
Grapher file4

(...) and the 3-Sphere itself

I just found this well done recent video on the 3-Sphere, however without mention of the Clifford Torus (in French):
https://www.youtube.com/watch?v=dy_MUfBuq2I


It inspired me to these torus-less approaches to visualise the 3-Sphere itself:

1) one by letting a sphere evolve through it along a diameter, growing and shrinking as it marries the 3-sphere's "border", in the same way that, in 3D, a circle would marry a sphere's border while sliding through it; and

2) one by letting a sphere rotate within a 3-sphere of the same diameter, in the same way that, in 3D, a circle would rotate within a sphere of the same diameter.
! except that this being a 4D case, the plane of rotation of the sphere would rotate, not around an axis as in 3D, but around 2 independent axes, both perpendicular to the plane of rotation, or in other words, around the plane of axes perpendicular to it. (Notice that the sphere, while rotating along the plane of u- and v-axes,  has fixed rotation points on the x- and y-axes).

(...) I hope you enjoy 4D as much as I do !

1) By translation of spheres:          Grapher file1 4D  , Grapher file1 3D

3-Sphere
                  visualisations     3-Sphere
                  visualisations



2) By rotation of a sphere:          Grapher file2 4D  , Grapher file2 3D

3-Sphere
                  visualisations     3-Sphere
                  visualisations



References on the net

Complex Functions :

"complex functions"+4D - Google zoeken Google lookup
Thomas Banchoff's Home Page a start page
Th. Banchoff: On-Line Mathematics some beyond the third dimension
Th. Banchoff: Complex Exponential and Logarithm see   complex net.pps
Graphing Calc. PacificTech: 24 Views of the Complex Exponential Function see   complex net.pps
Robert_Liebo_Final.pdf (application/pdf Object) see   complex net.pps
Meshview: 4Dim figures see   complex net.pps
Websites related to "Visual Complex Analysis" No more portal, but it does yield the book of Tristan Needham
Thinking Like a Mathematician talking about a book on complex space and its 50 or 60 pictures, but no pictures         out of order?
Dr William T. Shaw "Complex Analysis with Mathematica"       out of order?
Davide P. Cervone (CV/Art): Exponential Tetraview referenced by Th. Banchoff's page
Tetraview - Wikipedia, the free encyclopedia referenced by Th. Banchoff's page
exp z -4D see   complex net.pps       out of order?
sitov_sergei.pdf   see fig 4.8 see   complex net.pps
NEW 2017

Complex functions graph: YT Playlist by kojocho2
almost exactly some of "my" Complex function surfaces
Exponential function in 4D: YT by abacuswizard

Grapher - 4D Sine Wave Rotations : YT by Tony Burk






Other 4D visuals :

wikipedia : Clifford_torus The Clifford torus and its 3D Cyclide explained
https://www.youtube.com/watch?v=8oZhO9asPxI A 3D Cyclide
https://www.youtube.com/watch?v=oVryLr9rm8E The 4D Clifford torus itself
wikipedia : Thue-Morse_sequence About the endless and non-periodic abba baab sequence
Is-there-anyway-to-intuitively-visualize-a-function-of-4-variables-in-a-computer-as-in-f-R-4-rightarrow-R? The question of visualising z = z (x,y,u,v)
https://en.wikipedia.org/wiki/Talk:3-sphere Visuals like mine, of the Clifford torus, not withheld for "not understood"
https://www.youtube.com/watch?v=dy_MUfBuq2I The 3-Sphere explained and visualised