True 4D graphs of Complex Functions
Given complex variables z=x+iy and w=u+iv, a complex function w=f(z) can be rendered visually by a surface in (x,y,u,v) space. Most if not all ‘traditional’ visualisations scratch attempts to visualise 4D from the start (°) and confine themselves to 2D and 3D visualisations, separating z and w variables, omitting the 4th variable, or using colour coding for the 4th.
(°) And yet the tesseract is commonly represented in its ‘full’ 4D appearance, using 4 axes. So, why not complex functions?
Since the 70ies (!) I have been doing ‘true 4D’ graphs of complex functions. Lately I have discovered and used Desmos3D as a generator of the 4D space for them. BTW the 4D rendering method is of course also appropriate for other 4D surfaces than complex functions, and I have included some: Clifford tori, and constituents of the 3-sphere.
This gallery is my first attempt in Imaginary.org to give an impression of how complex functions look like when represented ‘full 4D’. Sorry if the images are not sufficient HD, but I think they’re worth enough to study as such. I’d welcome work on it by others to produce more impressive pictures! BTW I run a math contest precisely inviting students to develop an app doing just that! See in the links for more detail.
About the ‘true 4D (and 3D)’ method:
https://www. wugi.be/mijndocs/compl-func-visu.4D3D.pdf
About the ‘True 4D and 3D’ math competition:
https://sites. google.com/view/true4d3d-contest
Main webpage:
QB-Complex
Desmos webpage:
Wugi’s DesmoComplex
Youtube channel:
https://www. youtube.com/@wugionyoutube/playlists
| See Desmos file link in description |
4D axes set in Desmos3D
Desmos file: https://www. desmos.com/3d/a4caeed740?lang=nl
(unselect the ‘Circle-Hyperbola w=1/z’ entry to see only the axis system)
| Desmos file see description |
The function w=1/z : a 'Circle-Hyperbola'
Contains the real function u=1/x, a hyperbola.
But as a complex surface it is of the same form (save for a factor of sqrt2) as the surfaces of the functions
z^2 +/- w^2 = +/-1,
ie, (real) circle, hyperbola and imaginary circle equations.
Desmos file: https://www. desmos.com/3d/a4caeed740?lang=nl
| Desmos file: see description |
The function w=1/z^2
Simple asymptote (one ‘blade’) w=0, double asymptote (‘double blade’) z=0.
Desmos file: https://www. desmos.com/3d/ccd62239f4?lang=nl
| Desmos file: see description |
The Exponential w=exp z
One period shown: y=-π to +π. Asymptote for negative x. Exponential ‘blade’ for positive x.
Desmos file: https://www. desmos.com/3d/frvryuwj30?lang=nl
| Desmos file: see description |
The Cosine w=cos z
One period shown: x=-π to +π.
Is the combination of a half ‘exponential’ and its inverse. Both asymptotes disappear, absorbed by the counterparts’ ‘blades’: two opposite blades, joining along the real cosine curve.
Desmos file: https://www. desmos.com/3d/97l2phxgs3?lang=nl
| Desmos file: see description |
The Tangent w=tan z
One period shown: x=-π/2 to +π/2. Real tangent curve running ‘north-south’ in the middle.
Desmos file: https://www. desmos.com/3d/okfzaouikc?lang=nl
| Desmos file: see description |
3-Sphere, shown as a stack of growing-shrinking spheres.
Like a circle is generated by growing/shrinking pairs of points along a diameter segment in the 2nd dimension, and
a sphere by growing/shrinking circles along the 3rd dimension, so
the 3-sphere is generated by growing/shrinking spheres along the 4th dimension.
A set of red circles is ‘scanning’ each participating sphare in turns.
Desmos file: https://www. desmos.com/3d/8b9c08d5bd?lang=nl
| Desmos file: see description |
3-Sphere, shown as an 'orange' of rotating spheres.
Like a circle is generated by a rotating pair of points into the 2nd dimension, and
a sphere by a rotating circle into the 3rd dimension, so is
a 3-sphere by a rotating sphere into the 4th dimension.
The picture shows an ‘orange’ of successive positions of the rotating sphere, generating the 3-sphere.
Desmos file: https://www. desmos.com/3d/tbhzomiuk2?lang=nl
| Desmos file: see description |
Clifford torus with 4 sets of circles.
The Clifford torus has the equation
(x^2+y^2) = 1/2 =(u^2+v^2)
It has two x two sets of circles: a pair of ‘latitude’ and ‘altitude’ circles, and a pair of so-called ‘Villarceau’ circles.
The former pair is shown as the ‘horizontal and vertical’ ones,
the latter pair as the oblique ones.
Desmos file: https://www. desmos.com/3d/bc50fb564e?lang=nl
(zoom for larger picture)
