Bijecting 4D objects onto the complex plane

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• 3c
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Bijecting 4D objects onto the complex plane

« on: October 09, 2020, 03:07:59 PM »
Visualizing 4D objects like quaternion Julia sets is often done by slicing 3D cubes and displaying those on a 2D screen. This however loses information as a whole dimension is disregarded.

I tried something different here. As the 4D continuum R^4 has the same cardinality as the complex plane (already known to Cantor), one could take the 4D coordinates of every point in a 4D object (like e.g. non-escaping) and biject them onto the plane. That way the 2D image is the 4D object and not a slice - quite interesting.

I have no idea whether this could lead to artistically appealing images. Has anyone done explorations in that regard? Good or bad bijections to use?

The bijection attempt I came up with is the obvious one based on the decimal string expansion of numbers in the half-open unit interval [0.1).

If a,b are two such expansion with digits $$a=0.a_1a_2a_3...$$ and $$b=0.b_1b_2b_3...$$ (and infinite trailing zeros if needed), the bijected number is the intertwined digits $$0.a_1b_1a_2b_2a_3b_3...$$. This is injective as if a and a' differ by at least one digit, the intertwined numbers (a,b) and (a',b) differ at an odd position, and if two numbers c, c' differ by at least one digit, if they are split into (a,b) and (a',b') the two pairs cannot be identical.

However, I currently have not dealt with numbers like "0.199999999..." and "0.2" which are identical in R but have different decimal digit expansion, so they map here onto different images. So the bijection here is not actually from R^4 to R^2, but rather one from textstrings to textstrings interpreted as numbers. (Literature search awaits me).

The image below depicts a solid 4D hypersphere at center (0.5,0.5,0.5,0.5) with radius just below 0.5. As a plausibility check, the detected number of interior pixels was compared with the volume formula for hyperballs (wikipedia) and they match.

The image looks like someone has picked out points from the sphere and put them spaced onto a sheet of paper (which basically the intertwining does), so I cannot tell which points are adjacent in the 4D sphere with a black&white image.

Next step is sub-coloring parts of the hypersphere and transfer those colors onto the 2D image.

• 3c
• Posts: 840

Re: Bijecting 4D objects onto the complex plane

« Reply #1 on: October 10, 2020, 03:03:26 PM »
"The 4D basilica"

Here I used David Makin's formula for the 3D basilica (x^2-y^2-z^2-1, 2*x*y, 2*z*(x-y) ) extending to w_new := 2*w*(y-z), the coloring indicates the neighbourhood in the hypercube of non-escaping points (every axis was split into left and right and differently colored). Depicted is the [-1..1]^4 cube (coordinates transformed 1:1 into [0..1) before performing the intertwining bijection).

There are some interesting saw-tooth like features and a wave-y "height" of the 2D image, which looks very vaguely like the 2D basilica itself. The overall shape resembles a sound wave in some recording software.

• 3c
• Posts: 840

Re: Bijecting 4D objects onto the complex plane

« Reply #2 on: October 17, 2020, 10:43:32 PM »
I tried bijecting my old 3D Julia sets interweaving two coordinate axis while leaving the third intact.

The image below shows { x^5-y^5 , -0.25*x+2x^2*y^2 , z^2-y^2-x^2 } + { -0.67187 , -0.95315 , -1.78125 } which has a 2-cycle (upper left, turquois = immediate basin, blue = attraction basin).

The three bijections depict a nice stripe pattern (stems from the interweaving where one coordinate takes precedence as it determines the first decimal after the radix point giving the main location).

I particularly like the upper right - it looks like a DNA fingerprint with two fluorescent dyes.

Next is bijecting a 4D object into a 3D cube, checking different axis-mergings.

• 3c
• Posts: 840

Re: Bijecting 4D objects onto the complex plane

« Reply #3 on: October 18, 2020, 09:54:50 PM »
First impression of a 4D -> 3D -> 2D bijection chain. The stripe pattern as derived from the interweaving bijection is also seen in 3D. The object has several rotational symmetries (lower right). Coloring represents quadrant in the 4D hypercube but the points quite strongly get splattered around.

f_new := { x^2-y^2-z^2 , 2xy , 2w(x-y) , 2z(y-z) } + { -1,0,0,0 }

I'm now looking for a bijection that (partially) respects vicinity (probably an enumeration technique for Q^n).

• 3c
• Posts: 840

Re: Bijecting 4D objects onto the complex plane

« Reply #4 on: October 20, 2020, 07:53:30 PM »
Bijecting a Quaternion Julia set z^2+c from http://paulbourke.net/fractals/quatjulia/ with c={ -0.08,0.0,-0.8,-0.03 }.

The 4D->2D bijection shows circle-based shapes with increasing "holes". The upper middle 4D->3D shows a stack of planes in the form of a ball, quite interesting. Lower row shows - instead of bijecting - connected slices at specific axis values.

Color indicates from which 4D quadrant the pixel comes. The slices adhere to neighbourhood relations, the bijections not in all cases (e.g. 0.19 and 0.29 biject to 0.1299, whereas the nearby 0.20 and 0.30 biject to 0.230, so quite a different location).

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