The 4-Dimensional Mandelbrot Object

Introduction

This page shows images of the 6 2-dimensional orthogonal slices that illustrate the 4-dimensional structure of the combined Mandelbrot and Julia sets for a few points. The patterns are the result of a faily random walk, starting at (0,0,-1,0), the centre of the largest circle in the Mandelbrot plane. Co-ordinates are of the form (zx, zy, cx, cy), (zx, zy) being the initial Z value in the iterations and (cx,cy) the constant.

The Mandelbrot set is generated using the 'M' plane with (zx,zy) = (0,0), the julia sets by fixing (cx,cy) and varying (zx, xy). See the main Python page for the detailed explanation.

Patterns

This section contains examples of the sets of 6 intersecting planes for a number of points. Som images contain links to larger versions. This is indicated by the clickable links, such as "[big]", beneath the image.

(0,0,-1,0) : Magnification = 1

We start off on the Mandelbrot plane in the centre of the largest circle at (-1,0).

example example
A B
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C D
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M J

(0,0,-0.5,.575) : Magnification = 10

Still on the Mandelbrot plane we move up to one of the circles attached to the cardioid at (-0.5,0.575).

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A B
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C D
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M J

(0,0,-0.56,.65) : Magnification = 30

Still on the Mandelbrot plane we move NW toward the point where the 5 strands meet. (-0.56,0.65) is outside the mandelbrot set so now the Julia set (image J) has become disconnected.

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A B
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C D
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M J

(0,0,-0.53,0.669) : Magnification = 200

NE of the previous point is one of the mini-mandelbrots (-0.53,0.669). This takes us back inside the mandelbrot set, so the Julia set becomes reconnected. You can't really appreciate that from this view as we have zoomed in too far.

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A B
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C D
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M J

(0,0,-0.53,0.669) : Magnification = 2

So, staying at the same point we zoom right back out. Now the structure of the Julia set can be appreciated, the previous view was well inside the central black region. The structure in the other planes has also become clearer too.

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A B
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C D
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M J

(0,-0.5,-0.325,0.669) : Magnification = 4

Now we move off the mandelbrot plane in the zy direction, taking us closer to the spiral region at the top left of the B plane view.

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A B
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C D [Big]   [Bands]
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M J

(0,-.875,-0.075,0.669) : Magnification = 8

To produce these next views we continue moving NW in the B plane, getting closer to the spiral region at the top left of the B plane.

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A B [Big]
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C D [Big]
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M J [Big]

(0,-.894,-0.056,0.669) : Magnification = 20

We're moving closer to the centre of the spiral structure in the B plane. The Julia set at this position is quite spectacular, click on the [Big] link for a better view.

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A B
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C D
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M J [Big]

(0,-.894,-0.056,0.669) : Magnification = 200

Staying in the same place we zoom in by a factor of 10.

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A B
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C D
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M J

(0,-.902,-0.056,0.673) : Magnification = 1000

And by another 5, moving a bit to the SW on the B plane.

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A B
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C [Big] D
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M J

(0,-.902,-0.056,0.6716) : Magnification = 4000

We zoom into the above images by another factor of 4, aiming for the blue cluster at the bottom of the M plane view. Now all the views are very similar.

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A B
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C D
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M [Stars] J

(0,-.875,-0.069,0.6716) : Magnification = 200

Here we've zoomed back out by a factor of 20 and positioned ourselves inside one of the gaps that cut through the solid black regions in the magnification 20 view above.

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A B
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C D
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M J

(0,-.875,-0.0650,0.6707) : Magnification = 1000

Now we've zoomed back in again, aiming for the top right end of the upper black band in the previous B image, where the solid black pattern starts disintegrating. The centre point is inside the highly confused (unresolved) region away from the black and probably lies outside the set. The corresponding Julia image (J) is too complicated to tell if it's connected or not.

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A B [Big]
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C D
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M J

(0,-.875,-0.0650,0.6707) : Magnification = 1000

As if the previous image wasn't confusing enough we've taken a giant leap by another factor of 10 to see what lies beneath. Again as we zoom in the images start becoming more similar, a property I was aware of with the Julia and Mandelbrot sets, but wasn't expecting to occur in the other 4 orthogonal planes.

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A B
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C D
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M J

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(c) John Whitehouse 2010 - 2023