4-Dimensional Patterns

• Description
• Using the Python Source Code
• The Python Source Code
• Python Patterns Home Page

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 page for the detailed explanation.

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

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

A	B
C	D
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).

A	B
C	D
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.

A	B
C	D
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.

A	B
C	D
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.

A	B
C	D
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.

A	B
C	D [Big]   [Bands]
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.

A	B [Big]
C	D [Big]
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.

A	B
C	D
M	J [Big]

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

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

A	B
C	D
M	J

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

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

A	B
C [Big]	D
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.

A	B
C	D
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.

A	B
C	D
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.

A	B [Big]
C	D
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.

A	B
C	D
M	J

The Mandelbrot source code can be downloaded here - Mandelbrot.py and MandelbrotDriver.py. (Updated October 2008)

The batch file used to create the images shown on this page is shown below.

    @echo off

    python MandelbrotDriver.py -tA -s320x240 -c(0,0,-1,0) -fpics/a -l32
    python MandelbrotDriver.py -tB -s320x240 -c(0,0,-1,0) -fpics/b -l32
    python MandelbrotDriver.py -tC -s320x240 -c(0,0,-1,0) -fpics/c -l32
    python MandelbrotDriver.py -tD -s320x240 -c(0,0,-1,0) -fpics/d -l32
    python MandelbrotDriver.py -tM -s320x240 -c(0,0,-1,0) -fpics/m -l32
    python MandelbrotDriver.py -tJ -s320x240 -c(0,0,-1,0) -fpics/j -l32

    python MandelbrotDriver.py -tA -s320x240 -c(0,0,-0.5,.575) -fpics/a2 -l32 -m10
    python MandelbrotDriver.py -tB -s320x240 -c(0,0,-0.5,.575) -fpics/b2 -l32 -m10
    python MandelbrotDriver.py -tC -s320x240 -c(0,0,-0.5,.575) -fpics/c2 -l32 -m10
    python MandelbrotDriver.py -tD -s320x240 -c(0,0,-0.5,.575) -fpics/d2 -l32 -m10
    python MandelbrotDriver.py -tM -s320x240 -c(0,0,-0.5,.575) -fpics/m2 -l32 -m10
    python MandelbrotDriver.py -tJ -s320x240 -c(0,0,-0.5,.575) -fpics/j2 -l32 -m10

    python MandelbrotDriver.py -tA -s320x240 -c(0,0,-0.56,.65) -fpics/a3 -l32 -m30
    python MandelbrotDriver.py -tB -s320x240 -c(0,0,-0.56,.65) -fpics/b3 -l32 -m30
    python MandelbrotDriver.py -tC -s320x240 -c(0,0,-0.56,.65) -fpics/c3 -l32 -m30
    python MandelbrotDriver.py -tD -s320x240 -c(0,0,-0.56,.65) -fpics/d3 -l32 -m30
    python MandelbrotDriver.py -tM -s320x240 -c(0,0,-0.56,.65) -fpics/m3 -l32 -m30
    python MandelbrotDriver.py -tJ -s320x240 -c(0,0,-0.56,.65) -fpics/j3 -l32 -m30

    python MandelbrotDriver.py -tA -s320x240 -c(0,0,-0.53,0.669) -fpics/a4 -l32 -m200
    python MandelbrotDriver.py -tB -s320x240 -c(0,0,-0.53,0.669) -fpics/b4 -l32 -m200
    python MandelbrotDriver.py -tC -s320x240 -c(0,0,-0.53,0.669) -fpics/c4 -l32 -m200
    python MandelbrotDriver.py -tD -s320x240 -c(0,0,-0.53,0.669) -fpics/d4 -l32 -m200
    python MandelbrotDriver.py -tM -s320x240 -c(0,0,-0.53,0.669) -fpics/m4 -l32 -m200
    python MandelbrotDriver.py -tJ -s320x240 -c(0,0,-0.53,0.669) -fpics/j4 -l32 -m200

    python MandelbrotDriver.py -tA -s320x240 -c(0,0,-0.53,0.669) -fpics/a5 -l32 -m2
    python MandelbrotDriver.py -tB -s320x240 -c(0,0,-0.53,0.669) -fpics/b5 -l32 -m2
    python MandelbrotDriver.py -tC -s320x240 -c(0,0,-0.53,0.669) -fpics/c5 -l32 -m2
    python MandelbrotDriver.py -tD -s320x240 -c(0,0,-0.53,0.669) -fpics/d5 -l32 -m2
    python MandelbrotDriver.py -tM -s320x240 -c(0,0,-0.53,0.669) -fpics/m5 -l32 -m2
    python MandelbrotDriver.py -tJ -s320x240 -c(0,0,-0.53,0.669) -fpics/j5 -l32 -m2

    python MandelbrotDriver.py -tA -s320x240 -c(0,-.5,-0.325,0.669) -fpics/a6 -l32 -m4
    python MandelbrotDriver.py -tB -s320x240 -c(0,-.5,-0.325,0.669) -fpics/b6 -l32 -m4
    python MandelbrotDriver.py -tC -s320x240 -c(0,-.5,-0.325,0.669) -fpics/c6 -l32 -m4
    python MandelbrotDriver.py -tD -s320x240 -c(0,-.5,-0.325,0.669) -fpics/d6 -l32 -m4
    python MandelbrotDriver.py -tM -s320x240 -c(0,-.5,-0.325,0.669) -fpics/m6 -l32 -m4
    python MandelbrotDriver.py -tJ -s320x240 -c(0,-.5,-0.325,0.669) -fpics/j6 -l32 -m4

    python MandelbrotDriver.py -tD -s1280x1024 -c(0,-.5,-0.325,0.669) -fpics/d6x -l64 -m4
    python MandelbrotDriver.py -tDB -s1280x1024 -c(0,-.5,-0.325,0.669) -fpics/d6xb -l64 -m4

    python MandelbrotDriver.py -tA -s320x240 -c(0,-.875,-0.075,0.669) -fpics/a7 -l32 -m8
    python MandelbrotDriver.py -tB -s320x240 -c(0,-.875,-0.075,0.669) -fpics/b7 -l32 -m8
    python MandelbrotDriver.py -tC -s320x240 -c(0,-.875,-0.075,0.669) -fpics/c7 -l32 -m8
    python MandelbrotDriver.py -tD -s320x240 -c(0,-.875,-0.075,0.669) -fpics/d7 -l32 -m8
    python MandelbrotDriver.py -tM -s320x240 -c(0,-.875,-0.075,0.669) -fpics/m7 -l32 -m8
    python MandelbrotDriver.py -tJ -s320x240 -c(0,-.875,-0.075,0.669) -fpics/j7 -l32 -m8

    python MandelbrotDriver.py -tJ -s1280x1024 -c(0,-.875,-0.075,0.669) -fpics/j7x -l32 -m8 -i1200
    python MandelbrotDriver.py -tB -s1280x1024 -c(0,-.875,-0.075,0.669) -fpics/b7x -l32 -m8 -i1200
    python MandelbrotDriver.py -tD -s1280x1024 -c(0,-.875,-0.075,0.669) -fpics/d7x -l32 -m8 -i1200

    python MandelbrotDriver.py -tA -s320x240 -c(0,-.894,-0.056,0.669) -fpics/a8 -l32 -m20
    python MandelbrotDriver.py -tB -s320x240 -c(0,-.894,-0.056,0.669) -fpics/b8 -l32 -m20
    python MandelbrotDriver.py -tC -s320x240 -c(0,-.894,-0.056,0.669) -fpics/c8 -l32 -m20
    python MandelbrotDriver.py -tD -s320x240 -c(0,-.894,-0.056,0.669) -fpics/d8 -l32 -m20
    python MandelbrotDriver.py -tM -s320x240 -c(0,-.894,-0.056,0.669) -fpics/m8 -l32 -m20
    python MandelbrotDriver.py -tJ -s320x240 -c(0,-.894,-0.056,0.669) -fpics/j8 -l32 -m20

    python MandelbrotDriver.py -tJ -s1280x1024 -c(0,-.894,-0.056,0.669) -fpics/j8x -l32 -m20 -i1200

    python MandelbrotDriver.py -tA -s320x240 -c(0,-.894,-0.056,0.669) -fpics/a9 -l32 -m200
    python MandelbrotDriver.py -tB -s320x240 -c(0,-.894,-0.056,0.669) -fpics/b9 -l32 -m200
    python MandelbrotDriver.py -tC -s320x240 -c(0,-.894,-0.056,0.669) -fpics/c9 -l32 -m200
    python MandelbrotDriver.py -tD -s320x240 -c(0,-.894,-0.056,0.669) -fpics/d9 -l32 -m200
    python MandelbrotDriver.py -tM -s320x240 -c(0,-.894,-0.056,0.669) -fpics/m9 -l32 -m200
    python MandelbrotDriver.py -tJ -s320x240 -c(0,-.894,-0.056,0.669) -fpics/j9 -l32 -m200

    python MandelbrotDriver.py -tA -s320x240 -c(0,-.902,-0.056,0.673) -fpics/a10 -l32 -m1000
    python MandelbrotDriver.py -tB -s320x240 -c(0,-.902,-0.056,0.673) -fpics/b10 -l32 -m1000
    python MandelbrotDriver.py -tC -s320x240 -c(0,-.902,-0.056,0.673) -fpics/c10 -l32 -m1000
    python MandelbrotDriver.py -tD -s320x240 -c(0,-.902,-0.056,0.673) -fpics/d10 -l32 -m1000
    python MandelbrotDriver.py -tM -s320x240 -c(0,-.902,-0.056,0.673) -fpics/m10 -l32 -m1000
    python MandelbrotDriver.py -tJ -s320x240 -c(0,-.902,-0.056,0.673) -fpics/j10 -l32 -m1000

    python MandelbrotDriver.py -tC -s1280x1024 -c(0,-.902,-0.056,0.673) -fpics/c10x -l32 -m1000 -i2400

    python MandelbrotDriver.py -tA -s320x240 -c(0,-.902,-0.056,0.6716) -fpics/a11 -l32 -m4000 -i2400
    python MandelbrotDriver.py -tB -s320x240 -c(0,-.902,-0.056,0.6716) -fpics/b11 -l32 -m4000 -i2400
    python MandelbrotDriver.py -tC -s320x240 -c(0,-.902,-0.056,0.6716) -fpics/c11 -l32 -m4000 -i2400
    python MandelbrotDriver.py -tD -s320x240 -c(0,-.902,-0.056,0.6716) -fpics/d11 -l32 -m4000 -i2400
    python MandelbrotDriver.py -tM -s320x240 -c(0,-.902,-0.056,0.6716) -fpics/m11 -l32 -m4000 -i2400
    python MandelbrotDriver.py -tJ -s320x240 -c(0,-.902,-0.056,0.6716) -fpics/j11 -l32 -m4000 -i2400

    python MandelbrotDriver.py -tMS -s1280x1024 -c(0,-.902,-0.056,0.6716) -fpics/m11xs -l32 -m4000
                               -i2400 -b0.0005

    python MandelbrotDriver.py -tA -s320x240 -c(0,-.875,-0.069,0.6716) -fpics/a12 -l32 -m200 -i2400
    python MandelbrotDriver.py -tB -s320x240 -c(0,-.875,-0.069,0.6716) -fpics/b12 -l32 -m200 -i2400
    python MandelbrotDriver.py -tC -s320x240 -c(0,-.875,-0.069,0.6716) -fpics/c12 -l32 -m200 -i2400
    python MandelbrotDriver.py -tD -s320x240 -c(0,-.875,-0.069,0.6716) -fpics/d12 -l32 -m200 -i2400
    python MandelbrotDriver.py -tM -s320x240 -c(0,-.875,-0.069,0.6716) -fpics/m12 -l32 -m200 -i2400
    python MandelbrotDriver.py -tJ -s320x240 -c(0,-.875,-0.069,0.6716) -fpics/j12 -l32 -m200 -i2400

    python MandelbrotDriver.py -tA -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/a13 -l32 -m1000 -i4800
    python MandelbrotDriver.py -tB -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/b13 -l32 -m1000 -i4800
    python MandelbrotDriver.py -tC -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/c13 -l32 -m1000 -i4800
    python MandelbrotDriver.py -tD -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/d13 -l32 -m1000 -i4800
    python MandelbrotDriver.py -tM -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/m13 -l32 -m1000 -i4800
    python MandelbrotDriver.py -tJ -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/j13 -l32 -m1000 -i4800

    python MandelbrotDriver.py -tB -s1280x1024 -c(0,-.875,-0.0650,0.6707) -fpics/b13x -l32 -m1000 -i4800

    python MandelbrotDriver.py -tA -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/a14 -l32 -m10000 -i4800
    python MandelbrotDriver.py -tB -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/b14 -l32 -m10000 -i4800
    python MandelbrotDriver.py -tC -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/c14 -l32 -m10000 -i4800
    python MandelbrotDriver.py -tD -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/d14 -l32 -m10000 -i4800
    python MandelbrotDriver.py -tM -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/m14 -l32 -m10000 -i4800
    python MandelbrotDriver.py -tJ -s320x240 -c(0,-.875,-0.0650,0.6707) -fpics/j14 -l32 -m10000 -i4800