# Julia-Mandelbrot Space

Robert P. Munafo, 2023 Jun 23.

A 4-dimensional space in which the Julia and Mandelbrot iterations are performed.

Two dimensions correspond to the Parameter values C used to start the Iteration; the other two dimensions correspond to the Z values.

Julia Sets are cross-sections in the Z plane. The Mandelbrot Set is a cross-section in the C plane passing through the origin.

The Buddhabrot article shows a set of 12 images, in six pairs. The images each have two axes, which are two of the four axes of the 4-dimensional space. There are six pairs because there are six ways to choose 2 axes out of 4.

### Similarity Along the Main Diagonal

If you are viewing an area of the Mandelbrot Set near a point P, then the most obvious features of the filaments in the area will resemble the Julia Set for C=P. Furthermore, the region near that point P in that Julia Set will resemble a region near P in the Mandelbrot Set, except for a scaling and rotation. This is the Main Diagonal Symmetry.

revisions: 20021129 oldest on record; 20230623 relevant discussion of Buddhabrot images

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Jun 26. s.27