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# Deepest

Robert P. Munafo, 2023 Jul 3.

The words "deep", "depth" etc. are often used to refer to the amount of magnification in a view.

The definition of what constitutes "deep zooming" evolves with time, similarly to the definition of e.g. "supercomputer". Hardware and software capabilities improve, causing the "frontier" (defined by what is difficult or takes a lot of work) to progressively move towards ever-higher magnifications.

Records for deepest views and zoom sequences have been set year after year for the entire period since the Mandelbrot set discovery. These have been advanced both by direct improvements in computer hardware (Moore's law, GPUs, etc.) and by algorithmic improvements such as the various perturbation methods.

### Early Deepzooming History

The popular program FRACTINT allows arbitrarily deep views, which it implements by using extended precision math routines.

The 1991 book Mm - Much Ado About Nothing - Vol. 1, (A.G. Davis Philip, Adam Robucci, Michael Frame & Kenelm Philip, LC catalog number 91-092943) discusses the islands on the spike of the sequence R2F(1/2B1)S (period 3), R2F(1/2B1)FS[2]S (period 4), R2F(1/2B1)FS[2]FS[2]S (period 5), etc. (see Utter West)

The last island in the sequence they picture has period 300, and the image of it is at magnification 1.6×10359, requiring about 362 decimal digits or 1202 binary digits to compute. This is the deepest view I have seen, but with FRACTINT one could easily go deeper.

### More Recent Developments

For some time, the 10308 exponent limit of double precision limited the depth of views. During this period many "deep zoom" videos ended at about that magnification. Most used cubic extrapolation or similar, single-point extrapolation methods.

Very high dwell limits continued to be a problem for rendering many frames of a movie, until the advent of differential estimation together with bivariate linear approximation. Together with the use of floatexp formats, and GPUs for iterating the pixels in an image using only the reference point's iterates and derivatives at machine precision, zoom mivoes to magnifications of 101000 have become commonplace.

Here are some Internet pages related to very deep imaging:

Richard Voss' Avogadro Minibrot, as the name suggests, involves a magnification of about 6×1023.

This zoom movie (wmv video, WMV1 codec), one of several at fractal-animation.net, ends at a magnification of about 3×1027. Coordinates: -1.7499357218920984460646651243594 + 0.0000000890808697365708495087578 i @ 7.1e-28.

Adam Robucci's image of a minibrot close to R2t, magnification 10359. It is the leftmost island of period 300.

Deepzooming with Fractint, includes images up to about 101500. The arbitrary Precision algorithms are described here. The 101500 image, located at exactly 0.0 + 1.0 i, exploits a special property of those coordinates which makes deep zooming easier.

revisions: 20080218 oldest on record; 20230619 improvements fueled by perturbation methods; 20230703 add more recent history

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

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Jul 08. s.30