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High-Precision Deep Zoom

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40.3 MB WMV

Ununennius (UUE1) "de Moivre"

The deepest Mandelbrot set zoom ever

Date Generated:	11 Jun - 27 Jun 2008
Final Image Size:	9e-120
Resolution:	640x480
Video Length:	4:00 of fractal, 4:56 total
Frames:	7200
Rendering Time:	120:40 hours
Method:	Escape counts
Audio:	Custom using Acid Pro 6
MP4 File Formats:	320x240 768 kbps 18.5 MB quick draft quality 640x480 4Mbps 87.1 MB extreme quality
WMV File Formats:	320x480 768 kbps 21.1 MB quick draft quality 640x480 VBR 40.3 MB moderate quality 640x480 VBR 115.6 MB extreme quality

This is, to date, the deepest no-nonsense, full-resolution, full-speed animation ever created that hpdz.net is aware of. It zooms to a final size of 9e-120, which is equivalent to a FractInt magnification of 2.2e119.

By no-nonsense I mean it is not done with any kind of frame interpolation or "tweening" of frames, but rather each frame is individually calculated individually. Full-resolution means each frame is 640x480, and full-speed means there are 30 raw data frames per second.

Here are some still image frames from the video to give an idea what it looks like. Each thumbnail below links to a full-sized image.

Abraham de Moivre

This video is also named in honor of Abraham de Moivre (1667-1754). De Moivre was a French mathematician most famous for his discovery of the relation in complex arithmetic that bears his name:

(cos a + i sin a)ⁿ = cos na + i sin na

Errata: The closely-related formula below, which I previously had shown as De Moivre's formula, is actually Euler's formula:

cos a + i sin a = e ^ia.

De Moivre also worked in probability theory and number theory, and was the first to discover the closed-form expression for the Fibonacci numbers that for some reason today is known as Binet's formula:

F(n) = [φⁿ - (-φ)^-n]/√5, where φ is the Golden Ratio, φ = ½(1+√5).