## Powers of 10

This is an image my brother Doug and I made. A blanket is at the far left; then grass, streets, city, shoreline, clouds, the earth, orbits of the moon, Earth, Mars, Jupiter, Saturn, Neptune, nearby stars, the Milky Way, and distant galaxies. (Click for a much bigger view.)

Can you guess how we made it?

Robert Munafosaid, on 2011 March 10 at 10:01 pmI guess that you made it by interpreting the vertical and horizontal coordinates as imaginary and real (respectively) components of a complex number (with the vertical height of the image equal to 2 pi) then performing the transformation Z’ = exp(Z) using the complex exponential function. The pixel at Z shows whatever is at Z’ in the universe. Thus we see parts of the Earth on the left end, where Real(Z) is small and therefore Radius(Z’) is relatively small, and distant galaxies at the right end where Real(Z) is large and Radius(Z’) is very very large.

I created an image like this by hand when I was 10 years old, shortly after seeing the Charles and Ray Eames short film (1968 version). I used a roll of blank adding machine tape and the scale was approximately one power of ten per foot of adding-machine tape. Thus I did not show the full 360 degrees. The drawing included a human figure whose head appeared smaller than his big toe. I’m pretty sure I used a slide rule to aid in measurement and placement.

You may be interested in similar Mandelbrot set views, see mrob.com/pub/muency/exponentialmap.html for one example (-:

reperiendisaid, on 2011 March 10 at 11:23 pmI suppose the effect would be the same, but we used a very simple transformation of Eames’ movie to make this, one frame at a time.

mrob27said, on 2015 December 12 at 1:59 pmOh yes, of course. I see that now. Just take a tiny strip off the bottom of each frame of the movie, and assemble the strips together.