Jim Muth wrote:
In a recent letter to the Fractint list, Vortex S. asked what is the hyperspiral I keep talking about. The hyperspiral is simply the set of all the fractals that can be created with the same fractional exponent of Z.
Many fractals are possible from the same fractional exponent because the complex logarithm function is multi-valued. The formula for the complex natural logarithm is:
(1/2)ln(x^2+y^2)+i(atan(y/x)+2kPI)
Some links on the Internet which mention such things: http://web.mit.edu/18.02-esg/www/mapins/hyperspiral http://web.mit.edu/18.01-esg/18.01A/hyperspiral http://www.midcoast.com/~bo/SimpSpirOnSpir.html http://inprosys.bizland.com/website/8cyberbat.htm http://www.tcm.phy.cam.ac.uk/~dek12/tales/Advert.html http://canyoninstitute.org/resources/URBibliography/022_general_ideas_p.htm http://www.ebizasialink.com/Databank/Issue1/article3282/hyperspiral.jpg P.N.L. ------------------------------------------------- http://home.att.net/~Paul.N.Lee/PNL_Fractals.html http://www.Nahee.com/Fractals/