Fractals Generated by Jacobi Theta Functions

m

1

2

3

4

Re q

0.5

Imq

0.

initial x

0.

initial y

0.

escape radius r

10.

resolution δ

0.03

The definition of the Mandelbrot set is based on the mapping

z

n+1

=

2

z

n

+c

. This Demonstration uses a generalization of the mapping,

z

n+1

=

ϑ

m

(

z

n

,q)+c

, where

ϑ

m

is a Jacobi theta function,

m=1,2,3,4

, and

z

,

q

(over the unit disk) and

c

are complex. The escape radius is

r

, the initial value is

x+iy

, and the function is computed at resolution

δ

.