Newton's method on
xn-1
including fractional exponents

These images are of a style that was described in Nils B. Lahr and C. A. Reiter, Visualizing Newton's Method on Fractional Exponents, The Visual Computer, 11 2 (1994) 82-86 [Abstract] [Preprint].

Creating Images of the basins of attraction of Newton's method on xn - 1 is a lot of fun and is a great programming project. Here are some images for n=3,4,5,6,7,8.

65k
n=3
133k
n=4
152k
n=5
179k
n=6
210k
n=7
220k
n=8
Creating images for fractional exponents shows some interesting bifurcations. Here we see a cool introduction of an attractive 2-cycle near 3.88221 as we consider fractional exponents between 3 and 4.
75k
n=3.1
88k
n=3.2
127k
n=3.5
147k
n=3.7
171k
n=3.8
193k
n=3.88221
189k
n=3.88222
167k
n=3.9
Less dramatic, but also interesting is following the smooth edges between new attractive roots as they develop into chaotic boundaries. Here we consider fractional exponents between 4 and 5.
144k
n=4.1
135k
n=4.4
134k
n=4.6
138k
n=4.8
146k
n=4.9
152k
n=4.99
Goto link:
Cliff's Front Page
Cliff's Gallery of Fractals, Chaos and Symmetry