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YOU ARE HERE: Homepage > Visual Math Patterns > Secret Inner Life of Mirrored Spheres: Wada Basins

Reflective Spheres of Infinity: Wada Basin Fractals

(In physical reality and digital visualization)


Perhaps the Best Example
of a very rich and interesting natural fractal
is the infinite series of inter-reflections when mirrored spheres
(such as christmas ornaments) are packed together.
It may also illuminate properties of other processes
because the fractal is produced by perfect spheres
arranged into the most economical packing arrangement
used by nature and most waves are actually spherical.



Simplest Set-Up for Wada Basin Fractal
Is a Tetrahedron of FOUR Reflective Spheres

Bottom Three Spheres -
This one is Somewhat Monochrome. For a more high-contrast Fractal It's advisable to use 4 Differently Colored Balls.
Close Up Image of Bottom Three Spheres,
Showing Primary Wada Fractal Pattern




There Is Some Difficulty In Getting
a PERFECT Photo of a Wada Basin

(because the ideal viewpoint is the center of four tangent reflective spheres lit by a single point source light - the camera has to to be small and positioned somehow in the center without casting a reflection into the fractal)

Look in the middle and we see a beautiful fractal vortex!
making a REAL fractal reflection


Wada Photo
(left) with similarities to Mandelbrot Set (right) :
Mandelbrot-like wada basin image    mandelbrot set



Here is a shot i found from a digital photography "DCP Challenge" contest online - nice color choices!



"Wada basins are certain special basins of attraction studied in the mathematics of non-linear systems. A basin having the property that
every neighborhood of every point on the boundary of that basin intersects at least three basins is called a Wada basin, or said to have the
Wada property. Unlike the Lakes of Wada, Wada basins are often disconnected.
An example of Wada basins is given by
the Newton-Raphson method applied to a cubic polynomial with distinct roots, such as x3 − 1:" -from Wiki (simple Wada fractal in 2 dimensions)

2 dimensional wada fractal




The 3 dimensional Wada Basin example below is from a cover story in Scientific American 2002



Photography is limited by equipment quality and grain of the film, WADA Basins of INFINITE Depth and Detail
can be calculated & rendered Digitally by Ray Tracing Programs
(POV-ray & probably other programs)
that are used to simulates shapes and trace very accurate simulations of the path light takes.
This also allows for the option of using color palettes that highlight certain prosperities of the are being inspected.


Below:

Better focus and color control in Digital Wada Basins.

Gorgeously rendered Wada Basin in blues, gray, white, yellow and black
This and the next 10 images are from the prolific work of
by Paul Bourke, Swinburne University AU


Below: Zoom on area in Wada Fractal above ...




Below:
Nice Result With Rainbow Hues in the Wada Fractal

by Paul Bourke, Swinburne University AU


Below: Crisp Contrast in the basic pattern
is achieved by using black and low-saturation hues.
Spheres arranged as six vertices of an octahedron
Image credit: Dave Matthews, Mathematics Instructor Minnesota West Community and Technical College



Below: Another interesting display method
just showing the wada reflections and leaving the spheres blank ....
Next 3 by Paul Bourke, Swinburne University AU

-

Below:
Wada with the spheres at the eight vertices of a cube
Image by
Dave Matthews, Mathematics Instructor Minnesota West Community and Technical College

Another Analog Photograph of a "Real" Wada Basin made of Christmas Balls




Alternate Set-Up for Producing
Analog Wada Basin Photographs in PHI Ratio


When zooming deep into the Pi Wada Basin we see more asymmetry and chaos,
the shapes become eerily organic and dream-like.
Check Back Soon For Updates!


An Introduction to the Fascinating Patterns of Chaos & Visual Math
Platonic & Vectoral Geometry Simple Iterative Fractals Natural fractals
Multi Dimensional Forms The Famous Mandelbrot Set Fractal Types & Categories
PHI - the Golden Ratio Mandelbulb & 3 Dimensional Fractals Fractal "Wada" Basin Reflections
Minimal Surfaces Fractal Technology & Historical Fractals Other Fractals & Cool Patterns



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