Caleb Wherry

Dr. Ben Ntatin

Austin Peay State University

Four Dimensional Julia Sets

Introduction to Julia SetsMathematical properties

Connectedness & self-symmetry2 dimensional construction over the complex planeRelationship to Mandelbrot SetsExamples (single point & self-symmetry)

Four Dimensional Julia SetsQuaternion (hyper-complex) constructionThree dimensional visualization

Computational ComplexityMATLAB parallelization

Future Work

Outline

Introduction to Julia SetsGaston Julia – 1910s

Collaborated with Pierre Fatou (Fatou Sets)Commonly referred to as fractalsFormed by using a simple function

f(z) = z2 + cApply iterations and the function takes the form: zn+1 = zn

2 + c

Form two broad set categoriesTotally connected (dendrites – inside Mandelbrot Set)Totally disconnected (Cantor dusts – outside Mandelbrot Set)

Exhibit self-symmetry, real axis symmetry & rotational symmetry (depending on c)

2D Julia sets live in the complex planezn+1 = zn

2 + c z,c ε C (c = r + ai)For our construction, c is held constant and z is

iteratedDefine a hyperfine grid

Normalize our graphing plane to a cube with sides length 1 around the origin.

Grid size defined by how many step sizes we want

More step sizes = more computation! Use a simple “escape time” method

If the |zn| is under a certain escape value, keep iterating

Construction of a Julia Set

This process yields two sets of numbers1) The numbers that escape from the escape value

after a certain amount of iterations (Escape set)2) The numbers that converge to some number inside

of the escape value (Prisoner Set) and thus never escape

Each one of these sets contain basins in which all points within converge to a central point.

The actual Julia set is the boundary between the prisoner and escape sets!

We use the values from the escape set to make our pictures pretty and colorful (one-to-one mapping to color table values)

Construction of a Julia Set cont…

Relationship to Mandelbrot Sets

Examples of Julia Sets

c = (-0.687,0.312) c = (-0.500,0.563) c = (-0.75,0.00)

c = (0.285,0.535) c = (0.276,0.000) c = (-0.125,0.750)

Self-symmetry in action!

Use the quaternions (hyper-complex) for construction

zn+1 = zn2 + q z,q ε H (q = r + ai + bj +

ck)Quaternions

i2 = j2 = k2 = -1More complex relationship between i, j, & k

ij = k & ji = -kjk = i & kj = -iki = j & ik = -j

What do the above tell us?Quaternions do not form an abelian group

Much more computer time needed!

Four Dimensional Julia Sets

Four Dimensional Julia Sets cont…

(Bourke)

q = (0.0, -0.8, 0.8, 0.32)

3 dimensional - easiest way to visualizeMake three dimensions dependant on the fourthMove the selected cube along the independent axisAnimations with seamless transitions can be made

2 dimensional visualizationNot at exciting as the above methodCreates many more files to sort throughNo spacial sense is achieved

I have no perfected 3D visualization yetTransparencies and boundary checking are needed

to produce accurate 3D plots of quaternion Julia sets

Three Dimensional Visualization

Written in MATLAB Parallelized to 8 cores

Potentially expandable to 16, supercomputer problems currently Advantages of MATLAB

Very easy to code in Substantial user base Graphing made easy!

Disadvantages of MATLAB High-level language means slower process time

10004 quaternion grid (semi-fine) = 1 trillion floating point operations Can’t have too much overhead with the above number of calculations!

$$$$$ Alternative Languages

C/C++ Fast and free!

Java Create computational interfaces online and easily share research Also free!

Computational Complexity

Analysis of more complex functionszn= zn+1

k + c with k > 2

Finer gridsNeed more processing powerRecode in more versatile programming

language?Higher dimensional analysis

Quinternions , Septernions, Oct0rnions, etc.N-ternion code library development

Chance to share reseachEasy n-dimensional visualization

Future Work

Bourke, Paul. “Quaternion Julia Fractals.”

Elert, Glenn. “Strange and Complex.”

Ntatin, Ben. “The Cantor Set and Function.”

Rosa, A. “Methods and applications to display quaternion Julia sets.”

Rosa, A. “On a solution to display non-filled-in quaternionic Julia sets.”

References

Questions?

Comments?