Four Dimensional Julia Sets


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Four Dimensional Julia Sets

  1. 1. Four Dimensional Julia Sets<br />Caleb Wherry<br />Dr. Ben Ntatin<br />Austin Peay State University<br />
  2. 2. Introduction to Julia Sets<br />Mathematical properties<br />Connectedness & self-symmetry<br />2 dimensional construction over the complex plane<br />Relationship to Mandelbrot Sets<br />Examples (single point & self-symmetry)<br />Four Dimensional Julia Sets<br />Quaternion (hyper-complex) construction<br />Three dimensional visualization<br />Computational Complexity<br />MATLAB parallelization<br />Future Work<br />Outline<br />
  3. 3. Introduction to Julia Sets<br />Gaston Julia – 1910s<br />Collaborated with Pierre Fatou (Fatou Sets)<br />Commonly referred to as fractals<br />Formed by using a simple function<br />f(z) = z2 + c<br />Apply iterations and the function takes the form: zn+1 = zn2 + c <br />Form two broad set categories<br />Totally connected (dendrites – inside Mandelbrot Set)<br />Totally disconnected (Cantor dusts – outside Mandelbrot Set)<br />Exhibit self-symmetry, real axis symmetry & rotational symmetry (depending on c)<br />
  4. 4. 2D Julia sets live in the complex plane<br />zn+1 = zn2 + c z,cεC (c = r + ai)<br />For our construction, c is held constant and z is iterated<br />Define a hyperfine grid<br />Normalize our graphing plane to a cube with sides length 1 around the origin. <br />Grid size defined by how many step sizes we want<br />More step sizes = more computation! <br />Use a simple “escape time” method<br />If the |zn| is under a certain escape value, keep iterating<br />Construction of a Julia Set<br />
  5. 5. This process yields two sets of numbers<br />1) The numbers that escape from the escape value after a certain amount of iterations (Escape set)<br />2) The numbers that converge to some number inside of the escape value (Prisoner Set) and thus never escape<br />Each one of these sets contain basins in which all points within converge to a central point.<br />The actual Julia set is the boundary between the prisoner and escape sets!<br />We use the values from the escape set to make our pictures pretty and colorful (one-to-one mapping to color table values)<br />Construction of a Julia Set cont…<br />
  6. 6. Relationship to Mandelbrot Sets<br />
  7. 7. Examples of Julia Sets<br />c = (-0.687,0.312)<br />c = (-0.500,0.563)<br />c = (-0.75,0.00)<br />c = (0.285,0.535)<br />c = (0.276,0.000)<br />c = (-0.125,0.750)<br />
  8. 8. Self-symmetry in action!<br />
  9. 9. Use the quaternions (hyper-complex) for construction<br />zn+1 = zn2 + q z,qεH (q = r + ai+ bj + ck)<br />Quaternions<br />i2 = j2 = k2 = -1<br />More complex relationship between i, j, & k <br />ij = k & ji = -k<br />jk = i & kj = -i<br />ki = j & ik = -j<br />What do the above tell us?<br />Quaternions do not form an abelian group<br />Much more computer time needed!<br />Four Dimensional Julia Sets<br />
  10. 10. Four Dimensional Julia Sets cont…<br />q = (0.0, -0.8, 0.8, 0.32)<br />(Bourke)<br />
  11. 11. 3 dimensional - easiest way to visualize<br />Make three dimensions dependant on the fourth<br />Move the selected cube along the independent axis<br />Animations with seamless transitions can be made<br />2 dimensional visualization<br />Not at exciting as the above method<br />Creates many more files to sort through<br />No spacial sense is achieved<br />I have no perfected 3D visualization yet<br />Transparencies and boundary checking are needed to produce accurate 3D plots of quaternion Julia sets<br />Three Dimensional Visualization<br />
  12. 12. Written in MATLAB<br />Parallelized to 8 cores<br />Potentially expandable to 16, supercomputer problems currently<br />Advantages of MATLAB<br />Very easy to code in<br />Substantial user base<br />Graphing made easy!<br />Disadvantages of MATLAB<br />High-level language means slower process time<br />10004 quaternion grid (semi-fine) = 1 trillion floating point operations<br />Can’t have too much overhead with the above number of calculations!<br />$$$$$<br />Alternative Languages<br />C/C++ <br />Fast and free!<br />Java<br />Create computational interfaces online and easily share research<br />Also free!<br />Computational Complexity <br />
  13. 13. Analysis of more complex functions<br />zn= zn+1k + c with k > 2<br />Finer grids<br />Need more processing power<br />Recode in more versatile programming language?<br />Higher dimensional analysis<br />Quinternions , Septernions, Oct0rnions, etc.<br />N-ternion code library development<br />Chance to share reseach<br />Easy n-dimensional visualization <br />Future Work<br />
  14. 14. Bourke, Paul. “Quaternion Julia Fractals.”<br />Elert, Glenn. “Strange and Complex.”<br />Ntatin, Ben. “The Cantor Set and Function.”<br />Rosa, A. “Methods and applications to display quaternion Julia sets.”<br />Rosa, A. “On a solution to display non-filled-in quaternionic Julia sets.”<br />References<br />
  15. 15. Questions?Comments?<br />