Beginning in the early 1870s, though, a 50-year crisis transformed mathematical thinking. Weierstrass described a function that was continuous but not differentiable -- no tangent could be described at any point. Cantor showed how a simple, repeated procedure could turn a line into a dust of scattered points, and Peano generated a convoluted curve that eventually touches every point on a plane. These shapes seemed to fall "between" the usual categories of one-dimensional lines, two- dimensional planes and three-dimensional volumes. Most still saw them as "pathological" cases -- a "monstrous" branch of inquiry with no tractable methods for uinderstanding them.
In the 1970's, Benoit Mandelbrot at IBM research began using computers to create graphs of chaotic dynamical systems. What he found in the pictures of these graphs was pattern, instead of chaos -- self-similarity with a shift in scale -- and infinite detail. It is fundamentally unknown from whence these properties arise.
The Mandelbrot set is an iterative procedure: Z -> Z2 + C; where Z and C are points in the complex plane of the form: a + ib, where i = sqrt(-1). The Mandelbrot set is computed for values of C corresponding to every pixel in a color-mapped, rectangular image in which the raster pixel addresses represent a regular subdivision of the complex plane. The color value of each pixel is proportional to the number of iterations of the procedure at which the value Z converges to a finite value.
The Julia set is the compliment of the Mandelbrot set -- it is the convergence of Z -> Z2 + C; where C is a chosen constant and pixel raster address corresponds to the initial value of Z. There is a unique Julia set for every point on the Mandelbrot set.
Alan Norton, also of IBM Research, explored generating Julia sets in the four-dimensional complex space of numbers known as the Quaternions. Quaternions were developed by Hamilton to solve problems of transforming orientations in three dimensions. For example, vector division is undefined in three-space, but is well-defined in Quaternion, complex four-space.
Computing Quaternion Julia sets is similar to those in C2, except that we render convergence values into an X-Y-Z three-dimensional pixel volume instead of a two-dimensional color-mapped image, and we use volume rendering to visualize them. We "project" the Quaternion four-space into R3 orthographically simply by ignoring one of the Quaternion axes.
These images show a 512 x 320 x 320 x 8 bit per pixel volume convergence map of the procedure Z -> ei R Z2 + e-i R (-0.745,0.113,0.01,0.01), R -> 0 where Z is a complex quaternion four-vector of the form (a, b i, c j, d k), where i2 = j2 = k2 = ijk = -1. The pixel volume represents the three-space spanned by (1, i, j, 0).
Software/Hardware used: Quaternion Julia Set calculator by Stewart Dickson; Kitware's Visualization ToolKit; Apple dual-2.7GHz G5 PowerMac
Close-up of the surface polygon structure produced by the Vtk Marching-Cubes IsoSurfacer.
The Isosurface of the Quaternion Julia Set rendered in Fullcure 720 acrylic
photopolymer resin on a Stratasys
Eden 333 PolyJet 3D printer at the
Western Carolina University
Center for Rapid Product Realization.
The following Vtk data pipeline was used to produce the STL file for the Stratasys Eden:
vtkImageReader [raw UnsignedChar; DataSpacing 0.00585938 -- Data size: 52,428,800 BYTES] -> vtkContourFilter [Value 0 16; ComputeScalarsOn] -> vtkCleanPolyData -> vtkTriangleFilter -> vtkSTLWriter [FileType 2/Binary -- Data size 63,127,284 bytes, 1,262,544 triangles Physical dimensions: 3.0 X 1.875 X 2.0]
Note that there is a complex rotation factor (ei R + e-i R in the expression we are computing. This is where Julia sets in the Quaternions get really interesting. Whereas two-dimensional cross-sections of Julia sets computed in the quaternions can resemble Julia sets computed in complex 2-space, Quaternion Julia sets are extended outward in space from the complex plane in complicated ways and rotations in four-space are not projected orthographically into three-space. The three-dimensional result of a complete rotation in Quaternion four-space is a continuous, complex metamorphosis of the unrotated form, above into the rotated form, shown below.
512 x 512 x 512 x 8 bit per pixel volume convergence map of the procedure Z -> ei R Z2 + e-i R (-0.745,0.113,0.01,0.01), R -> π/2
The following AVI movies show one full, continuous rotation in sixty frames.
The following is the proposal for the four-dimensional Zoetrope of the Julia set, computed in the quaternions, seeded at (-0.745,0.113,0.01,0.01), rotating in complex four-space. This movie shows the isosurfaces extracted from the previous movie, mounted on numbered, standard bases for installation into the Zoetrope.
The following images depict the sixty sculptures of the complete rotation of the Quaternion Julia set mounted on the Zoetrope wheel. At 1.3 million polygons per phase of the animation, the following images represent a rendering from on the order of 78 million polygons. The ASCII Open Inventor file for this composition is on the order of five Gigabytes in size. It was rendered using Coin3D on an Intel Core2Duo PC with nine gigabytes of RAM, running the 64-bit version of Redhat Fedora Core 7 Linux.
The following is a similar series of the procedure Z -> ei R Z2 + e-i R (0.2809,0.53,0.01,0.01), R -> (0, π)
(8.8 MBytes AVI) or
(11.7 MByte Quicktime)
(8.8 Mbyte AVI) or
(2.6MByte Quicktime)
The behavior of deterministic fractals, arising from chaotic processes in complex number spaces are fundamentally mysterious. A four-dimensional, complex number space such as the Quaternions is fundamentally difficult to visualize, even when projected into three-space from a mapping into Euclidean four-space. Yet this is what we are seeing here: A "monstrous", chaotic graph with self-similarity at shifts in scale with infinite detail, which takes place in a four-dimensional space of imaginary numbers -- and we see a complete rotation of the four-dimensional object, exposing us to the object in its entirety, via this fantastic metamorposis in physical three-space and over time.
And the Four-Dimensional Zoetrope allows us to see the exposure of this metamorphosis in three physical dimensions, in physical materials, over time.
(436Kbyte AVI)