Clifford A. Pickover

Fractal characterization of speech waveform graphs

Abstract Mandelbrots fractal geometry has provided a new qualitative and quantitative approach for understanding the complex shapes of nature. In this paper, the fractal structure of speech waveforms is studied at time scales where important phonetic and prosodic information reside. We have found, using methods commonly applied to complex shapes such as coastlines, that speech exhibits fractal characteristics. We have made measurements of the fractal dimension ( D ) for sentences and have found that D ∼ 1.66 with little change between speakers and sentences.

Pattern formation and chaos in networks

Chaos theory involves the study of how complicated behavior can arise in systems that are based on simple rules, and how minute changes in the input of a system can lead to great differences in the output. Using computer graphics, the dynamic behavior of chaos-producing networks is explored, and convergence maps reveal a visually striking and intricate class of displayable objects.

Biomorphs: Computer displays of biological forms generated from mathematical feedback loops

A computer graphics algorithm is used to create complicated forms resembling invertebrate organisms. These natural morphologies are generated through the iteration of mathematical transformations. Several illustrations are chosen as examples of the diversity of biological structures which result from this technique.

Computer graphics generated from the iteration of algebraic transformations in the complex plane

Abstract Algorithms for the generation of intricate shapes resulting from the iteration of algebraic transformations are presented. A special convergence test makes possible the production of a visually striking class of displayable objects. Several illustrations are chosen as examples of the diversity of forms which result from this technique.

A note on rendering 3-D strange-attractors

Abstract A simple graphics technique is described for enhancing features in drawings of 3-D chaotic attractors. Numerical recipes are included to encourage reader involvement.

Automatic generation of general quadratic map basins

Abstract In this informal article, we describe simple approaches whereby a computer can automatically select parameters and generate a large collection of diverse, aesthetically appealing fractal patterns based on general quadratic map basins. Computational recipes are included to encourage reader involvement. In essence, we describe methods for teaching a computer to be both an artist and a critic of its own art.

Mathematics and beauty: Time-discrete phase planes associated with the cyclic system, {x(t) = -f(y(t)), y(t) = f(x(t))}.

Abstract Portraits of time-discrete phase planes associated with the cyclic system [ x (t) = −f(y(t)), y (t) = f(x(t)) } are presented, where f is a frequency modulation term of the general form, f(x) = sin[x + sin(ϱx)]. The resulting trajectories are of interest artistically and mathematically, and they reveal a visually striking and intricate class of patterns ranging from stable points to hierarchies of stable cycles and to apparently random fluctuations. The computer-based system presented is special in its primary focus on the fast characterization of simple cyclic systems using an interactive graphics system with a variety of controlling parameters.

Frequency spectra of DNA sequences: application to a human bladder cancer gene

Abstract Two useful ways of describing base content and periodicity for nucleic acid sequences are the spectrogram and 3D power spectrum, representations similar to those frequently used in the field of digital signal processing. A description of a vector graphics facility for coordinated computation and display of such functions is presented. The interactive nature of its user interface and the variety of input parameters available to the user greatly facilitate the characterization of a particular nucleic acid sequence. In this paper, calculations are performed for a human bladder oncogene.

Generating irregularly oscillating fossil seashells

The generation and representation of unusual growth patterns characteristic of certain extinct mollusks are discussed. The mathematical methods for biological pattern generation to render seashells on the computer and the software tools and hardware for the visualization of mathematical models of nature are described.<<ETX>>

A short recipe for seashell synthesis

Several artistic examples of seashell-like forms produced on a graphics supercomputer are provided. The shapes were created using a simple graphics primitive and rendered using lighting and shading facilities of 3-D extensions to X Windows or the PHIGS+ proposed standard. It is expected that the techniques, equations, and systems will provide useful tools and stimulate future studies in the graphics characterization of morphologically rich spiral shapes produced by relatively simple generating formulas.<<ETX>>