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     

Electronic Kaleidoscopes for the Mind

The goal ofthis article is to present an informal introduction and tutorial on aestheticallypleasing kaleidoscopic images. The article is intended for the non-mathematical reader interested in computer art. Simple generating formulas and recipes are included.     

A note on visualizing the Omega Prism

Simple techniques are described for demonstrating graphically interesting behavior of puzzles defined on finite lattices. The simple puzzles should be of interest to students and teachers who may wish to design and solve similar problems. In particular, this note describes computer graphics used to gain insight into the behavior of the solution space.     

Exploring infinite realms

How does one “visualize” infinity? The author describes how infinitudes of many sorts-purely mathematical or in chaotic dynamical systems-can be brought to stunningly beautiful life with computer graphics     

Mathematics and beauty V: Turbulent complex curls

To help characterize complicated physical and mathematical structures and phenomena, computers with graphics can be used to produce visual representations with a spectrum of perspectives. In this paper, algorithms are described for the computer graphics rendering of a particular class of chaotic structures created from complex iteration. This paper differs from others in that its focuses on one small region of the complex plane and on nonstandard convergence tests. Reader involvement is encouraged by giving “recipes” for the various turbulent forms, and the resulting maps reveal a visually striking and intricate class of shapes.     

Graphics, Bifurcation, Order and Chaos

Chaos theory involves the study of how complicated behaviour can arise in systems which are based on simple rules, and how minute changes in the input of a system can lead to large differences in the output. In this paper, bifurcation maps of the education X_t+1=αλX_t [1+X_t] ^-β, where α= 1 or α=e^-Xi, are presented, and they reveal a visually striking and intricate class of patterns ranging from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. The computer-based system presented is special in its primary focus on the fast characterization of simple "chacs equation" data using an interactive graphics system with a variety of controlling parameters.     

Biom orphs: 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

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 Monte Carlo Approach for ɛ Placement in Fractal-Dimension Calculations for Waveform Graphs

Many diverse and complicated objects of nature and math possess the quality of self-similarity, and algorithms which produce self-similar shapes provide a way for computer graphics to represent natural structures. For a variety of studies in signal processing and shape-characterization, it is useful to compare the structures of many different "objects". Unfortunately, large amounts of computer time are needed as prerequisite for rigorous self-similarity characterization and comparison. The present paper describes a fast computer technique for the characterization of self-similar shapes and signals based upon Monte Carlo methods. The algorithm is specifically designed for digitized input (e.g. pictures, acoustic waveforms, analytic functions) where the self-similarity is not obvious from visual inspection of just a few sample magnifications. A speech waveform graph is used as an example, and additional graphics are included as a visual aid for conceptualizing the Monte Carlo process when applied to speech waveforms.