Generation of volumetric escape time fractals

Abstract

Aesthetically appealing two-dimensional escape time fractals are often based on complex-function transformations. The same methods of analysis and rendering can be extended to create fractal solids. Methods for generation of these objects very often use quaternion representations for transforming points in three or four dimensions. In this tutorial and artistic statement, we describe a simple yet effective approach for visualizing escape times of general quadratic fractal maps in the third dimension. The approach allows us to automatically generate diverse and beautiful volumetric shapes in real time on currently available personal computers.

Introduction

This tutorial and artistic statement describes a simple method for automatically producing a wide range of artistically interesting forms. As is well known, the role of 3-D graphics is becoming increasingly important in the world of personal computers. Almost all modern operating systems provide graphics drivers and a set of low-level application programming interfaces (APIs) for creating advanced 3-D graphics. Increasingly, advanced 3-D graphics cards are continually being created by hardware companies. Also, a 3-D graphics engine is a vital part of a virtual reality system, a computer environment likely to be pervasive in the near future.

In the paper, we present a new shape modeler of complex 3-D objects. As is well known, fractal modeling can yield a diversity of shapes that are stunningly beautiful and similar to natural patterns [1], [2]. The method presented in the paper is based upon the idea of the automated fractal art generators, previously advanced by Sprott and Pickover [3]. Here, we extend the concept to higher dimensions and fractal solids. As in two-dimensional sets of nonlinear functions, in 3-D we will select artistically intriguing forms from a virtually infinite set of possible solids. In particular, we often enjoy exploring visually interesting shapes either similar to natural phenomena or that present unique geometries. The approach here may be extended to a wide range of applications, including real-time computer graphics (e.g. computer games), 3-D modelers (e.g. software like 3-D Studio Max), and geometry compression.