Book contents
- Frontmatter
- Contents
- Foreword: New Directions in Computer Graphics: Experimental Mathematics
- Preface to the German Edition
- 1 Researchers Discover Chaos
- 2 Between Order and Chaos: Feigenbaum Diagrams
- 3 Strange Attractors
- 4 Greetings from Sir Isaac
- 5 Complex Frontiers
- 6 Encounter with the Gingerbread Man
- 7 New Sights – new Insights
- 8 Fractal Computer Graphics
- 9 Step by Step into Chaos
- 10 Journey to the Land of Infinite Structures
- 11 Building Blocks for Graphics Experiments
- 12 Pascal and the Fig-trees
- 13 Appendices
- Index
2 - Between Order and Chaos: Feigenbaum Diagrams
Published online by Cambridge University Press: 30 March 2010
- Frontmatter
- Contents
- Foreword: New Directions in Computer Graphics: Experimental Mathematics
- Preface to the German Edition
- 1 Researchers Discover Chaos
- 2 Between Order and Chaos: Feigenbaum Diagrams
- 3 Strange Attractors
- 4 Greetings from Sir Isaac
- 5 Complex Frontiers
- 6 Encounter with the Gingerbread Man
- 7 New Sights – new Insights
- 8 Fractal Computer Graphics
- 9 Step by Step into Chaos
- 10 Journey to the Land of Infinite Structures
- 11 Building Blocks for Graphics Experiments
- 12 Pascal and the Fig-trees
- 13 Appendices
- Index
Summary
First Experiments
One of the most exciting experiments, in which we all take part, is one which Nature carries out upon us. This experiment is called life. The rules are the presumed laws of Nature, the materials are chemical compounds, and the results are extremely varied and surprising. And something else is worth noting: if we view the ingredients and the product as equals, then each year (each day, each generation) begins with exactly what the previous year (day, generation) has left as the starting-point for the next stage. That development is possible in such circumstances is something we observe every day.
If we translate the above experiment into a mathematical one, then this is what we get: a fixed rule, which transforms input into output; that is, a rule for calculating the output by applying it to the input. The result is the input value for the second stage, whose result becomes the input for the third stage, and so on. This mathematical principle of re-inserting a result into its own method of computation is called feedback (see Chapter 1).
We will show by a simple example that such feedback is not only easy to program, but it leads to surprising results. Like any good experiment, it raises ten times as many new questions as it answers.
The rules that will concern us are mathematical formulas. The values that we obtain will be real numbers between 0 and 1. One possible meaning for numbers between 0 and 1 is as percentages: 0% ≤ p ≤ 100%.
- Type
- Chapter
- Information
- Dynamical Systems and FractalsComputer Graphics Experiments with Pascal, pp. 17 - 54Publisher: Cambridge University PressPrint publication year: 1989