Looking for fractals texts suitable for guided self-study.

In summary, the conversation is about finding suitable books for an honors research/reading course on fractals. The person has a collection of older layperson's books on fractals but needs something more mathematically advanced. They are considering Barnsley's Superfractals but are unsure about its rigor compared to Falconer's work. They also mention being interested in fractals generated by nonlinear dynamical systems and are looking for inspiration for a project. The recommended books for their level are "Nonlinear Dynamics" by Strogatz and "Chaos in Dynamical Systems" by Ott. The latter book is more advanced and focuses on discrete mappings.
  • #1
tiohn
3
0
I have finally convinced one of my professors to sponsor an honors research/reading course over the summer, but I need to find a suitable book. I own quite a number of older layperson's books on fractals including the standards by Mandelbrot and Barnsley and even Peitgen/Jürgens/Saupe's Chaos and Fractals. However, I need something much more mathematically advanced than what I own. I'm looking at Barnsley's Superfractals, since it appears to be much more aesthetically appealing (hey, fractals are beautiful), but I'm unsure about whether or not it is as rigorous as something like Falconer's work.

Of course, I really need to use several books, so I guess the real question is, what books should I be looking at? I'm working at an advanced undergraduate level with a firm grasp on analysis and will be starting graduate coursework next fall. In addition to the mathematics, I'm also looking for something to inspire an interesting project.
 
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  • #2
Maybe you are interested in the kinds of fractals that are generated by nonlinear dynamical systems (smooth differential equations and/or discrete mappings), also known as Chaos theory. There is a lot of work that can be done with the fractal phase portraits of these systems.

The book everyone uses for this subject at your level is 'Nonlinear Dynamics' by Strogatz, who is a professor at MIT. Characteristic of the difficulty of this subject there are 6 preliminary chapters and and six more chapters about chaos and fractals. A more advanced book that goes straight into Chaos and deals more with discrete mappings is written by Ott called 'Chaos in Dynamical Systems'.
 

1. What are fractals?

Fractals are geometric shapes or patterns that exhibit self-similarity at different scales. This means that the same pattern is repeated at smaller and smaller scales, creating an infinite complexity. They are found in nature and can also be created using mathematical equations.

2. Why should I study fractals?

Studying fractals can help us better understand the complexity and beauty of the natural world. It also has practical applications in fields such as physics, biology, and economics. Additionally, learning about fractals can help improve our spatial and mathematical reasoning skills.

3. What are some good texts for learning about fractals through self-study?

Some popular texts for learning about fractals through self-study include "The Fractal Geometry of Nature" by Benoit Mandelbrot, "Chaos and Fractals: New Frontiers of Science" by Heinz-Otto Peitgen, and "The Beauty of Fractals: Images of Complex Dynamical Systems" by Heinz-Otto Peitgen and Peter H. Richter.

4. Are there any online resources for learning about fractals?

Yes, there are many online resources available for learning about fractals, including interactive websites, video lectures, and online courses. Some popular websites include Fractal Foundation, Fractal Foundation, and Fractal World.

5. Can I learn about fractals without a strong mathematical background?

Yes, while a basic understanding of math is helpful, many texts and resources on fractals are designed for beginners and do not require advanced mathematical knowledge. However, some advanced concepts and applications of fractals may require a deeper understanding of math.

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