Where Can I Find Resources to Study Peano Curves?

[Ed Quanta]

Can anyone point me towards an online link or some books where I can study these bad boys? I am supposed to write like a 5-10 page paper on Peano curves in which I prove a few interesting things regarding them.

 

[saltydog]

Can anyone point me towards an online link or some books where I can study these bad boys? I am supposed to write like a 5-10 page paper on Peano curves in which I prove a few interesting things regarding them.

"Chaos and Fractals", by Peitgen, et.al. I quote:

"In Nature, the organization of space-filling structures is one of the fundamendal buliding blocks of living beings". You know, lungs, kidneys, vascular system. But I digress. Suppose you just want to prove how they fill the plane. Looks like a good N-epsilon proof. Like, for any (x,y) in the plane, curve comes epsilon distance away from it whenever the iteration is greater than N. Thus as N goes to infinity, distance goes to zero. Never proved it though. Suppose Peano and Hilbert did.

 

[M98Ranger]

There are many great resources available online and in books for studying Peano curves. Here are a few suggestions to help you get started:

1. Online Links:
- The Wolfram MathWorld website has a comprehensive entry on Peano curves, including definitions, properties, and examples: http://mathworld.wolfram.com/PeanoCurve.html
- The Cut the Knot website has a page dedicated to Peano curves, with interactive applets and visualizations to help you understand their construction: https://www.cut-the-knot.org/curriculum/Geometry/PeanoCurves.shtml
- The Plus Magazine website has an article on Peano curves that discusses their history and applications in computer graphics: https://plus.maths.org/content/peano-curves

2. Books:
- "The Fractal Geometry of Nature" by Benoit Mandelbrot is a classic text on fractals and includes a chapter on Peano curves.
- "The Beauty of Fractals" by Heinz-Otto Peitgen, Hartmut Jürgens, and Dietmar Saupe is another well-known book that covers Peano curves and other fractal constructions.
- "Fractals Everywhere" by Michael F. Barnsley is a popular introduction to fractal geometry and also includes a chapter on Peano curves.

I hope these resources will be helpful in your study of Peano curves. Best of luck with your paper!