Ed Quanta said: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.
A Peano curve is a continuous space-filling curve that was first described by the Italian mathematician Giuseppe Peano in 1890. It is a one-dimensional curve that passes through every point in a two-dimensional plane, and it has the property of being non-self-intersecting.
Peano curves are significant in mathematics because they provide a concrete example of a continuous function that maps a one-dimensional interval onto a two-dimensional space. This concept has important applications in topology, geometry, and analysis.
Peano curves are constructed using an iterative process known as the "Peano-Gosper construction." This involves dividing a line segment into smaller segments and connecting them in a specific pattern. The resulting curve becomes increasingly complex as more iterations are performed.
Peano curves have been used in computer graphics to create fractal images and in computer algorithms for data compression. They have also been studied in physics, specifically in the field of fluid dynamics, to model the behavior of fluid flows.
Some good books for studying Peano curves include "Topology" by James Munkres and "Fractal Geometry: Mathematical Foundations and Applications" by Kenneth Falconer. Additionally, there are many online resources and interactive demonstrations available for exploring Peano curves, such as Wolfram MathWorld and Geogebra.