space filling curve

In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the “epsilon-delta” definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the “gaps” between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the “size” of the set of discontinuities of real functions.

Also, “monsters” (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

vimeo

Submitted by Willgallia:

Drawing portraits of Mathematicians with space filling curves. (Some are only discrete space filling).
The final drawings looks like the image below.

Thanks for the submission, Will Gallia! These are amazing!!! For those of you who don’t know what space filling curves are, I wrote a post about them a while back

To see more of Will Gallia’s cool drawings, click here. :D

Giuseppe Peano, born on 27th August 1858, was an Italian mathematician. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation ( in his first book dealing with mathematical logic, the modern symbols for the union and intersection of sets appeared for the first time.). The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction.

Moreover, Peano’s famous space-filling curve appeared in 1890 as a counterexample. He used it to show that a continuous curve cannot always be enclosed in an arbitrarily small region. This was an early example of what came to be known as a fractal.

He has done so many things ^_^ 

On space-filling curves

Formal definition.

Let \(\phi :[0,1]\to T\) (where \( T\) is a topological space) be a continous function (i.e. \(\phi ^{-1}\) of any open set \( O\) of \( T\) is an open set of the unit interval) from the unit interval to a topological space. Such map is said to be an space-filling curve if \(\phi\) is onto (i.e. if \phi passes through every element of \( T\) ).

Note

Generally, space filling curves are represented on spaces homeomorphic to \( E^{n}\) euclidian spaces. So to say, the plane, or any geometric solid, since these are the easiest ways to visualize a space filling curve. The most popular example, perhaps, is the two-dimensional Hilbert curve.

Images

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A Hilbert curve from the unit interval to the unit cube

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Peano Curve from the unit interval to the unit cube.

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Flow Snake sapce filling curve.