Hilbert Curves

Hilbert Curves is a unique app in the form of a book that shows, explains, and lets you explore and play with one of the most famous and elegant constructions in mathematics, Hilbert's space-filling curve. It shows how to map the points from a line segment to every point inside a square, as a continuous curve. You can do an animated infinite zoom into its construction, or explore the mapping interactively at different levels of precision to understand how it works as a continuous curve.

But that's just the first dozen pages. This 160-page app/book then proceeds to explore and illustrate the ways the Hilbert Curve construction can be generalized into an amazing array of space-filling curve designs, paths, patterns, and tiles, ranging from fractal pinwheel curves to half-domino curves with almost-everywhere linear borders to fuzzy-topped squares. Along the way, it explains or proves the mathematical journey taken.

It presents an animated Meta-Hilbert construction that shows how the borders of an infinite sequence of space-filling curve tiles with fractal borders can converge to a set of Hilbert Curves. There are even never-before-seen half-domino constructions here some of whose patterns might easily be mistaken for Native American or other indigenous rug, basket, or pottery designs. One illustration alone shows more than 77,000 possible space-filling curve, half-domino constructions, each millions of lines long, based on enumerating all possible self-negative "tendril" motifs. Most of these beautiful, eye-catching patterns are mathematical discoveries presented here for the first time.

Designed for high-school or college students, math professionals, and any math-curious person interested in two-dimensional design patterns and space-filling curves and/or fractals, the app presents you with over 130 illustrations in the form of a 160-page electronic document entitled "Outside-In and Inside-Gone". The illustrator, author, researcher, and programmer is Douglas McKenna, an award-winning software developer, mathematical artist, and fractal pioneer and space-filling curve researcher whose career started by drawing space-filling curve illustrations for the father of fractals, Benoît Mandelbrot.

But this app does not (and cannot) use any PDF or other vector files for its resolution-independent illustrations. Instead, it internally self-typesets the text and math at application launch time to the highest (TeX-quality) standards at the same time as presenting the reader with independently zoom-able pictures embedded in the text. Most of these pictures are parameterized with various controls or are animated so that you can interact with and better understand the visual constructions. Many illustrations with tens of millions of connected lines in them can be constructed on the fly and deeply zoomed into, independently of the text, so that you can explore and understand the myriad tiny details and structure.

Representing about ten years of writing, research, typesetting, and other software development, this app is a definitive visual treatise on a subject that is uniquely suited to being accurately drawn only by computer.