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Five Easy Pieces

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Monodromy

Perelman proved that the “cigars” that had troubled Hamilton could not actually occur, and he showed that the “neck” problem could be solved by performing an intricate sequence of mathematical surgeries: cutting out singularities and patching up the raw edges.

Stable Manifold

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The Penrose Triangle

The Penrose triangle, also known as the Penrose tribar, is an impossible object. It was first created by the Swedish artist Oscar Reutersvärd in 1934. The mathematician Roger Penrose independently devised and popularised it in the 1950s, describing it as “impossibility in its purest form”.

Above: Each of the black shapes above are variations of the Penrose triangle, and all of them are a little bit impossible in some way… Source.

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Hilbert Curves German mathematician David Hilbert discovered the curve that bears his name in the early 1900’s. It is an example of a “space-filling” curve: it literally covers every point in a square. Like fractals, it is generated in iterations. Hilbert with Ladders(below) is output generated from a modified Hilbert algorithm that also draws other shapes and ladders at points along the Hilbert Curve. The Hilbert space filling curve is a one dimensional curve which visits every point within a two dimensional space. It may be thought of as the limit of a sequence of curves which are traced through the space. The basic pattern is a curve which starts near the bottom left corner of a box and terminates near the bottom right corner. It has a kink in it, the kink takes it into the top left and top right of the box. Don Relyea, based on Professor Cumming of Napier University, Edinburgh, UK works and with a small amount of effort adding several of his drawing routines, was able to generate fantastic results. http://www.donrelyea.com/hilbert_algorithmic_art_gallery.htm If you like to try yourself, it is possible to it here:

http://www.donrelyea.com/space_curve_generator.htm

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3D Printed Mathematical Sculptures by Bathsheba Sculpture LLC

In order of appearance: the tesseract or hypercube, a Voronoi network wrapped onto a Möbius toroid, the Klein bottle, the hyperdodecahedron, and the Menger sponge. All of these objects are very cool, so feel free to click the links to learn more about them!