Beauty in Patents

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In an earlier post, I wondered about the hidden beauty of scientific illustrations in books which are read only by very few specialists. A similar situation is found in the context of patents. They usually come with clarifying sketches or illustrations which range from sloppy to artistic.

Not later than from the moment Richard Buckminster Fuller filed his patent on cartography in 1944, mathematical beauty had found its way into patent applications. How many hidden gems may there be in patents? At least, having patents available online makes it a lot easier to spot some.

Soap Bubbles: a Paradise for Kids and Math Nerds

Soap bubbles are simply irresistible. As is the heading of Section 4.3 of Roberto Piazza‘s book Soft Matter, so I had to borrow it for this post. In the book, you can learn about some of the wonderful physical properties of soap bubbles. For example, I was not aware that each bubble is a double shell enclosing water in between. In this way, the hydrophilic heads of the soap molecules point inside whereas the hydrophobic tails point outside. The colours are then created by interference of light reflected from the two soap films. As water flows and evaporates between the films, colours change depending on the distance between the two films.

Perspectiva Corporum Regularium

Most of the time, I enjoy how one book leads to another in a seemingly endless bibliophilic journey into the past (it is a pity that books usually cannot refer to future books). This time it is the book Shaping Space (see short review) referring to a book with very few words and many wonderful drawings of polyhedra: Wenzel Jamnitzer’s Perspectiva Corporum Regularium from 1568.

Zeolites

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One of the books I am currently reading is Roberto Piazza‘s Soft Matter (see here and here for reviews) . In the book, there is a short paragraph on zeolites which left me wishing for more information. In particular, there was no illustration of a zeolite. A quick image search convinced me that this topic is a perfect fit for my blog.

So what is a zeolite? Zeolites form a class of minerals composed of aluminium, silicon, and oxygen. They form a very regular arrangement of pores which makes them highly useful in industrial applications.

The Groups of the Tetrahedron, the Octahedron, and the Icosahedron

After discussing the dihedral group, it is time to post my images of how Klein introduces the symmetry groups of the tetrahedron, the octahedron and the icosahedron.

By duality, this also handles the case of the dodecahedron (it is the dual of the icosahedron) and that of the cube (it is the dual of the octahedron) and thus all the Platonic solids are covered.

Mathematical Inspiration from the Sea

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The Census of Marine Life is one of the truly big scientific endeavours of our time. Together with loads of scientific results, it produced incredible images of an alien world, largely unknown and full of wonders.

The NOAA maintains a website with many quotes along the lines of the following:

» Man’s perpetual curiosity regarding the unknown has opened many frontiers. Among the last to yield to the advance of scientific exploration has been the ocean floor. Until recent years much more was known about the surface of the moon than about the vast areas that lie beneath three-fourths of the surface of our own planet. «

— In Submarine Geology (1948) by F. P. Shepard. p. 1.

The situation was not much different when the Census of Marine Life started and this is one of the reasons why life from the deep can still astonish and inspire us.

The Dihedral Group

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Snow flakes are one of nature’s beauties which are easy to appreciate even for the more mathematically or technically minded. Kenneth G. Libbrecht produces wonderful photographs of them; some are available online.

The symmetry of snow flakes is described by the dihedral group. This is one of the first groups described in Felix Klein’s book I have advertised before. Here is my illustration of how Klein described this group geometrically.

Lectures on the Icosahedron

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Some weeks ago, I was looking for examples giving me a quick overview on how to control transparency in the raytracer POV-Ray. This took me to the website of David Dumas who has beautiful illustrations of limit sets. (A very good and accessible introduction to the beauty of the Kleinian groups behind this is given by the book “Indra’s Pearls” [1].) What took me as even more beautiful were the pages from old mathematics books he is using as a background.