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Tuesday, November 5, 2013

Johannes Kepler and (3,3,4,3,4)

Who doesn't know the name of Johannes Kepler? Kepler (1571 - 1630 ) formulated the laws of planetary motion and his work provided the foundation for Isaac Newton's theory of gravity. My point being that Kepler was a scientific giant in his days and his name will live on forever.

My current mathematical project ( personal challenge if you like ) is focused towards tilings of the plane, creating ( Mathematica ) software to print and generate tilings. And ultimately -find- new tilings I haven't seen before. There aren't many textbooks on the subject, the classic work is very recent ( in mathematical terms ), it was published in 1986: Tilings and Patterns, by Branko Grunbaum and Geoffrey C. Shephard

Earlier today I was doodling on the tiling (3,3,4,3,4) of which I uploaded a picture.
Doodle of (3,3,4,3,4).
Then, to my surprise I read in Grunbaum / Shephard that it was Johannes Kepler (!) who started the mathematical research on tilings and patterns. One of the beautiful books Kepler wrote is called the Harmony of the World, originally published in 1619 but recently translated into English by Aiton, Duncan and Field and published the American Philosophical Society.

The Harmony of the World consists of 5 books.
- Book 1: On the construction of regular figures
- Book 2: On the congruence of regular figures
- Book 3: On the origins of the harmonic proportions, and on the nature and differences of those things which are concerned with melody
- Book 4: Preamble and explanation of the order
- Book 5: ( No title ).

This is part of a picture ( drawing ) from book 2 which has a drawing of (3, 3, 4, 3, 4 )  just like my doodle rype  marked as O ( top right ).

This proves once more that mathematics, by itself, does not change over time, its timeless. At least this part ( if not all ) of mathematics has to be discovered. The tilings of the plane have always been there, it just takes us to see them so that we can ultimately categorize them.

The fact that Kepler worked on this makes him human ( but still a giant of course ) to me, I can imagine the joy and excitement he must have felt drawing the illustrations especially since they take a lot of ( behind the scenes ) calculations.

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