Saturday, November 16, 2013

Mathematician's toy or Klein Quartic

Here it is - a mathematician with stuffed toy.

It all started with this animation. Then I learned - it can be made with 24 heptagons. No problem - 24 heptagons were crocheted and then we both started to figure out what would be the best way to put them together. I followed David's suggestion and sewed them together in symmetric way. Then it was put away and when I picked up this fall again I wanted to stress this nice heptagonal tiling, so I put the white lines on the model. Since it is impossible to finish this surface in three dimensions I decided to add handles that would indicate how heptagons will fit together in fourth dimension. David was not happy with the surface because; It is not smooth! - that was topologist speaking. Can you iron it to make it smooth? We do have iron at home and I pressed all heptagons to make them smooth. As soon I finished doing it I regretted what have I done - yes, surface was smooth but it was not holding shape anymore! It looked like wrongly put together afghan.;-( David suggested to stuff it. And here it is - stuffed toy for the mathematician.

What is this surface? Klein quartic is named after Felix Klein who first described it in 1878. It is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed.

Klein's quartic occurs in many branches of mathematics - some say it is one of the most important mathematical structures. Which mathematical branches use Klein quartic? Representation theory, octonion multiplication, homology theory, it was used to prove Fermat Last theorem... Sounds very complex - but you do not need to know all that to enjoy this nice stuffed structure!;-)

I would like one day to see artistic representation of this surface by Halaman Ferguson. His sculpture The Eightfold Way is at the Mathematical Sciences Research Institute in Berkeley, CA. It - is made of marble and was unveiled on November 14th, 1993- 20 years and two days ago!

The acquisition of the sculpture led in due course to the publication of a book of papers (Levy 1999), detailing properties of the quartic and containing the first English translation of Klein's paper.

Thurston explained this sculpture

Halaman Ferguson about creating The Eightfold Way

Mathematician's toy made it to Scientific American blog

1 comment:

  1. Ciao mi chiamo Lia ed insegno matematica....amo anche il crochet...leggendobro ti ho scoperta e mi hai incantata col tuo uncinetto iperbolico...ho fatto la sfera all'uncinetto è l'ho mostarla ai miei studenti...amazing!Ti seguo è troppo bello il tuo blog.Un caro saluto dall'italia Lia