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© Cordon Art-Baarn-the Netherlands

Many of you must be aware of the works of the famous Dutch artist Maurits Cornelis Escher who explored the notions of symmetry and  infinity and depicted visual paradoxes and impossible worlds through his art. He worked extensively on regular divisions of the plane, also called tessellations, which are arrangements of closed, interlocking planar shapes which cover the whole plane without any gaps. Escher distorted basic planar figures such as triangles, squares and other polygons into organic forms to construct his tessellations. Even though he was not mathematically-trained, Escher displayed a keen intuition and creativity which appeals to mathematicians and non-mathematicians alike. You can visit Escher in the Classroom, a fantastic site which walks you through some of Escher’s constructions.

Inspired by a drawing by H. S. M. Coxeter, Escher created his Circle Limit series, which uses the Poincare disk model of Hyperbolic space

Math and the art of M. C. Escher and   the mathematical art of M. C. Escher are also wonderful sites which talks in detail about the mathematical nature of Escher’s work.

One of Escher’s most fascinating works is the Print Gallery. It shows a young man standing in an exhibition gallery, viewing a print of a Mediterranean seaport. As his eyes follow the buildings shown on the print from left to right and then down, he discovers among them thevery same gallery in which he is standing. A circular white patch in the middle of the lithograph contains Escher’s monogram and signature. Artists and mathematicians have wondered if at all this white patch could  be filled. It was only in 2002 that Henrik Lenstra, a professor of Mathematics, came up with a mathematical explanation for how the Print Gallery can be constructed and his solution provided him with a way to fill the mysterious hole in the center. For more details you can refer to the excellent article titled Artful Mathematics: The heritage of M. C. Escher in the Notices of the American Mathematical Society, and visit the website of this project titled Escher and the Droste Effect, in which a step-by-step process at arriving the solution is explained.

Achieving the Unachievable is an award-winning film that unravels the mystery behind the Print Gallery. It has been screened in many universities across the world and is a must-watch for everyone. You can enjoy the film at the screening organized by Singularity at 11:00 a.m. on Saturday, March 26 in L1. For now, here’s a trailer:

I have talked about some of the major open problems in mathematics with my MTH 101 students (either in class or outside) and these include the Riemann Hypothesis, the Goldbach Conjecture and the Collatz Conjecture. One important problem which was resolved recently is the Poincare Conjecture . As you may know, Grigory Perelman provided a sketch of the proof in his arxiv papers which was subsequently verified by various prominent mathematicians and at the 2006 International Congress of Mathematicians held in Madrid, Spain, he was awarded the Fields medal (which he declined; more recently, he also declined the million dollar prize from Clay Mathematics Institute). I have been wanting to mention this for some time but it was not easy to find a context to talk about this problem in a Calculus-1 class…..until now.

I read the OU Math Club blog regularly and over there I found a news item which I am sure my MTH 101 class will find interesting (particularly since many of them recently participated in a Fashion Show at the annual fest EnthuZia 2010).  Now, Perelman actually proved a more general result, called Thurston’s Geometrization Conjecture which implies the Poincare conjecture. Cornell math professor William Thurston proposed this conjecture in 1982 and it says that, roughly speaking, a three dimensional geometric object (more precisely, a closed, oriented 3-manifold) can be cut into geometric pieces, each of which has one of the eight geometries in dimension three. These consists of spherical, hyperbolic and flat geometries and five other kinds which are somewhat difficult to explain here. For his seminal contributions, Thurston was awarded the Fields Medal in 1982.

Inspired by Thurston’s work, Issey Miyake fashion designer Dai Fujiwara created his fall-winter 2010-2011 collection which attempts to illustrate the eight geometries that are sufficient to describe a three-dimensional form.  The collection was displayed in Paris in March this year and Thurston, wearing a designer jacket, attended the show. Here’s an ABC news story on the event. You can watch the interview with Fujiwara and Thurston here:

and the fashion show can be seen here: