How Algebra Sheds Light on Things

causticEgbert Brieskorn‘s mathematics books contain a lot more flesh than those structured strictly by definitions, lemmas, and theorems. What might be a hindrance to those looking for a quick grasp of a theory is a treasure trove for others who relish an extra of motivation, history, and examples. Currently, I am reading his book Plane Algebraic Curves (authored together with Horst Knörrer) which, already in the introductory chapter, contains many wonderful examples such as linkages, envelopes, and the Hopf fibration. I will come back to some of these topics in later posts. Today’s post contains a quick glimpse of caustics.

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Beautiful Binomials

binomialsWhile leafing through the book Geometric Trilogy I – An Axiomatic Approach to Geometry, I came across two nice geometric depictions that are probably widely known but which I would love to have seen back in school. The first one is the geometric illustration of the algebraic fact that (a+b)^2 = a^2+2ab+b^2, the quadratic case of the binomial theorem. This can already be found in Book II of Euclid’s Elements. From this, it is easy to come up with a three dimensional version of the construction giving (a+b)^3=a^3+3a^2b+3ab^2+b^3. Creating a good picture of this is somewhat tedious so I am very glad that I can use this opportunity to link to a wonderful blog where this and other wonderful mathematical illustrations and animations can be found: Hyrodium’s Graphical MathLand.

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Permanent Determinant

permanent_graphThe determinant and the permanent of a matrix are central characters in an endeavour to bring the powerful weapons of modern geometry to a battle in the epic war of computer science: the P vs. NP problem.

JM Landsberg has recently written a wonderful introduction to geometric complexity theory which is how the corresponding research field is called. This has inspired me to borrow some of it and write about the permanent and the determinant of a matrix.

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Euler and the Kids

euler-o-ballBeing on parental leave, somewhat naturally, gave me more time to spend in the kids’ room. After a chance encounter with Mr Pythagoras there some time ago, it was now time  for a scheduled meeting with Mr Euler. The configuration in the image to this post is known to young parents as an Oball. To mathematicians, it is known as the truncated icosahedron. An interesting fact is that because it is constructed entirely from pentagons and hexagons, it has to have exactly 12 pentagons. Let’s see how Mr Euler can convince us of this necessity. Continue reading

Regular Viruses

As long as they do not infect us, viruses are nothing but fascinating. One aspect of this fascination is based on the shapes of viruses. Looking at electron microscopy images of viruses or at computer models based on X-ray crystallography, my impression is that mother nature has copied from a geometry book. This impression is echoed by what you find in books and papers on virology. Why do biologists think about possible polyhedra with icosahedral symmetry? Why is it that many viruses have the shape of such polyhedra? For some years, it seemed like biologists had a very accurate theory of the construction of such regular viruses. Advances in imaging have left them less confident but with an even higher appreciation of the formation of biological shapes.

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