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.

Continue reading

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.

Continue reading