# Beautiful Binomials

While 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.

Let me close with a second figure inspired by the Geometric Trilogy books. It shows two different ways of (literally) seeing that the sum of the first $n$ odd numbers equals $n^2$:

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