The Dirichlet principle and a parabola problem

Hermann Minkowski is less famous than Albert Einstein, partly because Minkowski died young, from appendicitis; but Minkowski developed the maths for the theory of relativity. One of the displays in 2B6 is Minkowski’s statement of an idea that goes back to the mid-19th century mathematician Dirichlet:

“Face problems with a minimum of blind calculation, a maximum of seeing thought”.

Below (click on the diagram to see it bigger) is a solution to FP1 Review Exercise no.60 following Minkowski’s principle. I owe the idea for this solution to Brenda Kuekia, who suggested drawing in the vertical PX.

The angles marked α are all equal, and β = π/2 − α.

Triangle NPT is similar to triangle SYD

So PT/PN = DY/YS = 2at/2a = t

The advantage of this solution over the algebra offered in Edexcel’s Solution Bank is there the neat answer PT/PN = t comes as a surprise at the end of the calculation (a page and a half of it), and you can see no reason beyond good luck why it’s not some much more complicated formula.

Here, “seeing thought” can show the answer with much less working than the Edexcel method, and also at least an idea of why and how the answer comes to be so neat.

No disrespect to calculation skills. Being able to work through complicated sequences of algebraic calculations is as important in maths as being able to punctuate and spell correctly (without needing to check) is in writing. But becoming a good writer is about more than being able to punctuate and spell fluently. It is also about learning to express ideas compactly, vividly, and imaginatively.