This is very small when the pendulum is long and the subtended angle of the swing small (see diagram on left), but there is an error component. Various branches of mathematics have mathematical beauty. Why thin metal foil does not break like a metal stick? Next time you have to roll a stone down a hill, I hope you'll remember those 17th Century mathematicians! To calculate the optimal path does not just require vanilla calculus (where you are minimizing a variable in a function), but instead requires minimizing a function that minimizes some other variables. The above pattern is nothing but area of the rectangle formed by joining the squares (continued fibonacci squares sum). The Fibonacci sequence is a recursive sequence, generated by adding the two previous numbers, the first two numbers of the sequence being 0 and 1.So, Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …An interesting fact is that the number of petals on a flower always turns out to be a fibonacci number. You can find a complete list of all the articles here. Objects placed near the top will travel faster past the finish line than objects placed near the bottom, but all will finish at the same time. I am mostly intrested in any examples for High School , but any other K12 examples would be appreciated as well. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.”. A near vertical drop at the beginning builds up the speed of the bead very quickly so that it is able to cover the horizontal distance faster to result in an average speed that is the quickest. As I understand it, there is no defined standard for ski-jump profiles (many being built-into the surrounding terrain and using the natural gradient), but if you were going to construct one, you could give it a brachistochronic profile. Could students learn a lot more from school if they're only taught number theory until way later? Fibonacci spiral recurs throughout the nature — in the seed heads of sunflower, the petals of a rose, the eye of the hurricane, the curve of a wave, even the spiral of galaxies! The animation on the left shows a Brachistochrone path in red, and two other possible paths in blue with representations of balls rolling down tracks. Remembering back to high-school physics, the period of a pendulum is determined by it's length, but this is an approximation. 15 – Snowflakes, You can’t go past the tiny but miraculous snowflake as an example of symmetry in nature.