Monday, November 13, 2017

Hebrew Wikipedia's Democratic Structure (Hebrew)

AlgCalcOne: The Mystery of "Circular Area" (English)

What is the area of a circle of radius 1?..The truth is that even the definition of the "area" of a unit circle is problematic. While we can get at this in an approximate fashion as laid out for us by Archimedes, getting an exact definition is illusory.

The main calculation that approximates pi due to Archimedes is to approximate the "length" of the circumference of a unit circle by inscribed and circumscribed regular polygons with 6,12,24,48 and then 96 sides. [Note: In the video I mistakenly discuss areas instead of arclengths around 17:29, but the conclusion is the same anyway.]

Some key Calculus ideas will be examined in this video, as we come to appreciate the huge gulf between the exact area computation of a parabolic arc and the approximate area computation for a circular arc. This understanding represents a major departure from established orthodoxy, so hold onto your hats!

AlgCalcOne: Parabolic Splines and Archimedes (English)

This is our first serious integral calculus computation: we get the formula for the signed area of a parabolic arc, parallel to what Archimedes remarkably accomplished more than 2000 years ago, but with quite a different approach. We recover Archimedes' relation to the signed area of a maximal triangle inscribed in a parabolic arc: at least for the case of y=x^2..

Mistakes and Biases in Understanding Data (Hebrew)