My favorite bit of trivia is related to the following game:

Start with 2 numbers, a and b and calculate HM and GM Now you have 2 numbers again, so you can play the game again with the new values Every step brings the results together, one from above, the other from below, sandwiching the value in the limit. That value is called Geometric-Harmonic Mean

This works for all 3 pairs of means (HM-GM, GM-AM, HM-AM). The fun fact I was talking about is about the last combination: playing the game with two "extremal" means, the AM and HM, the value they converge to is GM !!

There’s a whole pile of math like this that kind of lies in this nether land between more advanced than you’ll get in most high school math¹ but less advanced than you’ll get in most college high school math that I was only ever exposed to when I took the classes for my teaching credential. One of my favorite was how cos/sin, tan/cot and sec/csc all can be derived from right triangles on a unit circle with the first setting the hypotenuse to the radius, the second with a vertical side tangent to the circle at x = ±1 and the third with the horizontal side tangent to the circle at y = ±1 (you can use similarity and Pythagoras to get all the standard identities like tan = sin/cos, etc.)

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1. I kind of did a speed run through high school math, taking essentially 5+ years of math in three years, so it’s likely that I ended up missing/glossing over stuff that people who were learning at a more rational pace did learn, although I think some of my teachers were too intimidated by me to try actually teaching me, much to my detriment.

In case people aren't aware, the inequality of these specific four means is a special case of the more general power mean inequality: https://en.wikipedia.org/wiki/Generalized_mean#Generalized_m...

The geometric representation of AM/GM is very cool, but the first animation seems wrong to me, it should be varying the value of `b`, not the location of the circle, for it to make sense, no?

My favorite geometric proof of an inequality is the one I read on Terry Tao's blog. Interestingly, it's not presented as a geometric proof, but it is very much one: if you have two vectors x, y, you just shrink the longer one and grow the shorter one until they reach the same size, without changing the LHS and the RHS of the inequality. Then you expand the norms of ||x - y||^2>=0 and ||x + y||^2>=0 and see -||x||^2 - ||y||^2 <= 2<x,y> <= ||x||^2 + ||y||^2, and since ||x||=||y|| you get the result.

The animated visuals are very cool, but I desperately want to turn them off in order to understand what they depict and reason about it geometrically. A pause button would be greatly appreciated.

The AM/GM inequality is why the world switched from (what in the UK is called) the Retail Price Index (AM) to the Consumer Price Index (GM).

The book "When Less is More: Visualizing Basic Inequalities" by Claudi Alsina might also be of interest.

The first chart is super confusing. The OP line is changing size as the circles move, yet (a-b)/2 is a constant.

Oh, these are really nice Andrei! Thanks for posting them.

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