[HN Gopher] The Shape of Inequalities
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       The Shape of Inequalities
        
       Author : nomemory
       Score  : 85 points
       Date   : 2026-03-19 14:37 UTC (8 hours ago)
        
 (HTM) web link (www.andreinc.net)
 (TXT) w3m dump (www.andreinc.net)
        
       | incognito124 wrote:
       | 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 !!
        
       | dhosek wrote:
       | 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.)
       | 
       | [?]
       | 
       | 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.
        
         | epgui wrote:
         | Weird, in Canada (at least some provinces) I think that's a
         | pretty standard part of both high school and undergraduate
         | maths.
        
           | dhosek wrote:
           | The relationships between the functions are pretty standardly
           | taught, but their derivation from the right triangles on the
           | unit circle less so (other than sin and cos).
        
       | Sniffnoy wrote:
       | 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...
        
         | abnry wrote:
         | Which IIRC are all a consequence of Jensen's inequality.
        
           | arjvik wrote:
           | This I didn't know!
        
         | cubefox wrote:
         | I think this is not quite right as stated there because the
         | root mean square (quadratic mean) is always positive or 0 while
         | the arithmetic mean can be negative, making it smaller. I guess
         | the inequality only holds for positive numbers.
         | 
         | That's actually one argument for not calling the root mean
         | square a "mean", because a mean should arguably have the
         | property that it is always a number between the largest and
         | smallest value. But the RMS of two negative numbers is
         | positive. (On the other hand, the median would qualify as a
         | mean in this sense, even though it is not a "power mean".)
        
       | foldU wrote:
       | 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?
        
         | nomemory wrote:
         | Thanks for spotting this. I've mixed two ideas. Need to
         | comeback to it. The smaller circle has to increase its size as
         | b grows. As it is now it works because o triangle degeneration.
        
       | StefanKarpinski wrote:
       | 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.
        
         | nomemory wrote:
         | That's actually a good advice. Does a separate, "static"
         | screenshot also work?
        
           | StefanKarpinski wrote:
           | Sure, that would work just as well. Plus, then you get to
           | pick a "good" placement instead of making the user try to
           | find one.
        
             | nomemory wrote:
             | Ok. I will add them tomorrow. Also I will fix the first
             | animation (b doesn't grow there as it should).
        
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