Nonlinear interactions

A friend sent several though-provoking quotes:

• If you find no one to support you on the spiritual path, walk alone. There is no companionship with the immature. - Buddha from The Dhammapada

• Stay with friends who support you in these. Talk with them about sacred texts, and how you are doing, and how they are doing, and keep your practices together. - Rumi 

• Retire into yourself as much as possible. Associate with people who are likely to improve you. Welcome those whom you are capable of improving. The process is a mutual one. People learn as they teach. - Seneca 

• You are the average of the five people you spend the most time with. - Jim Rohn

The response in our group was interesting. We're the inverse of Lake Wobegon, ("where all the children are above average") - we're all below average and our friends are bringing us up.  We all "average up."

It occurred to me that this is similar to what I call the 60/40 split. I'm the kind of person that needs marriage to be a 50/50 relationship - not that each duty should be split in half, but that both parties should be pulling equally (and hopefully the same direction).  I imagine there are couples out there that are happy with some other arrangement, but it wouldn't work for me.  However, the problem with a 50/50 relationship is that we have imperfect knowledge of what the other person is doing, and we have absolutely perfect knowledge of what we're doing (well, perhaps perfectly overestimated).  So when I think things are 50/50 it means I'm probably not pulling my own weight. Thus, my 60/40 split - things are really 50/50 when you feel like they're 60/40. So no need to drop into the resentment dungeon just because I think I'm doing more, because I'm probably not.  

I doubt that my thought is original, but I don't know when/where I heard it or thought it.  I Googled the idea and found some similar marriage advice, proposing that you give 60% and take 40%. Although they pitch it differently, I think the results are the same: you overestimate what you give and underestimate what you take, so a perceived 60/40 give/take is probably a 50/50 actuality.

Back to the group of friends who are all averaging up.  We could propose an inverse relation for friendship - we have excellent knowledge of what we get from our friends but imperfect knowledge of what we give.  So its reasonable, even desirable, for a group of friends to all perceive themselves as below average relative to the group. I don't know anything about social theory, but I suspect that dysfunctional groups exist if one or two people believe themselves superior or are believed by the others to be superior.  There's probably something to say about our political parties in that.

Although mathematically impossible for everyone to be above or below average, there are cases where the average, or arithmetic mean, is simply meaningless.  For example, the average amplitude over one sine wave period is exactly zero - a truth that has relatively little value but is merely inherent in the nature of sine waves.  However, the root-mean-square (RMS) of a sine wave can be computed by successively squaring the amplitude, taking the average of the square amplitude, and the the square root of that average (sounds complicated for non-math types, but it really isn't).  For sine waves of a given frequency, the RMS value describes one unique wave - providing a useful description that you cannot get from the simple average.  

The value of the RMS approach is that the squaring will accentuate the highs and minimize the lows.  This brings us to the idea of nonlinear interactions. If you take two different sine waves and compute a point by point product of the two waves (instead of squaring only one wave) and the take the mean and the square root as done in the RM of RMS, you get another unique wave whose shape depends on the amplitudes and frequencies of the underlying waves but may have a completely unusual shape. It is possible for waves to be out of phase and destructive, or in phase and supportive. 

If I could only move from pedantic to pithy there might be a quotable idea in there somewhere.