The Whole and the Parts – Memory and Hysteresis

Imagine I’ve got a sheet of four (square) postage stamps, like this:

postage-stamps-hysteresisIf I need a single postage stamp, I need to do some tearing, and that tearing takes a bit of time and effort.  To tear along a single edge over the length of a stamp, we’ll call that a single unit of effort (or time).

To tear off the first stamp, I need to tear along two edges, so this takes 2 units of effort.  To tear off a second stamp only requires a single unit of effort.  To tear off the third stamp also requires a single unit of effort, but then the fourth stamp has been liberated also, so getting the fourth stamp requires 0 units of effort.


If we make a graph of the number of freed stamps in relation to total effort expended, we get:


This graph may look strange, not the kind of graph students typically see in secondary school.   A smooth version of this graph shows up if I turn my attention to a retractable ballpoint pen, as in the picture below:

From my secondary school days, I have a clear muscle memory of the relationship between the amount of effort with which you push on the silvery top, and the distance that the ballpoint tip is freed from the blue plastic casing.  The graph below shows a (somewhat simplified) version of this:

ballpoint-tip-graph1Let’s look first at the lower blue path, from a to b to c to d to e.  This represents pushing the tip out, over a hump, till it comes out no further.  This part of the graph is essentially the same as the graph for the postage stamps above.  If I then back off on the pressure, the tip will retract a bit, but only to point c.  If I back off even further, the tip will not retract more, and we will end up at point f.  At this point, the ballpoint pen is stable, and I can write with it.  I don’t need to keep pushing on the tip to have the point stay out.  In fact, the pen in this state can withstand quite a bit of pressure against the paper without the tip disappearing back into the plastic.  In this state, to get the tip to retract, I need to apply effort in a roughly symmetrical way.  This is shown in the green path, from f to g to h.  Again I push over a hump, and no amount of additional effort past g will have the tip extend more.  But if I now back off on the pressure, the tip will follow the path from h to i to j back to a.  We have now completed an entire round trip and we are back in the same situation we started in, with the tip fully retracted.

What we have here is a situation with memory.  The full graph represents two stable states (a and f).  For the ballpoint pen, these represent the two situations without my finger applying pressure to the silvery top, and with the tip either extended or retracted.  When effort is applied, the path followed depends on the situation we start out in, either blue or green.  Somewhere along the way, when applying more effort, the state gets flipped, and this shows up when we back off on the pressure: we don’t end up the same place we started.  A graph like I’ve shown is a variation on a type of graph called a hysteresis loop.  The hysteresis effect has many important applications – it underlies the entire world of digital electronics.  It is the principle that allows us to copy something without loss.  It allows a single physical device, like a memory stick or a disc drive, to hold on to many different stable states even when all electrical power is removed – and yet have its content be changed at will when powered up.

Stable states turn a bunch of parts into a stable whole.  If there are multiple discrete stable states, the whole can be a mechanism or organism capable of cycling through various states in response to outside stimuli.

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1 Response to The Whole and the Parts – Memory and Hysteresis

  1. Pingback: The Whole and the Parts – The Series « Learning and Unlearning Math

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