I was teaching this lesson, when I came across one of the children (V) showing another (C) that the parallelogram they had made out of card did not have any lines of symmetry. C had conjectured that there was a line of symmetry from corner to corner. V was folding the card in half, showing that there was no fold that produced two perfectly overlapping halves. C took the card and tried folding it again and again to find such a fold, but could not, and then became convinced that it wasn't possible.
They, and everyone else in the class, had been using a mirror to test lines of symmetry up to this point, but using mirrors to test lines of symmetry isn't always that conclusive. Sometimes it is necessary to feel something, as opposed to only seeing it.
C's conjecture is understandable. Looking at the parallelogram, we can see it has symmetry, but this symmetry is of a different kind to reflection symmetry. We have not yet discussed rotational symmetry. The parallelogram looks the same if we turn it through a half-circle, or if we walk around to the opposite side of it (which I think is also true of shapes with reflection symmetry), but it doesn't have that folding quality.
I only remembered this today, two days after the lesson. I started thinking about the nature of reflection symmetry. Which comes first, a line of symmetry creating equal and opposite lengths and angles, or the equal and opposite lengths and angles creating the line of symmetry? Then my mind began to wander.
Mathematics imposes an order on things, and perhaps suggests a desire in some humans to impose order and symmetry on the things around them that might not be there. I started thinking about perceived symmetry in the world around me, such as the symmetry of the human body. There is only (approximate) symmetry on the surface, but asymmetry in many of the organs underneath. That said, where there is perceived asymmetry there may also be symmetry; underneath the appearance of difference I often find much similarity.
Nature contains examples of symmetry, but mostly asymmetry and chaos. Looking out to sea (which I do often!), I see the surface symmetry of 'living' things (such as birds), but in 'non-living' things I see little reflection symmetry, mostly only chaos with momentary cyclical repetition - perhaps suggesting a rotational symmetry of a different kind.
In my training as a counsellor, I have found it useful not to assume symmetry, particularly regarding mine and others' thoughts, values, and feelings. There is a counselling technique - although it is not really accurate to call it a technique - called 'reflection of feeling', in which the counsellor attempts to hold up a mirror to the client's thoughts and feelings. I have found this difficult to do; it requires careful looking and listening. Often my own thoughts and feelings influence where I put the mirror. What of the folding analogy here? Perhaps a bringing into contact, or an unfolding to reveal the image.