Consider a square : four points in a plane constructed with classical means (compass and straightedge). Since no point is different from others (no coordinates, no labels...) it seems that we can not define any (some) symmetry without additional mathematical structure.
With distance involved, the two nearest points to a given point are equivalent and so this new object has two lines of symmetry only.
In case all four points are taken as different we have our four lines of symmetry...
The question is : what is the best way to introduce the minimum additional mathematical structure needed, to go from 1 to 2 and from 2 to 3 without reaching to physics and other extra-mathematical illustrations (paper folding, physical rotation etc.) ? Maybe there are more than one good road to achieve this goal ? What are the key / most persistent / misconceptions in introducing symmetry of a square to elementary graders as well as the silent assumptions underlaying such introductions ?
Any reference would be very much appreciated.
Let me try to clarify myself.
The goal is to explain, in simple but hopefully, mathematically correct way, why and under which conditions one may say that a square has four lines of symmetry. So, let start with those four points that geometrically define square. They are all indistinguishable as lay on a plane. We may label them (and so make them physically distinguishable) but lets try not to step outside mathematics. We can not rotate this "square", can not translate it - it is quite a primitive object.
Introduce, at a step No. 2, a simple metric (distance) in this set X of our four points. (map d: X x X -> R) Question: what is the most simple metric here ? The one in which the two adjacent point of any given point of our square are still indistinguishable (equivalent) and the fourth one, is the furthest one from our vantage point. Question: with this most simple metric, how many lines of symmetry does this (new) object (four points plus simple metric) has ? I say two, one along the diagonal and the other cutting the opposite sides at half. We can not discuss any rotation at this level as we did not have coordinates for all points of our plane. Now we may select another point as reference point of a coordinate system, make axes, units along these axes and label the points with coordinates in respect of this coordinate system. This is a lot of additional mathematical structure. Now we have square with distinguishable points, rotations and - voila : four lines of symmetry.
So, that is. From geometry, to metric spaces, real numbers, continuous symmetry etc. Quite a lot, i would say, to be introduced at one step to.
Key question : how to make these steps, from a most basic square (four points only) to our useful, ordinary square (with circumference and area) as simple and as precise as possible.