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  1. 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.

  2. With distance involved, the two nearest points to a given point are equivalent and so this new object has two lines of symmetry only.

  3. 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.

With regards.

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    $\begingroup$ Your question is quite obscure to me. First, using a compass already implies distance is involved (a compass is nothing else than an instrument that traces the set of points at the same distance to a given point than the distance separating two given points), and the very notion of square relies on distances. Second, item 2 makes no sense to me: what is "the new object" you are writing about? Third, what do you mean by "different points"? Do you mean that are made distinguishable? Or that they are not equal? Or something else? Last, your ultimate goal is not very clear to me, I must say. $\endgroup$ – Benoît Kloeckner Jun 1 '16 at 12:14
  • $\begingroup$ thank you for helping me. I will keep here one short comment only; I am intrigued by your remark that using a compass (pure geometry) implies distance (real numbers); please clarify, as we may met at some point on this.. $\endgroup$ – Dragan Jun 1 '16 at 15:08
  • $\begingroup$ Sorry, but I still can't make sense out of this question. In my remark on the compass, the distance I have in mind is of course the Euclidean distance (sometimes called a metric, but the two words are only one in French and have a large intersection in mathematical English, hence a possible misunderstanding). For a discrete set such as you seem to consider in you step two, the notion of line of symmetry makes no sense. I vote to close for I don't feel your question is articulated properly in mathematical language. $\endgroup$ – Benoît Kloeckner Jun 1 '16 at 15:28
  • $\begingroup$ if i understand you correctly, you claim that a square or a triangle, for that matter, must have be defined only with full blown mathematical structure of my step 3 to be candidate for a geometric figure with a symmetry ? $\endgroup$ – Dragan Jun 1 '16 at 18:21
  • $\begingroup$ You write: "So, let start with those four points that geometrically define square.". Please clarify what definition of "square" you are using here. You seem to be (implicitly) saying that any set of four points is a square. Surely that's not what you mean? $\endgroup$ – mweiss Jun 1 '16 at 22:13
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The indistinguishability of the points is precisely the reason why the square has symmetries.

Let's labels one of the four points of the square $A$. To this you object (I think, if I have understood you correctly): "But there is no way to tell the points apart, so how do we know which one is $A$?" And to this I respond: "I agree that we don't know which one is $A$. It is an arbitrary choice. Remember that for later."

Next, let's label one of the remaining three points $B$ -- but let's be sure to choose one of the points with the property that the segment joining $A$ to $B$ is one of the sides of the square, not its diagonal. (Notice that having made a choice of $A$, we are now able to partly distinguish among the remaining three points, because two of them are adjacent to $A$ and the third one is diagonally across from $A$.) We still don't have any basis for distinguishing between the two candidates for $B$, so just pick one of them arbitrarily, and remember for later that we made this choice.

Having chosen $A$ and $B$, let $C$ be the point diagonally across from $A$ and let $D$ be the point diagonally across from $B$. Notice that this identification is now unambiguous; having chosen $A$ (arbitrarily) and $B$ (partly arbitrarily), the identification of $C$ and $D$ is now completely determined.

Now, what if two different people, looking at the same (unlabeled, "naked" square in a bare plane with no coordinates) had followed the same set of instructions above, but made different initial choices about which point to call $A$ and which point to call $B$? Then they would have labeled the squares differently, of course. All together there are $8$ different possible labelings they could have chosen. Those $8$ labelings are precisely the $8$ symmetries of the square.

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  • $\begingroup$ o.k. i think i understand your point here. 1st: Please clarify 'opposite' (point); how to find (constructively) which of the three other points are 'opposite' to A (arbitrary chosen point) and which adjacent if you have only points, no angles, no distances ? 2nd: the second labeling of the "naked" square will be exactly the same as the first ! The difference will pop up only if you compare the second labeling with the first one which means that the square is not naked or that the points are already distinguishable. And 3rd: please try formally to define "labeling" - no chalk, no pencil $\endgroup$ – Dragan Jun 1 '16 at 18:18
  • $\begingroup$ "Label" does not require an act of physical labeling, just the decision to call an arbitrarily-chosen member of a set by a generic name. "There are three elements in a set; call one of them $x$" is a perfectly reasonable mathematical move. $\endgroup$ – mweiss Jun 1 '16 at 22:05
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    $\begingroup$ As for your claim that two such assignments of names are "the same", the fact that the points are not distinguishable does not mean that they are not distinct. If there are 3 points in a set, and the points are not distinguishable, there are nevertheless 3 points in the set, and 6 permutations of the set. That is: If I decide to call one of them $A$, and one of them $B$, and one of them $C$, there are 6 different ways to do that. There may operationally be no way to tell one such set of choices from another, but that doesn't change the fact that there are 6 sets of choices. $\endgroup$ – mweiss Jun 1 '16 at 22:06
  • $\begingroup$ How can you even define "square" if you don't have angles or distances? More specifically: one doesn't need to be able to assign a measurement to segments or angles, but one does need to be able to determine whether two segments are congruent, and whether two angles are congruent to each other. Otherwise there is no way to tell the difference between a square and an arbitrary quadrilateral. $\endgroup$ – mweiss Jun 1 '16 at 22:09
  • $\begingroup$ Finally: Whether a given point $P$ is "opposite from $A$" or "adjacent to $A$" can be determined quite simply by checking whether the segment from $A$ to $P$ passes through the interior of the square or lies along one of its edges. (If you don't have a sufficiently robust geometry to distinguish between "interior" and "edge", then I ask again, how can you even define "square" in the first place?) $\endgroup$ – mweiss Jun 1 '16 at 22:12

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