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Mathematics LibreTexts

2.0: Symmetry and Functions


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Now we will work towards a precise definition of a group. In mathematics, we often begin explorations with some general concept (in our case, symmetry) and then work on those concepts until we can arrive at a very precise definition. Having precise definitions allows us better abstractions and generalizations; the more precise our definitions, the better the mathematics we can expect to derive.

Recall our definition of the symmetries of a geometric object: all of the ways of moving that object back onto itself without changing it. Each of these different ways of moving the object back onto itself we can identify as a function. Let's call our object \(X\). Then every symmetry of \(X\) we can identify as a certain special function \(f:X\rightarrow X\).

Earlier we noted that leaving \(X\) alone should also be a symmetry of \(X\). This is the identity function! \(id:X\rightarrow X\), with \(id(x)=x\) for all points \(x\in X\).

Next, we notice that composition of functions is a helpful operation: Indeed, if we have two different symmetries \(f\) and \(g\) of \(X\), then their composition \(g\circ f\) will also be a symmetry. The first function applied to \(X\) 'moves \(X\) onto itself without changing it,' and then the second does as well. Thus, the composition is also a symmetry.

Finally, we note in passing that the composition of functions is associative: for three symmetries \(f,g,h\), we have \((f\circ g)\circ h = f\circ (g\circ h)\).

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Composition of two symmetries of a rectangle.

We really need some examples here! So let's consider the perfectly symmetrical face. There are only two symmetries: the identity and the left-to-right flip (reflection over the vertical axis). Call the identity \(e\) and the flip \(f\). Then we see that, considered as functions from the face to itself, \(f\circ f=e\).

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The red arrows describe the flip over the vertical axis. Do it twice, and you end up where you started: \(F\circ F=id\).

Now remember our 'bumpy' hexagon, which only had rotational symmetries. Call the identity \(e\) and let \(r\) be the clockwise rotation by \(60^\circ\). All of the other rotations we can think of as \(r\) composed with itself some number of times; we'll just write this as \(r^k\). So all of the symmetries of the bumpy hexagon are \(\{e, r, r^2, r^3, r^4, r^5\}\). We notice that \(r^5\circ r=r^6=e\). This is quite interesting! In fact, we can make a 'composition table' to keep track of what happens when we compose any two of the symmetries.

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The six rotational symmetries of the bumpy hexagon. The blue arrow represents the rotation \(r\); \(r^6=id\).

And now we can make a slightly less obvious observation: For any of these functions, we can find another symmetry taking the object 'back' to its original orientation. Thus, for any symmetry \(f\), there exists a \(g\) such that \(f\circ g=e\). The function \(g\) is then called the inverse of \(f\). We've already seen two examples of this.

Exercise 2.0.0:

Write down all of the symmetries of an equilateral triangle. Make a 'composition table' of the symmetries, showing what happens when any two of them are composed. Then make a list of each symmetry and its inverse. What do you observe?

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