Proposition 9.4.1. If F is a figure with rotational symmetry then:
a) all rotations must have the same center.
b) if Rc ∈ Iso(F) then Rc,2 ∈ Iso(F), Rc,3 ∈ Iso(F), ... In general, for any integer m, Rc,m ∈ Iso(F).
c) if F is not a circle (or union of circles with the same centers) then there exists a real number θ such that Rc,θ ∈ Iso(F) and for any α with Rc,α(F) = F we have α = nθ for some integer n. (We say that Rc,θ is the "smallest" rotational symmetry.)
d) If θ is as in c), then (360)/θ = n is a positive integer.
Proof.
a) To prove that all rotations must have the same center, we note that if we did have Rc, Rp ∈ Iso(F) then Rc ◦ Rp ∈ Iso(F). But the composition of two rotations with different centers is a translation (we will take this result for granted since we didn't prove it). On the other hand, we cannot have translations in Iso(F).
b) The result in b) follows based on the fact that Iso(F) is a group. (Why?)
c) Since F is not a circle, we can consider the smallest angle for which rotating in the center C through that angle is a symmetry of F. Let's call this angle θ. We want to show that for any other angle α with Rc,α(F) = F, we must have α = nθ for some integer n. Indeed, for such an angle α, we can find an integer k such that kθ ≤ α < (k + 1)θ. Let β = α - kθ and note that 0 ≤ β < θ. We claim that Rc,β ∈ Iso(F). This is so because of the following:
i) Rc,kθ ∈ Iso(F) (Why?)
ii) Rc,-kθ ∈ Iso(F) (Why?)
iii) Rcβ = Rc,α - kθ = Rc,α ◦ Rc,-kθ ∈ Iso(F). (Why?)
Now, since β < θ and Rc,θ is the smallest rotational symmetry, the only way that iii) above can hold is if β = 0. But this implies α = kθ.
d) To prove that (360)/θ is an integer, let's note that we can find an integer n such that nθ ≤ 360 < (n + 1)θ. But then, by using the same argument as above, Rc,360-nθ ∈ Iso(F). Again, this would show that Rc,360-nθ is a smaller rotational symmetry than Rc,θ, unless 360 - nθ = 0. But this is equivalent to (360)/θ = n is an integer.
Exercise 9.4.2. Give a precise proof of b) above.
Exercise 9.4.3. How many rotational symmetries does a circle have? Is the result in c) valid for circles? Why?
Exercise 9.4.4. Find the integer k for which k * 1.8 ≤ 7 ≤ (k + 1) * 1.8.
Exercise 9.4.5. Can a bounded figure have rotational symmetry through 90 degrees? How about 240 degrees? If Rc,α(F) = F, what can you say about (360)/α? (For the last question, please see parts c) and d) of Proposition 9.4.1)
Exercise 9.4.6. Consider the figures with rotational symmetry from Exercise 9.2.3 and for each one of them identify θ (the smallest rotation angle) and n(= 360/θ).
Exercise 9.4.7. Based on part b) of Proposition 9.4.1, it may seem that a figure F with rotational symmetry has infinitely many rotational symmetries. Show that if F is not a circle (or union of circles centered at the same point) then Iso(F) contains only finitely many rotations.
