Chapter 2: Round and Round the Merry-Go-Round - The Beauty That Emerges From Rotations

In the previous chapter, the judge was tasked to look at some of the foreign visitors to determine the extent of their reflection symmetry.

What he has found, to his dismay, is that each and every one of them are at least more symmetrical than the vain queen. How so? Well, the queen, being human-shaped, has only one line of symmetry. But some of the mysterious visitors have as many as eight or twelve lines of symmetry!

While he ponders on the possible ways that he can deliver this news to the queen without being locked in the dungeons for life, the arrival of a second entourage of the mysterious beings was announced. They too were summoned by the queen for judgement.

Internally resigning himself to his fate, the judge turned to look at the next wave of beings here to be checked. Expecting yet another round of crushing despair at the perfect reflection symmetries of the strange beings, he was rather surprised by what fell upon his eyes.

While a few of them did have reflection symmetry as expected, his checks revealed that some of them had no reflection symmetry whatsoever. However, what should be a happy revelation turned into puzzling stares at the strange beings. Sure, some of them weren't symmetrical in the mirror, but they had some sort of... otherworldly beauty. So much so that it made the judge tilt his head in confusion.

And then it clicked.

                                                                                                                                                                                                                                                                       

In the following simulation, we explore another form of symmetry, called rotational symmetry. Similar to reflection symmetry, instead of an object looking the same under a mirror reflection, this symmetry is the property of an object looking the same under some rotation. Even though they are both symmetries, there are some main differences between reflection and rotational symmetry.

The first notable difference is that there is no concept of a "line of symmetry" in rotational symmetry. After all, a line of symmetry tells us where to reflect the object in order to see the symmetry, but there is no reflection in rotational symmetry.

The second difference is that instead of lines of symmetry, rotational symmetry has a different concept that reflection symmetry does not have. This concept is called the order of a rotational symmetry. More informally, the order of a rotational symmetry is the number of rotational symmetries that a shape has.

We can illustrate this by looking at a simple example - that of an equilateral triangle. To make it easier for us to count, we can mark one corner of the triangle with a point, like so: