We’ve learned that one object can be more symmetrical than another, based on the number of rotational symmetries it has — that is, the number of rotations that preserve its appearance and shape. It turns out that the group of symmetries of an object is in fact a mathematical object itself. If we plot each of these symmetries, a distinct mathematical graph is produced. In the demonstration below, you can rotate a cube and a sphere around their central axes using the slider. On the right, the group of symmetries is plotted for both objects. Notice how the sphere has an infinite number of rotational symmetries — and so, its symmetry group is a circle; the shape that emerges from plotting all points from 0° to 360°. The cube, however, only has four rotational symmetries, at {0º, 90°, 180°, 270°}; its symmetry group forms a square.

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