Rotational Symmetry
Geometry & MeasurementA figure has rotational symmetry if it looks identical to itself after being rotated by some angle less than 360 degrees about its center.
Formula
\text{Minimum rotation angle} = \frac{360}{\text{order}}
Definition
A shape has rotational symmetry if you can rotate it less than a full turn and it looks exactly the same as before. The number of times it looks the same in one full rotation is called its order.
Example
A square has rotational symmetry of order $4$: it looks the same after $90$, $180$, $270$, and $360$ degree turns. An equilateral triangle has order $3$ ($120$, $240$, $360$ degrees). A regular hexagon has order $6$.
Key Insight
Every shape has "trivial" rotational symmetry of order $1$ at $360$ degrees. When we say a shape has rotational symmetry, we usually mean order $2$ or higher. A circle has infinite rotational symmetry because any rotation maps it to itself.
Definition
A figure has rotational symmetry of order $n$ if a rotation of $360/n$ degrees (about the center) maps the figure to itself, and $n$ is the largest such integer. The minimum angle of rotation is $360/n$ degrees. All regular n-gons have rotational symmetry of order $n$. A figure with rotational symmetry of order $2$ also has point symmetry.
Example
A regular pentagon has rotational symmetry of order $5$. The minimum rotation is $360/5 = 72$ degrees. A rectangle (not a square) has order $2$ ($180$ degrees only). The letter "S" has order $2$; the letter "N" has order $2$; the letter "Z" has order $2$.
Key Insight
Rotational symmetry of order $n$ means the shape is invariant under the cyclic group $\mathbb{Z}_n$ of rotations. Combined with reflections, the full symmetry group becomes the dihedral group $D_n$, governing the complete symmetry of a regular n-gon.
Definition
A figure $F$ has rotational symmetry of order $n$ if the rotation $R_{360/n}$ is an element of the symmetry group $\text{Sym}(F)$, and $n$ is the maximum such integer. The cyclic subgroup generated by $R_{360/n}$ is isomorphic to $\mathbb{Z}_n$ and forms the rotational symmetry group. For 3-D objects, rotational symmetry about different axes generates groups such as cyclic ($C_n$), dihedral ($D_n$), tetrahedral ($T$), octahedral ($O$), and icosahedral ($I$) groups.
Example
The five Platonic solids have the following rotational symmetry groups: tetrahedron $T$ ($12$ elements), cube and octahedron $O$ ($24$ elements), dodecahedron and icosahedron $I$ ($60$ elements). These are also the rotation groups of the five exceptional regular polytopes.
Key Insight
The classification of all finite subgroups of $SO(3)$ (3-D rotational symmetry groups) is a theorem: they are cyclic groups $C_n$, dihedral groups $D_n$, and the three exceptional groups $T$, $O$, $I$ corresponding to the symmetries of the Platonic solids. This classification connects group theory to solid geometry in a complete and beautiful way.