Point Symmetry
Geometry & MeasurementA figure has point symmetry if it looks identical after a 180-degree rotation about a central point.
Definition
A shape has point symmetry if you can spin it exactly half a turn ($180$ degrees) around its center and it looks exactly the same as before. Every part of the shape has a matching part directly opposite the center.
Example
A rectangle has point symmetry: spin it $180$ degrees and it looks the same. The letters "S," "Z," "N," and "H" all have point symmetry. A regular hexagon and a circle also have point symmetry.
Key Insight
Point symmetry means the center point is special: for every point on the shape, the point directly across the center (the same distance on the other side) is also on the shape. The center is like a balancing pin.
Definition
A figure has point symmetry about a center $C$ if rotation by $180$ degrees about $C$ maps the figure to itself. Equivalently, for every point $P$ on the figure, the point $P' = 2C - P$ (the reflection of $P$ through $C$) is also on the figure. Point symmetry is rotational symmetry of order $2$.
Example
A parallelogram has point symmetry about the intersection of its diagonals: every vertex maps to the opposite vertex under $180$-degree rotation. A regular hexagon has both point symmetry and $6$-fold rotational symmetry. A non-square rectangle has point symmetry but only $2$ lines of symmetry (unlike a square's $4$).
Key Insight
Point symmetry is equivalent to rotational symmetry of order $2$ (minimum rotation $180$ degrees). Odd regular polygons (triangle, pentagon) do not have point symmetry because rotating $180$ degrees does not map them to themselves. Even regular polygons (square, hexagon, octagon) do.
Definition
Point symmetry about $C$ is the isometry $x \mapsto 2C - x$, which is a $180$-degree rotation. A figure $F$ has point symmetry about $C$ iff the antipodal map (with respect to $C$) is in $\text{Sym}(F)$. In higher dimensions, the analogous concept is central symmetry: $F$ is centrally symmetric about $C$ if $-x + 2C \in F$ for all $x \in F$. Centrally symmetric convex bodies are studied in convex geometry and number theory (Minkowski's theorem).
Example
Minkowski's theorem: every centrally symmetric convex body in $\mathbb{R}^n$ with volume greater than $2^n$ contains a nonzero lattice point. This result, fundamental in the geometry of numbers, uses central symmetry (point symmetry about the origin) as a hypothesis.
Key Insight
Central symmetry is a hypothesis in many geometric inequalities and number-theoretic results. The Minkowski sum of two centrally symmetric sets is centrally symmetric, and many extremal problems in convex geometry (isoperimetric, Mahler conjecture) involve centrally symmetric bodies, connecting point symmetry to deep questions in mathematics.