Point Symmetry
A figure has point symmetry if it looks identical after a 180-degree rotation about a central point.
Definition
A shape has point symmetry if spinning it exactly half a turn ($180$ degrees) around its center leaves it looking exactly the same, so every part of the shape has a matching part directly opposite the center. Formally, 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 $P' = 2C - P$ is also on the figure; point symmetry is just rotational symmetry of order $2$. In higher dimensions, the analogous concept is central symmetry: a figure $F$ is centrally symmetric about $C$ if $-x + 2C \in F$ for all $x \in F$, a hypothesis studied in convex geometry and number theory (Minkowski's theorem).
Example
A rectangle has point symmetry, spin it $180$ degrees and it looks the same, and the letters "S," "Z," "N," and "H" all have it, as do a regular hexagon and a circle. A parallelogram has point symmetry about the intersection of its diagonals, every vertex maps to the opposite vertex under $180$-degree rotation; a non-square rectangle has point symmetry but only $2$ lines of symmetry, unlike a square's $4$. Minkowski's theorem states that every centrally symmetric convex body in $\mathbb{R}^n$ with volume greater than $2^n$ contains a nonzero lattice point, a fundamental result in the geometry of numbers.
Key Insight
The center acts like a balancing pin: for every point on the shape, the point directly across the center, the same distance on the other side, is also on the shape. Odd regular polygons (triangle, pentagon) do not have point symmetry, since rotating $180$ degrees does not map them to themselves, while even regular polygons (square, hexagon, octagon) do. 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.