Mystic Rose and Handshakes: Exploring Patterns and Combinatorics

①This example constructs a Mystic Rose—a star-like figure formed by connecting equally spaced points on a circle—and illustrates the handshake problem, counting pairwise interactions among n points. ②Key mathematical ideas include combinatorial counting (n choose 2), quadratic sequences, geometric transformations, symmetry, and the relationship between algebraic formulas and visual patterns. ③The GeoGebra construction uses commands Intersect, Point, Circle, and Sequence to generate points A and B, create chords via intersection, store coordinates in lists list1 and list2, control the number of points with a Number slider n, and display the handshake count in text1. ④Designed for high‑school students studying combinatorics, sequences, and geometry, and useful for teachers seeking a dynamic illustration of algebraic and geometric reasoning.