Visualizing Complex Roots: Circles, Vectors, and Fractal Geometry

① This GeoGebra example dynamically visualizes complex roots on the complex plane, turning algebraic solutions into intuitive geometric objects such as circles, vectors, and points, while incorporating fractal geometry aesthetics. ② Core mathematical concepts include complex number operations (multiplication, root extraction), roots on the unit circle, vector representation of complex numbers, and initial ideas for generating fractal patterns through sequences or iterations. ③ Construction highlights: uses Intersect to find intersections, Sequence to generate root sequences, Circle to construct unit or root circles, Vector to draw complex vectors, and Point to mark root points; objects include n (number of roots), O (origin), Z (complex point), i (imaginary unit), Roots (root set), ZCircle (root circle), ZVector (root vector), WCircle (multiplication result circle), W (multiplication result point), WVectors (multiplication vectors), RootsLabels (root labels), RootsVectors (root vectors), ShowRoots (toggle for roots), ShowMultiplications (toggle for multiplications), W_1, W_2 (specific multiplication results), and WCircles (set of multiplication circles). ④ Suitable for high school to undergraduate complex number instruction, as well as mathematical art or fractal geometry exploration, helping students understand root symmetry, geometric meaning of multiplication, and iterative generation of complex visual patterns.