Constructing Golden Rectangles and Golden Spirals

This GeoGebra example demonstrates how to construct a golden spiral from golden rectangles, highlighting geometric properties and different construction approaches (with and without GeoGebra assistance). It centers on the golden ratio via the defining split of a rectangle into a smaller similar golden rectangle, then uses successive circular arcs to form the spiral. Key concepts include similarity, parallel and perpendicular constraints, midpoint/intersection-based construction, and the way circle arcs stitch together to produce the characteristic spiral shape. The build proceeds by defining control points, using distance and midpoint constraints to locate the golden subdivision, then finding intersection points and constructing orthogonal lines to determine circle centers and arc endpoints. Finally, polygons are used to visualize golden rectangles (e.g., multiple stages) while circles and CircleArc objects draw the spiral consistently. GeoGebra commands used include Point, PointIn, Distance, Midpoint, Polygon, Circle, Ray, Intersect, OrthogonalLine, and CircleArc, with objects such as goldPoly, poly1, and poly2 organizing the rectangle/spiral components. Suitable for secondary math enrichment through early university geometry and dynamic geometry exploration.