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Golden Spiral and Golden Rectangles

Golden Spiral and Golden Rectangles
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2011-10-25 20:20
TrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the geometric construction of the golden spiral and golden rectangles. A golden rectangle has a width-to-length ratio of the golden section, while the golden spiral is formed by expanding outward from a series of golden rectangles. The core mathematical concepts involved include the golden ratio, golden rectangles, and golden spirals. Using GeoGebra commands such as Point, Rotate, Line, and Intersect, the construction process of the golden rectangles and golden spirals can be dynamically displayed. Suitable for students at the junior high school level and above to learn about geometric construction and the golden ratio.

GeoGebra Commands & Parameters

  1. Point(xAxis) → P
  2. Rotate(P, -anggrad, Q) → P'
  3. Line(Q, P') → a
  4. OrthogonalLine(P, a) → b
  5. Intersect(a, b) → A
  6. Intersect(OrthogonalLine[Q, a], Line[A, (0, 0)]) → B
  7. Angle(P', Q, P) → α
  8. Sequence(Polygon[Element[L_A, i], Element[L_E, i], Element[L_E, i + 1], Element[L_E, i + 2]], i, 1, 7) → rectangulos
  9. CurveCartesian(x(A7) * aureo^(2 * t / 3.141592653589793) * cos(t), x(A7) * aureo^(2 * t / 3.141592653589793) * sin(t), t, 0, 9.42478) → d
  10. Element(rectangulos, 5) → ultim
  11. Segment(E1, E3) → c
  12. Segment(A2, A4) → f
  13. Segment(E4, E2) → e
  14. CircleArc(E4, A2, A1) → g
  15. CircleArc(E5, P, A2) → h
  16. CircleArc(E6, A4, P) → k
  17. CircleArc(E7, A5, A4) → p
  18. CircleArc(E8, A6, A5) → q
  19. Element(rectangulos, visible) → rectangulovisible
  20. Intersect(c, e) → O

AI Drawing Prompt Reference

Draw a figure containing golden rectangles and golden spirals. Golden rectangles have a width-to-length ratio of the golden ratio, and golden spirals are formed by expanding outward from a series of golden rectangles. The construction order is to first draw several golden rectangles, then generate the golden spiral from them. The figure should clearly show these geometric relationships, suitable for geometry learning and demonstration.
The prompt above was compiled from the original drawing commands and is for reference only.