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Golden Rectangle and Spiral

Golden Rectangle and Spiral
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2016-07-12 11:22
CirclePolygonVectorCoordinate GeometryGeometric Construction
This example demonstrates the construction and properties of a golden rectangle and its spiral. Key math concepts include the golden ratio, geometric construction, and coordinate geometry. Construction highlights involve commands like Polygon, Line, OrthogonalLine, Intersect, Circle, Vector, Sequence, CircleArc, Point, and Length in GeoGebra. Suitable for high school and above students to understand the application and beauty of the golden ratio in geometry.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. Line(F, G) → j
  3. OrthogonalLine(H, j) → k
  4. OrthogonalLine(E, j) → l
  5. Intersect(j, k) → I
  6. Intersect(j, l) → J
  7. Circle(J, I) → c
  8. Intersect(c, l, 2) → K
  9. Line(K, j) → m
  10. Intersect(k, m) → L
  11. Vector(A, J) → u_1
  12. Sequence(Rotate[Dilate[Translate[poly1, u_1], b^(i)], (i * π / 2)], i, 0, 10, 1) → list1
  13. CircleArc(A, B, D) → d
  14. Sequence(Rotate[Dilate[Translate[d, u_1], b^(i)], (i * π / 2)], i, 0, 10, 1) → list2
  15. Point(c) → K_1
  16. Point(c) → K_2
  17. CircleArc(J, K_1, K_2) → e
  18. Length(Segment[I, J]) → n
  19. Length(Segment[G, F]) → o
  20. Length(Segment[E, F]) → p
  21. Length(Segment[E, F]) → r
  22. Length(Segment[G, H]) → s
  23. Vector(A, I) → u_2
  24. Vector(A, L) → u_3
  25. Vector(A, K) → u_4
  26. Sequence(Rotate[Dilate[Translate[poly1, u_3], b^(i)], (i * π / 2)], i, 0, 10, 1) → list3
  27. Sequence(Rotate[Dilate[Translate[poly1, u_4], b^(i)], (i * π / 2)], i, 0, 10, 1) → list4
  28. Sequence(Rotate[Dilate[Translate[poly1, u_2], b^(i)], (i * π / 2)], i, 0, 10, 1) → list5
  29. CircleArc(B, C, A) → q
  30. CircleArc(C, D, B) → t
  31. CircleArc(D, A, C) → d_1
  32. Sequence(Rotate[Dilate[Translate[q, u_2], b^(i)], (i * π / 2)], i, 0, 10, 1) → list6
  33. Sequence(Rotate[Dilate[Translate[d_1, u_4], b^(i)], (i * π / 2)], i, 0, 10, 1) → list7
  34. Sequence(Rotate[Dilate[Translate[t, u_3], b^(i)], (i * π / 2)], i, 0, 10, 1) → list8