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Golden Rectangle: The Beauty of Geometry and Proportion

Golden Rectangle: The Beauty of Geometry and Proportion
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2020-10-09 03:52
TrigonometryCoordinate GeometryTransformationGeometric ConstructionSlider Animation
The golden rectangle is a key concept in geometry, illustrating the harmony between proportion and shape. This example demonstrates how to construct a golden rectangle using GeoGebra, involving concepts such as rectangles, trigonometric functions, natural numbers, volume, dilation, and surface. The construction requires using commands like Segment, Midpoint, CircleArc, Circle, Point, Intersect, OrthogonalLine, Angle, and objects such as A, B, kép1, C, D, f, E, F, G, c, d, H, I, e, J, K, g, h, k, L, p, i, α, j, q, M, r, l, N, O, s, P, m, Q, n, t, a, b, c_1, R. It is suitable for middle school and above students to learn about geometric proportion and mathematical aesthetics.

GeoGebra Commands & Parameters

  1. Segment(C, D) → f
  2. Midpoint(f) → E
  3. CircleArc(C, F, G) → c
  4. Circle(D, Radius[c]) → d
  5. Point(d) → H
  6. Point(d) → I
  7. CircleArc(D, H, I) → e
  8. Intersect(c, e, 2) → J
  9. Intersect(c, e, 1) → K
  10. Segment(J, K) → g
  11. OrthogonalLine(D, f) → h
  12. Circle(D, E) → k
  13. Intersect(k, h, 2) → L
  14. CircleArc(D, L, E) → p
  15. Segment(D, L) → i
  16. Angle(L, D, E) → α
  17. Segment(C, L) → j
  18. Circle(L, D) → q
  19. Point(q) → M
  20. CircleArc(L, M, D) → r
  21. OrthogonalLine(C, f) → l
  22. Intersect(r, j, 1) → N
  23. CircleArc(C, N, O) → s
  24. Intersect(s, l, 1) → P
  25. OrthogonalLine(P, l) → m
  26. Intersect(h, m) → Q
  27. Segment(P, Q) → n
  28. Segment(Q, D) → t
  29. Segment(D, C) → a
  30. Segment(C, P) → b
  31. Circle(C, P) → c_1
  32. Intersect(c_1, f, 1) → R
  33. CircleArc(C, R, N) → d_1
  34. OrthogonalLine(R, f) → f_1
  35. Intersect(f_1, n) → S
  36. Segment(R, S) → g_1
  37. CircleArc(S, P, R) → e_1
  38. Circle(D, R) → h_1
  39. Intersect(h_1, i, 1) → T
  40. CircleArc(D, T, R) → k_1
  41. OrthogonalLine(T, i) → i_1
  42. Intersect(i_1, g_1) → U
  43. Segment(U, T) → j_1
  44. CircleArc(U, R, T) → p_1
  45. Circle(Q, T) → q_1
  46. Intersect(q_1, n, 1) → V
  47. OrthogonalLine(V, n) → l_1
  48. Intersect(l_1, j) → W
  49. Segment(V, W) → m_1
  50. CircleArc(Q, V, T) → r_1
  51. CircleArc(W, T, V) → s_1
  52. Circle(V, S) → t_1
  53. Intersect(t_1, m_1, 1) → Z
  54. Circle(S, V) → k_2
  55. Intersect(k_2, g_1, 1) → D_1
  56. CircleArc(S, D_1, V) → p_2
  57. CircleArc(V, S, Z) → c_2
  58. OrthogonalLine(Z, m_1) → n_1
  59. Intersect(n_1, g_1) → A_1
  60. Segment(A_1, Z) → a_1
  61. CircleArc(Z, V, A_1) → d_2
  62. Circle(U, A_1) → e_2
  63. Intersect(e_2, j_1, 1) → B_1
  64. CircleArc(U, B_1, A_1) → f_2
  65. OrthogonalLine(B_1, j_1) → b_1
  66. Intersect(b_1, a_1) → C_1
  67. Segment(B_1, C_1) → g_2
  68. CircleArc(C_1, A_1, B_1) → h_2

AI Drawing Prompt Reference

Draw a golden rectangle to illustrate the harmony between proportion and shape. Start by drawing a line segment and finding its midpoint. Draw circles with the segment's endpoints as centers, intersecting at points to create arcs. Connect the intersection points with the segment. Next, draw a circle with the midpoint as the center, intersecting at points to create arcs. Connect the intersection points with the segment again. Then, draw a perpendicular line to the segment, intersecting the circles at two points, and connect these points. Finally, draw a circle with the midpoint as the center, intersecting the perpendicular line at two points, and connect these points to complete the golden rectangle.
The prompt above was compiled from the original drawing commands and is for reference only.