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The Golden Rectangle

The Golden Rectangle
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2015-02-03 22:42
FunctionCoordinate GeometryGeometric Construction
This example demonstrates the construction of a golden rectangle, a special type of rectangle with a length-to-width ratio of the golden ratio. Key mathematical concepts include proportion and similarity. Using GeoGebra commands such as Polygon, Midpoint, Circle, etc., a dynamic construction of the golden rectangle is achieved. Suitable for middle school and above students to learn about the concept of proportion and the golden section.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Midpoint(A, B) → E
  3. Circle(E, C) → e
  4. Ray(A, B) → f
  5. Intersect(e, f, 1) → F
  6. Ray(D, C) → g
  7. OrthogonalLine(F, f) → h
  8. Intersect(h, g) → G
  9. Segment(C, G) → i
  10. Segment(G, F) → j
  11. Segment(F, B) → k
  12. Polygon(G, C, 4) → poly2, l, m, n, p, H, I
  13. Polygon(F, I, 4) → poly3, q, r, s, t, J, K
  14. Polygon(B, K, 4) → poly4, a_1, b_1, c_1, d_1, L, M
  15. Polygon(H, M, 4) → poly5, e_1, f_1, g_1, h_1, N, O
  16. Polygon(J, O, 4) → poly6, i_1, j_1, k_1, l_1, P, Q
  17. Polygon(L, Q, 4) → poly7, m_1, n_1, p_1, q_1, R, S
  18. Polygon(N, S, 4) → poly8, r_1, s_1, t_1, a_2, T, U
  19. Polygon(P, U, 4) → poly9, b_2, c_2, d_2, e_2, V, W
  20. Polygon(R, W, 4) → poly10, f_2, g_2, h_2, i_2, Z, A_1
  21. Polygon(T, A_1, 4) → poly11, j_2, k_2, l_2, m_2, B_1, C_1
  22. Polygon(V, C_1, 4) → poly12, n_2, p_2, q_2, r_2, D_1, E_1
  23. Polygon(Z, E_1, 4) → poly13, s_2, t_2, a_3, b_3, F_1, G_1
  24. Polygon(B_1, G_1, 4) → poly14, c_3, d_3, e_3, f_3, H_1, I_1
  25. Polygon(D_1, I_1, 4) → poly15, g_3, h_3, i_3, j_3, J_1, K_1
  26. Polygon(F_1, K_1, 4) → poly16, k_3, l_3, m_3, n_3, L_1, M_1
  27. Polygon(H_1, M_1, 4) → poly17, p_3, q_3, r_3, s_3, N_1, O_1
  28. CircleArc(N_1, O_1, M_1) → t_3
  29. CircleArc(L_1, M_1, K_1) → c_4
  30. CircleArc(J_1, K_1, I_1) → d_4
  31. CircleArc(H_1, I_1, G_1) → e_4
  32. CircleArc(F_1, G_1, E_1) → f_4
  33. CircleArc(D_1, E_1, C_1) → g_4
  34. CircleArc(B_1, C_1, A_1) → h_4
  35. CircleArc(Z, A_1, W) → k_4
  36. CircleArc(V, W, U) → p_4
  37. CircleArc(T, U, S) → q_4
  38. CircleArc(R, S, Q) → r_4
  39. CircleArc(P, Q, O) → s_4
  40. CircleArc(N, O, M) → t_4
  41. CircleArc(L, M, K) → c_5
  42. CircleArc(J, K, I) → d_5
  43. CircleArc(H, I, C) → e_5
  44. CircleArc(B, C, A) → f_5

AI Drawing Prompt Reference

Draw a golden rectangle by first creating a rectangle ABCD, then finding the midpoint E of AB, drawing a circle with E as the center and C as the radius, intersecting the extension of AB at F, then drawing a perpendicular line to FC intersecting the extension of DC at G, connecting CG, GF, and FB to form a new rectangle CGFB. Repeat this process, each time drawing a circle with the diagonal of the new rectangle as the diameter, intersecting the extensions of adjacent sides at new points, and connecting these points to form new rectangles until the shapes become sufficiently small.
The prompt above was compiled from the original drawing commands and is for reference only.