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Three Golden Rectangles Surprise

Three Golden Rectangles Surprise
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2017-09-03 11:45
TransformationGeometric Construction
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources
This dynamic and modifiable illustration demonstrates the fascinating properties of three golden rectangles. A golden rectangle has a length-to-width ratio of the golden ratio. The example uses geometric construction to show the properties of golden rectangles, including constructing rectangles, using the Dilate command for similar transformations, and verifying the golden ratio through rotation and distance calculations. Suitable for middle school and above students learning about geometric transformations and ratios.

GeoGebra Commands & Parameters

  1. Line(F, H) → f
  2. Line(F, G) → h
  3. Line(H, E) → i
  4. Circle(F, H, xOyPlane) → c
  5. Intersect(h, c) → M, N
  6. OrthogonalLine(M, h, xOyPlane) → j
  7. Intersect(i, j) → O
  8. Midpoint(H, O) → P
  9. Distance(P, M) → distancePM
  10. Distance(M, E) → distanceME
  11. Dilate(F, If[0 < b ≤ 1, 1 - b, 0], H) → F'
  12. Segment(F, F') → k
  13. Rotate(F, If[1 < b ≤ 2, ((-(b - 1)) * π / 2), ((-π)) / 2], H) → F'_1
  14. Rotate(F', If[1 < b ≤ 2, ((-(b - 1)) * π / 2), ((-π)) / 2], H) → F''
  15. Segment(F'_1, F'') → k'
  16. Rotate(F'_1, If[2 < b ≤ 3, ((-(b - 2)) * π / 2), ((-π)) / 2], O) → Q
  17. Rotate(F'', If[2 < b ≤ 3, ((-(b - 2)) * π / 2), ((-π)) / 2], O) → F'''
  18. Segment(Q, F''') → k''
  19. Rotate(Q, If[3 < b ≤ 4, ((-(b - 3)) * π / 2), ((-π)) / 2], M) → Q'
  20. Rotate(F''', If[3 < b ≤ 4, ((-(b - 3)) * π / 2), ((-π)) / 2], M) → R
  21. Segment(Q', R) → k'''
  22. Angle(M, O, H, xOyPlane) → α
  23. Angle(O, H, F, xOyPlane) → β
  24. Angle(F, M, O, xOyPlane) → γ
  25. Angle(H, F, M, xOyPlane) → δ
  26. Polygon(F, H, O, M) → poly1, f_1, h_1, o, m
  27. Dilate(H, If[4 < b ≤ 5, 5 - b, 0], P) → H'
  28. Segment(H, H') → l
  29. Rotate(H, If[5 < b ≤ 6, ((-(b - 5)) * π), (-π)], P) → H'_1
  30. Rotate(H', If[5 < b ≤ 6, ((-(b - 5)) * π), (-π)], P) → H''
  31. Segment(H'_1, H'') → l'
  32. Dilate(P, If[6 < b ≤ 7, 7 - b, 0], M) → P'
  33. Segment(P, P') → n
  34. Angle(E, P, M, xOyPlane) → ε
  35. Rotate(P, If[7 < b ≤ 8, ((-(b - 7)) * ε), (-ε)], P) → P'_1
  36. Rotate(P', If[7 < b ≤ 8, ((-(b - 7)) * ε), (-ε)], P) → P''
  37. Segment(P'_1, P'') → n'
  38. CircleArc(P, P'', M, xOyPlane) → d
  39. Dilate(E, If[8 < b ≤ 9, 9 - b, 0], G) → E'
  40. Dilate(M, If[8 < b ≤ 9, 9 - b, 0], G) → M'
  41. Segment(E, E') → p
  42. Segment(M, M') → q
  43. Angle(F, G, E, xOyPlane) → ζ
  44. Polygon(M, O, E, G) → poly2, m_1, o_1, e, g_1
  45. Dilate(G, If[9 < b ≤ 10, 10 - b, 0], A) → G'
  46. Vector(G, G') → u
  47. Translate(E, u) → E'_1
  48. Translate(E', u) → E''
  49. Segment(E'_1, E'') → p'
  50. Plane(C, A, E'_1) → r
  51. Rotate(E'_1, If[10 < b ≤ 11, ((b - 10) * π / 2), π / 2], A, r) → S
  52. Rotate(E'', If[10 < b ≤ 11, ((b - 10) * π / 2), π / 2], A, r) → E'''
  53. Segment(S, E''') → p''
  54. Segment(F, G) → s
  55. Dilate(F, If[11 < b ≤ 12, 12 - b, 0], C) → F'_2
  56. Vector(F, F'_2) → v
  57. Translate(F, v) → F'_3
  58. Translate(G, v) → G'_1
  59. Segment(F'_3, G'_1) → s'
  60. Plane(T, F'_3, G'_1) → t
  61. Rotate(F'_3, If[11 < b ≤ 12, ((b - 11) * π / 2), π / 2], F'_3, t) → U
  62. Rotate(G'_1, If[11 < b ≤ 12, ((b - 11) * π / 2), π / 2], F'_3, t) → V
  63. Segment(U, V) → s''
  64. Angle(B, C, A) → η
  65. Dilate(C, If[12 < b ≤ 13, 13 - b, 0], A) → C'
  66. Vector(C, C') → w
  67. Translate(U, w) → U'
  68. Translate(V, w) → V'
  69. Segment(U', V') → s'''
  70. Angle(C, A, D) → θ
  71. Dilate(B, If[13 < b ≤ 14, 14 - b, 0], D) → B'
  72. Segment(B, B') → i_1
  73. Polygon(B, D, A, C) → poly3, b_1, d_1, a_1, c_1
  74. Dilate(A, If[14 < b ≤ 15, 15 - b, 0], I) → A'
  75. Vector(A, A') → u_1
  76. Translate(S, u_1) → S'
  77. Translate(E''', u_1) → W
  78. Segment(S', W) → p'''
  79. Plane(Z, I, J) → e_1
  80. Rotate(S', If[15 < b ≤ 16, ((b - 15) * π / 2), π / 2], I, e_1) → S''
  81. Rotate(W, If[15 < b ≤ 16, ((b - 15) * π / 2), π / 2], I, e_1) → W'
  82. Segment(S'', W') → j_1
  83. Dilate(B, If[16 < b ≤ 17, 17 - b, 0], J) → B'_1
  84. Vector(B, B'_1) → v_1
  85. Translate(U, v_1) → U'_1
  86. Translate(V, v_1) → V'_1
  87. Segment(U'_1, V'_1) → s'''_1
  88. Plane(A_1, B'_1, U'_1) → k_1
  89. Rotate(U'_1, If[15 < b ≤ 17, ((b - 16) * π / 2), π / 2], V'_1, k_1) → B_1
  90. Rotate(V'_1, If[15 < b ≤ 17, ((b - 16) * π / 2), π / 2], V'_1, k_1) → C_1
  91. Segment(B_1, C_1) → l_1
  92. Angle(I, J, L) → ι
  93. Dilate(L, If[17 < b ≤ 18, 18 - b, 0], K) → L'
  94. Vector(L, L') → w_1
  95. Translate(B_1, w_1) → D_1
  96. Translate(C_1, w_1) → E_1
  97. Segment(D_1, E_1) → n_1
  98. Angle(K, I, J) → κ
  99. Segment(L, D_1) → p_1
  100. Polygon(L, J, I, K) → poly4, l_2, j_2, i_2, k_2
  101. Dilate(F, If[20 < b ≤ 21, 21 - b, 0], H) → F'_4
  102. Segment(F, F'_4) → q_1
  103. Plane(F, C, H) → r_1
  104. Rotate(F, If[21 < b ≤ 22, ((-(b - 21)) * π / 3), ((-π)) / 3], F, r_1) → F'_5
  105. Rotate(F'_4, If[21 < b ≤ 22, ((-(b - 21)) * π / 3), ((-π)) / 3], F, r_1) → F_1
  106. Segment(F'_5, F_1) → s_1
  107. Plane(K, C, F) → t_1
  108. Rotate(F'_5, If[22 < b ≤ 23, ((b - 22) * π / 3), π / 3], C, t_1) → G_1
  109. Rotate(F_1, If[22 < b ≤ 23, ((b - 22) * π / 3), π / 3], C, t_1) → H_1
  110. Segment(G_1, H_1) → f_2
  111. Plane(C, K, A) → a_2
  112. Rotate(G_1, If[23 < b ≤ 24, ((-(b - 23)) * π / 3), ((-π)) / 3], K, a_2) → I_1
  113. Rotate(H_1, If[23 < b ≤ 24, ((-(b - 23)) * π / 3), ((-π)) / 3], K, a_2) → J_1
  114. Segment(I_1, J_1) → g_2
  115. Plane(A, K, G) → b_2
  116. Rotate(I_1, If[24 < b ≤ 25, ((b - 24) * π / 3), π / 3], A, b_2) → K_1
  117. Rotate(J_1, If[24 < b ≤ 25, ((b - 24) * π / 3), π / 3], A, b_2) → L_1
  118. Segment(K_1, L_1) → h_2
  119. Plane(A, G, E) → c_2
  120. Rotate(K_1, If[25 < b ≤ 26, ((-(b - 25)) * π / 3), ((-π)) / 3], G, c_2) → M_1
  121. Rotate(L_1, If[25 < b ≤ 26, ((-(b - 25)) * π / 3), ((-π)) / 3], G, c_2) → N_1
  122. Segment(M_1, N_1) → m_2
  123. Plane(G, E, D) → d_2
  124. Rotate(M_1, If[26 < b ≤ 27, ((-(b - 26)) * π / 3), ((-π)) / 3], E, d_2) → O_1
  125. Rotate(N_1, If[26 < b ≤ 27, ((-(b - 26)) * π / 3), ((-π)) / 3], E, d_2) → P_1
  126. Segment(O_1, P_1) → n_2
  127. Plane(F, H, B) → e_2
  128. Rotate(F, If[26 < b ≤ 27, ((-(b - 26)) * π / 3), ((-π)) / 3], H, e_2) → F'_6
  129. Rotate(F'_4, If[26 < b ≤ 27, ((-(b - 26)) * π / 3), ((-π)) / 3], H, e_2) → Q_1
  130. Segment(F'_6, Q_1) → p_2
  131. Plane(E, D, J) → o_2
  132. Rotate(O_1, If[27 < b ≤ 28, ((-(b - 27)) * π / 3), ((-π)) / 3], D, o_2) → R_1
  133. Rotate(P_1, If[27 < b ≤ 28, ((-(b - 27)) * π / 3), ((-π)) / 3], D, o_2) → S_1
  134. Segment(R_1, S_1) → q_2
  135. Point(s_1) → T_1
  136. Plane(H, B, J) → r_2
  137. Rotate(F'_6, If[27 < b ≤ 28, ((-(b - 27)) * π / 3), ((-π)) / 3], B, r_2) → U_1
  138. Rotate(Q_1, If[27 < b ≤ 28, ((-(b - 27)) * π / 3), ((-π)) / 3], B, r_2) → V_1
  139. Segment(U_1, V_1) → s_2
  140. Rotate(U_1, If[28 < b ≤ 29, ((-(b - 28)) * π / 3), ((-π)) / 3], J, r_2) → W_1
  141. Rotate(V_1, If[28 < b ≤ 29, ((-(b - 28)) * π / 3), ((-π)) / 3], J, r_2) → Z_1
  142. Segment(W_1, Z_1) → t_2
  143. Rotate(M_1, If[28 < b ≤ 29, ((b - 28) * π / 3), π / 3], G, d_2) → A_2
  144. Rotate(N_1, If[28 < b ≤ 29, ((b - 28) * π / 3), π / 3], G, d_2) → B_2
  145. Segment(A_2, B_2) → f_3
  146. Intersect(q_1, t_2) → C_2
  147. Plane(H, C, A) → u_2
  148. Plane(F, B, D) → v_2
  149. Rotate(W_1, If[29 < b ≤ 30, (((-(b - 29)) * 108) * π / 180), (-108 * π / 180)], H, u_2) → D_2
  150. Rotate(Z_1, If[29 < b ≤ 30, (((-(b - 29)) * 108) * π / 180), (-108 * π / 180)], H, u_2) → E_2
  151. Segment(D_2, E_2) → g_3
  152. Rotate(D_2, If[30 < b ≤ 31, (((-(b - 30)) * 108) * π / 180), (-108 * π / 180)], C, u_2) → F_2
  153. Rotate(E_2, If[30 < b ≤ 31, (((-(b - 30)) * 108) * π / 180), (-108 * π / 180)], C, u_2) → G_2
  154. Segment(F_2, G_2) → h_3
  155. Rotate(F_2, If[31 < b ≤ 32, (((-(b - 31)) * 108) * π / 180), (-108 * π / 180)], A, u_2) → H_2
  156. Rotate(G_2, If[31 < b ≤ 32, (((-(b - 31)) * 108) * π / 180), (-108 * π / 180)], A, u_2) → I_2
  157. Segment(H_2, I_2) → i_3
  158. Rotate(H_2, If[32 < b ≤ 33, (((-(b - 32)) * 108) * π / 180), (-108 * π / 180)], E, u_2) → J_2
  159. Rotate(I_2, If[32 < b ≤ 33, (((-(b - 32)) * 108) * π / 180), (-108 * π / 180)], E, u_2) → K_2
  160. Segment(J_2, K_2) → j_3
  161. Rotate(A_2, If[29 < b ≤ 30, (((b - 29) * 108) * π / 180), (108 * π / 180)], D, v_2) → L_2
  162. Rotate(B_2, If[30 < b ≤ 31, (((b - 30) * 108) * π / 180), (108 * π / 180)], D, v_2) → M_2
  163. Segment(L_2, M_2) → k_3
  164. Rotate(L_2, If[31 < b ≤ 32, (((b - 31) * 108) * π / 180), (108 * π / 180)], B, v_2) → N_2
  165. Rotate(M_2, If[30 < b ≤ 31, (((b - 30) * 108) * π / 180), (108 * π / 180)], B, v_2) → O_2
  166. Segment(N_2, O_2) → l_3
  167. Rotate(N_2, If[31 < b ≤ 32, (((b - 31) * 108) * π / 180), (108 * π / 180)], F, v_2) → P_2
  168. Rotate(O_2, If[32 < b ≤ 33, (((b - 32) * 108) * π / 180), (108 * π / 180)], F, v_2) → Q_2
  169. Segment(P_2, Q_2) → m_3
  170. Rotate(P_2, If[33 < b ≤ 34, (((b - 33) * 108) * π / 180), (108 * π / 180)], K, v_2) → R_2
  171. Rotate(Q_2, If[32 < b ≤ 33, (((b - 32) * 108) * π / 180), (108 * π / 180)], K, v_2) → S_2
  172. Segment(R_2, S_2) → n_3
  173. Plane(E, I, J) → w_2
  174. Rotate(J_2, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], J, w_2) → T_2
  175. Rotate(K_2, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], J, w_2) → U_2
  176. Segment(T_2, U_2) → p_3
  177. Rotate(H_2, If[33 < b ≤ 34, ((-(b - 33)) * π / 3), ((-π)) / 3], E, Plane[A, E, I]) → V_2
  178. Rotate(I_2, If[33 < b ≤ 34, ((-(b - 33)) * π / 3), ((-π)) / 3], E, Plane[A, E, I]) → W_2
  179. Segment(V_2, W_2) → q_3
  180. Rotate(F_2, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], A, Plane[A, C, I]) → Z_2
  181. Rotate(G_2, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], A, Plane[A, C, I]) → A_3
  182. Segment(Z_2, A_3) → r_3
  183. Rotate(D_2, If[33 < b ≤ 34, ((-(b - 33)) * π / 3), ((-π)) / 3], C, Plane[H, C, I]) → B_3
  184. Rotate(E_2, If[33 < b ≤ 34, ((-(b - 33)) * π / 3), ((-π)) / 3], C, Plane[H, C, I]) → C_3
  185. Segment(B_3, C_3) → s_3
  186. Rotate(W_1, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], H, Plane[H, J, I]) → D_3
  187. Rotate(Z_1, If[33 < b ≤ 34, ((b - 33) * π / 3), π / 3], H, Plane[H, J, I]) → E_3
  188. Segment(D_3, E_3) → t_3
  189. Rotate(P_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], F, Plane[K, F, L]) → F_3
  190. Rotate(Q_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], F, Plane[K, F, L]) → G_3
  191. Segment(F_3, G_3) → a_3
  192. Rotate(R_2, If[34 < b ≤ 35, ((b - 34) * π / 3), π / 3], K, Plane[K, G, L]) → H_3
  193. Rotate(S_2, If[34 < b ≤ 35, ((b - 34) * π / 3), π / 3], K, Plane[K, G, L]) → I_3
  194. Segment(H_3, I_3) → b_3
  195. Rotate(A_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], G, Plane[D, G, L]) → J_3
  196. Rotate(B_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], G, Plane[D, G, L]) → K_3
  197. Segment(J_3, K_3) → c_3
  198. Rotate(L_2, If[34 < b ≤ 35, ((b - 34) * π / 3), π / 3], D, Plane[D, B, L]) → L_3
  199. Rotate(M_2, If[34 < b ≤ 35, ((b - 34) * π / 3), π / 3], D, Plane[D, B, L]) → M_3
  200. Segment(L_3, M_3) → d_3
  201. Rotate(N_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], B, Plane[F, B, L]) → N_3
  202. Rotate(O_2, If[34 < b ≤ 35, ((-(b - 34)) * π / 3), ((-π)) / 3], B, Plane[F, B, L]) → O_3
  203. Segment(N_3, O_3) → e_3
  204. Polygon(C, I, A) → poly5, a_4, c_4, i_4
  205. Polygon(I, E, A) → poly6, a_5, i_5, e_4
  206. Polygon(E, G, A) → poly7, a_6, e_5, g_4
  207. Polygon(G, K, A) → poly8, a_7, g_5, k_4
  208. Polygon(K, C, A) → poly9, a_8, k_5, c_5
  209. Polygon(K, F, C) → poly10, c_6, k_6, f_4
  210. Polygon(F, H, C) → poly11, c_7, f_5, h_4
  211. Polygon(H, I, C) → poly12, c_8, h_5, i_6
  212. Polygon(H, J, I) → poly13, i_7, h_6, j_4
  213. Polygon(J, E, I) → poly14, i_8, j_5, e_6
  214. Polygon(H, B, J) → poly15, j_6, h_7, b_4
  215. Polygon(H, B, F) → poly16, f_6, h_8, b_5
  216. Polygon(B, F, L) → poly17, l_4, b_6, f_7
  217. Polygon(F, K, L) → poly18, l_5, f_8, k_7
  218. Polygon(L, K, G) → poly19, g_6, l_6, k_8
  219. Polygon(L, G, D) → poly20, d_4, l_7, g_7
  220. Polygon(B, L, D) → poly21, d_5, b_7, l_8
  221. Polygon(D, E, J) → poly22, j_7, d_6, e_7
  222. Polygon(G, D, E) → poly23, e_8, g_8, d_7
  223. Polygon(B, D, J) → poly24, j_8, b_8, d_8
  224. Line((0, 0, 0), (1, 0, 0)) → m_4
  225. Line((0, 0, 0), (0, 1, 0)) → n_4
  226. Line((0, 0, 0), (0, 0, 1)) → p_4