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3D Optical Illusion Geometric Construction in GeoGebra

3D Optical Illusion Geometric Construction in GeoGebra
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2020-08-21 00:32
CircleSolid GeometryTransformationGeometric Construction
This example demonstrates a 3D optical illusion built from fundamental geometric objects and transformations in GeoGebra. Core mathematical concepts include circle properties, axial symmetry via mirror reflection, orthogonality of points, lines, and planes, angle bisectors, circumcircle arcs, and spatial incidence relationships. The construction employs GeoGebra commands such as Circle, Mirror, Segment, OrthogonalLine, AngularBisector, Intersect, Point, Midpoint, OrthogonalPlane, and CircumcircleArc, together with objects including points A, B, O, P, circles c, lines f through m, and planes n, q, r. It is well suited for high school geometry instruction, classroom demonstrations, and independent study of solid geometry, visual perception, and transformation-based geometric drawing.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Mirror(B, A) → B'
  3. Segment(B', B) → f
  4. OrthogonalLine(A, f) → g
  5. AngularBisector(g, f) → h, i
  6. AngularBisector(g, i) → j, k
  7. AngularBisector(i, f) → l, m
  8. Intersect(c, g, 2) → C
  9. Intersect(c, l, 2) → D
  10. Intersect(c, h, 2) → E
  11. Intersect(c, k, 1) → F
  12. Intersect(c, m, 1) → G
  13. Intersect(c, i, 1) → H
  14. Intersect(c, j, 1) → I
  15. Intersect(c, g, 1) → J
  16. Intersect(c, l, 1) → K
  17. Intersect(c, h, 1) → L
  18. Intersect(c, k, 2) → M
  19. Intersect(c, i, 2) → O
  20. Intersect(c, j, 2) → P
  21. Intersect(c, m, 2) → N
  22. Segment(C, J) → n
  23. Segment(P, I) → q
  24. Segment(D, K) → p
  25. Segment(L, E) → r
  26. Segment(F, M) → s
  27. Segment(B', B) → t
  28. Segment(G, N) → a
  29. Segment(O, H) → b
  30. Point(c) → Q
  31. Midpoint(A, Q) → R
  32. Circle(R, Q) → d
  33. Intersect(d, p) → S, T
  34. Intersect(d, r) → U, V
  35. Intersect(d, s) → W, Z
  36. Intersect(d, t) → A_1, B_1
  37. Intersect(d, a) → C_1, D_1
  38. Intersect(d, b) → E_1, F_1
  39. Intersect(d, q) → G_1, H_1
  40. Intersect(d, n) → I_1, J_1
  41. AngularBisector(p, r) → e, f_1
  42. AngularBisector(r, s) → g_1, h_1
  43. AngularBisector(s, t) → i_1, j_1
  44. AngularBisector(t, a) → k_1, l_1
  45. Intersect(c, i_1) → K_1, L_1
  46. Intersect(c, k_1) → M_1, N_1
  47. Intersect(c, f_1) → O_1, P_1
  48. Intersect(c, h_1) → Q_1, R_1
  49. Intersect(c, j_1) → S_1, T_1
  50. Intersect(c, l_1) → U_1, V_1
  51. Intersect(c, e) → W_1, Z_1
  52. Intersect(c, g_1) → A_2, B_2
  53. Segment(N_1, M_1) → m_1
  54. Segment(P_1, O_1) → n_1
  55. Segment(Q_1, R_1) → p_1
  56. Segment(S_1, T_1) → q_1
  57. Segment(U_1, V_1) → r_1
  58. Segment(W_1, Z_1) → s_1
  59. Segment(A_2, B_2) → t_1
  60. Segment(K_1, L_1) → a_1
  61. Intersect(d, s_1) → C_2, D_2
  62. Intersect(d, r_1) → E_2, F_2
  63. Intersect(d, q_1) → G_2, H_2
  64. Intersect(d, p_1) → I_2, J_2
  65. Intersect(d, n_1) → K_2, L_2
  66. Intersect(d, m_1) → M_2, N_2
  67. Intersect(d, a_1) → O_2, P_2
  68. Intersect(d, t_1) → Q_2, R_2
  69. OrthogonalLine(A, xOyPlane) → b_1
  70. Point(b_1) → S_2
  71. OrthogonalPlane(S_2, b_1) → o
  72. OrthogonalLine(B, xOyPlane) → c_1
  73. Midpoint(S_2, A) → U_2
  74. Mirror(B_2, U_2) → V_2
  75. Mirror(P, U_2) → P'
  76. Mirror(K_1, U_2) → W_2
  77. Mirror(C, U_2) → C'
  78. Mirror(N_1, U_2) → Z_2
  79. Mirror(D, U_2) → D'
  80. Mirror(P_1, U_2) → A_3
  81. Mirror(E, U_2) → E'
  82. Mirror(Q_1, U_2) → B_3
  83. Mirror(F, U_2) → F'
  84. Mirror(T_1, U_2) → C_3
  85. Mirror(B, U_2) → B'_1
  86. Mirror(U_1, U_2) → D_3
  87. Mirror(G, U_2) → G'
  88. Mirror(Z_1, U_2) → E_3
  89. Mirror(H, U_2) → H'
  90. Mirror(A_2, U_2) → F_3
  91. Segment(B_2, V_2) → e_1
  92. Segment(P', P) → f_2
  93. Segment(K_1, W_2) → g_2
  94. Segment(C', C) → h_2
  95. Segment(N_1, Z_2) → i_2
  96. Segment(D', D) → j_2
  97. Segment(P_1, A_3) → k_2
  98. Segment(E', E) → l_2
  99. Segment(Q_1, B_3) → m_2
  100. Segment(F', F) → n_2
  101. Segment(T_1, C_3) → p_2
  102. Segment(B'_1, B) → q_2
  103. Segment(U_1, D_3) → r_2
  104. Segment(G', G) → s_2
  105. Segment(Z_1, E_3) → t_2
  106. Segment(H', H) → a_2
  107. Segment(A_2, F_3) → b_2
  108. OrthogonalLine(A_1, xOyPlane) → c_2
  109. OrthogonalLine(G_2, xOyPlane) → d_2
  110. OrthogonalLine(Z, xOyPlane) → e_2
  111. OrthogonalLine(J_2, xOyPlane) → f_3
  112. OrthogonalLine(U, xOyPlane) → g_3
  113. OrthogonalLine(L_2, xOyPlane) → h_3
  114. OrthogonalLine(D_1, xOyPlane) → i_3
  115. OrthogonalLine(H_1, xOyPlane) → j_3
  116. OrthogonalLine(P_2, xOyPlane) → k_3
  117. OrthogonalLine(J_1, xOyPlane) → l_3
  118. OrthogonalLine(N_2, xOyPlane) → m_3
  119. OrthogonalLine(T, xOyPlane) → n_3
  120. OrthogonalLine(F_2, xOyPlane) → p_3
  121. OrthogonalLine(C_2, xOyPlane) → q_3
  122. OrthogonalLine(E_1, xOyPlane) → r_3
  123. OrthogonalLine(R_2, xOyPlane) → s_3
  124. OrthogonalLine(O_2, xOyPlane) → t_3
  125. OrthogonalLine(I_1, xOyPlane) → a_3
  126. OrthogonalLine(G_1, xOyPlane) → b_3
  127. OrthogonalLine(M_2, xOyPlane) → c_3
  128. OrthogonalLine(S, xOyPlane) → d_3
  129. OrthogonalLine(K_2, xOyPlane) → e_3
  130. OrthogonalLine(V, xOyPlane) → f_4
  131. OrthogonalLine(I_2, xOyPlane) → g_4
  132. OrthogonalLine(W, xOyPlane) → h_4
  133. OrthogonalLine(B_1, xOyPlane) → i_4
  134. OrthogonalLine(E_2, xOyPlane) → j_4
  135. OrthogonalLine(C_1, xOyPlane) → k_4
  136. OrthogonalLine(D_2, xOyPlane) → l_4
  137. OrthogonalLine(F_1, xOyPlane) → m_4
  138. OrthogonalLine(Q_2, xOyPlane) → n_4
  139. OrthogonalLine(H_2, xOyPlane) → p_4
  140. Intersect(n_2, h_4) → G_3
  141. Intersect(n_2, e_2) → H_3
  142. Intersect(m_2, g_4) → I_3
  143. Intersect(f_3, m_2) → J_3
  144. Intersect(l_2, g_3) → K_3
  145. Intersect(l_2, f_4) → L_3
  146. Intersect(e_3, k_2) → M_3
  147. Intersect(k_2, h_3) → N_3
  148. Intersect(j_2, d_3) → O_3
  149. Intersect(j_2, n_3) → P_3
  150. Intersect(c_3, i_2) → Q_3
  151. Intersect(i_2, m_3) → R_3
  152. Intersect(l_3, h_2) → S_3
  153. Intersect(a_3, h_2) → T_3
  154. Intersect(g_2, t_3) → U_3
  155. Intersect(k_3, g_2) → V_3
  156. Intersect(f_2, b_3) → W_3
  157. Intersect(f_2, j_3) → Z_3
  158. Intersect(p_2, p_4) → A_4
  159. Intersect(p_2, d_2) → B_4
  160. Intersect(c_2, q_2) → C_4
  161. Intersect(q_2, i_4) → D_4
  162. Intersect(j_4, r_2) → E_4
  163. Intersect(r_2, p_3) → F_4
  164. Intersect(s_2, i_3) → G_4
  165. Intersect(s_2, k_4) → H_4
  166. Intersect(q_3, t_2) → I_4
  167. Intersect(l_4, t_2) → J_4
  168. Intersect(e_1, n_4) → K_4
  169. Intersect(e_1, s_3) → L_4
  170. Intersect(a_2, r_3) → M_4
  171. Intersect(a_2, m_4) → N_4
  172. CircumcircleArc(A_2, Q_1, B_2) → c_4
  173. CircumcircleArc(V_2, B_3, F_3) → d_4

AI Drawing Prompt Reference

Draw a 3D optical illusion geometric construction. Start with a circle and a point outside it. Reflect that point across the circle's center to get its mirror, then connect the two points with a line. Through the center, draw a line perpendicular to this segment. Then construct angle bisectors between this perpendicular and the segment, and between the perpendicular and the bisectors, creating multiple radiating lines. These lines intersect the circle at several points. Connect opposite points on the circle to form chords. Pick another point on the circle, take its midpoint with the center as a new center, and draw a second circle through that point. This second circle intersects each chord at two points. Finally, draw angle bisectors between adjacent chords; these bisectors meet the first circle at additional points, completing a symmetric, intricate pattern.
The prompt above was compiled from the original drawing commands and is for reference only.