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Jitterbug Origami Model Animation: Dynamic Transformation from Octahedron to Ico

Jitterbug Origami Model Animation: Dynamic Transformation from Octahedron to Ico
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2020-07-17 00:39
PolygonSolid GeometryTransformationGeometric ConstructionSlider Animation
In collections Action Modular Origami
This example demonstrates an animation of a paper-folding model called 'Jitterbug', based on the book 'Action Modular Origami'. It visualizes the dynamic transformation between polyhedra, particularly the transition from an octahedron to an icosahedron through folding and rotation. The core mathematical concepts include solid geometry, polyhedra (octahedron, icosahedron, cube), centroids, planes, and geometric transformations such as rotation, dilation, and reflection. The construction uses GeoGebra commands like Point, Intersect, Octahedron, Centroid, Line, Dilate, Rotate, Icosahedron, Plane, Cube, LineBisector, Mirror, Conic, Polygon, Midpoint, AngularBisector, Circle, Segment, Translate, Sequence, and OrthogonalLine, creating multiple faces (e.g., faceABC, faceACF) and edges (e.g., edgeAC, edgeBC) to form the animated structure. This example is suitable for high school or early college students to explore polyhedral geometry, spatial transformations, and dynamic visualization, and can serve as an engaging resource for origami mathematics and STEAM education.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Intersect(yAxis, zAxis) → OO
  3. Point(zAxis) → C
  4. Point(xAxis) → B
  5. Octahedron(C, B, A) → a, D, faceAEF, faceDEF, edgeAC, edgeBC, edgeAB, edgeCF, edgeAF, edgeCD, edgeDF, edgeBD, E, edgeBE, edgeDE, edgeAE, edgeEF, F, faceABC, faceACF, faceCDF, faceBCD, faceBDE, faceABE
  6. Centroid(faceABC) → G
  7. Line(OO, G) → f
  8. Dilate(A, (sqrt(5) - 1) / 2, B) → A'
  9. Rotate(A', (-(120°)), f) → A''_1
  10. Rotate(A', 120°, f) → A''
  11. Icosahedron(A''_1, A'', A') → b, H, face1, face2, face5, face4, face3, faceA''HL, faceA''IL, faceA'A''I, faceA'IM, faceA'JM, I, faceILO, faceIMO, faceMOP, faceJMP, faceLNO, faceNOP, faceKNP, faceJKP, faceHLN, faceHKN, J, edge1, edge2, edgeA'A'', edge4, edgeA'J, edge5, edgeJK, edge3, edgeHK, edgeA''H, K, edgeA''L, edgeHL, edgeA''I, edgeIL, edgeA'I, edgeA'M, edgeIM, edgeJM, edgeIO, edgeLO, L, edgeMO, edgeMP, edgeOP, edgeJP, edgeNO, edgeLN, edgeNP, edgeKP, edgeKN, edgeHN, M, N, O, P
  12. Point(Circle[Q, Distance[T, Q], Segment[T, Q]]) → U
  13. Plane(U, A, C) → d
  14. Cube(T, Q, U) → c, V, face8, face9, edgeTV, edgeQT, edgeQU, edgeUV, edgeTW, edge7, edge8, edgeWZ, W, edgeQZ, edge9, edge6, edge10, Z, A_1, B_1, faceQTVU, face7, faceQTWZ, face6
  15. LineBisector(A_1, R, d) → m
  16. Rotate(m, (-ang), OO, d) → m'
  17. Mirror(A''_1, m) → H_1
  18. Conic(S, A''_1, B_1, H_1, V) → e
  19. Intersect(m', e) → G_1, I_1
  20. Rotate(C_1, 120°, f) → D_1
  21. Rotate(C_1, (-(120°)), f) → E_1
  22. Polygon(C_1, E_1, D_1) → t3, d_1, c_1, e_1
  23. Midpoint(c_1) → L_1
  24. Plane(C_1, L_1, U) → o
  25. Intersect(f, t3) → F_1
  26. Midpoint(e_1) → K_1
  27. Midpoint(d_1) → J_1
  28. Polygon(C_1, K_1, F_1, J_1) → q1, c_2, k_1, f_1, j_1
  29. Line(I_1, E_1) → l_1
  30. Mirror(F_1, l_1) → S_1
  31. Mirror(K_1, l_1) → Q_1
  32. AngularBisector(S_1, I_1, Q_1) → n_1
  33. Line(Q_1, S_1) → q_1
  34. Intersect(n_1, q_1) → T_1
  35. Polygon(I_1, J_1, S_1, T_1) → q3, i_2, j_3, s_1, t_1
  36. Rotate(T_1, (-ang2), l_1) → Z_1
  37. Mirror(Z_1, o) → D_2
  38. Mirror(D_2, d) → G_2
  39. Plane(G_2, D_2, C_1) → w
  40. Mirror(Z_1, d) → B_2
  41. Plane(Z_1, C_1, B_2) → u
  42. Rotate(q3, (-ang2), l_1) → q3'
  43. Mirror(q3', o) → q3''
  44. Rotate(Q_1, (-ang2), l_1) → V_1
  45. Mirror(V_1, w) → J_2
  46. Plane(V_1, J_2, D_2) → a_1
  47. Circle(D_2, V_1, a_1) → e_2
  48. Intersect(d, e_2) → M_2, N_2
  49. Polygon(C_1, N_2, D_2) → t10, d_3, c_8, n_2
  50. Mirror(V_1, u) → C_2
  51. Polygon(Z_1, C_2, C_1) → t7, c_3, z_1, c_4
  52. Polygon(C_1, C_2, B_2) → t8, b_3, c_5, c_6
  53. Rotate(I_1, (-ang2), l_1) → U_1
  54. Segment(U_1, V_1) → r_1
  55. Rotate(C_2, (-(180°)), r_1) → H_2
  56. Polygon(H_2, D_2, C_1) → t9, c_7, h_2, d_2
  57. Mirror(C_2, o) → E_2
  58. Mirror(B_2, o) → F_2
  59. Mirror(t7, d) → t7'
  60. Polygon(Q, R, S) → t1, s, q, r
  61. Polygon(B_1, T, Z) → t2, g, h, i
  62. PointIn(face1) → Pointface1
  63. PointIn(t2) → Pointt2
  64. Plane(B, U, C) → j
  65. PointIn(t3) → Pointt3
  66. Mirror(t3, d) → t3'
  67. Mirror(t3', j) → t3''
  68. Translate({t7, t8, t10, q1, q3', q3'', Mirror[{q1, q3', q3'', t10}, d]}, Vector[A, A]) → list1
  69. Mirror(t3, j) → t3'_1
  70. Mirror(t3', xOyPlane) → t3''_1
  71. Mirror(t3'_1, xOyPlane) → t4
  72. Mirror(t3, xOyPlane) → t3'_2
  73. Mirror(t3'', xOyPlane) → t3'''
  74. AngularBisector(yAxis, xAxis) → k, l
  75. Sequence(Rotate[list1, (ii * 180°), zAxis], ii, 0, 1, 1) → list1'
  76. Sequence(Rotate[list1', (ii * 180°), yAxis], ii, 0, 1, 1) → L_2
  77. Rotate(L_2, 120°, f) → L_3
  78. Rotate(L_2, (-(120°)), f) → L_4
  79. Mirror(q1, i) → q1'
  80. OrthogonalLine(B_1, g, q1) → n
  81. Mirror(J_1, n) → M_1
  82. Mirror(F_1, n) → N_1
  83. Line(M_1, B_1) → p
  84. AngularBisector(p, n) → t, g_1
  85. OrthogonalLine(M_1, p, q1) → h_1
  86. OrthogonalLine(J_1, i, q1) → i_1
  87. Intersect(n, h_1) → O_1
  88. Intersect(h_1, t) → P_1
  89. Polygon(B_1, J_1, O_1, P_1) → q2, b_1, j_2, o_1, p_1
  90. Polygon(B_1, P_1, M_1) → t5, m_1, b_2, p_2
  91. Mirror(K_1, l_1) → R_1
  92. Mirror(q1, l_1) → q1'_1
  93. Polygon(I_1, Q_1, T_1) → t6, t_2, i_3, q_2
  94. Rotate(t6, (-ang2), l_1) → t6'
  95. Rotate(S_1, (-ang2), l_1) → W_1
  96. Mirror(W_1, d) → A_2
  97. Mirror(F_1, d) → I_2
  98. Mirror(q1, d) → q1'_2
  99. Mirror(t6', o) → t6''
  100. Plane(I_2, C_1, F_1) → v
  101. Line(J_2, D_2) → f_2
  102. Mirror(V_1, f_2) → K_2

AI Drawing Prompt Reference

Create an animated origami model showing a dynamic transformation from an octahedron to an icosahedron. Start with an octahedron whose vertices lie on the coordinate axes, and mark the centroid of one face. Use a line through the origin and this centroid as an axis, then scale one vertex by the golden ratio and rotate it around the axis to generate vertices of an icosahedron, which is then constructed. Add a cube with one face coplanar with a face of the octahedron. Using angle bisectors, reflections, and rotations, construct foldable triangular panels on the cube that rotate around edges to simulate paper folding. The scene includes multiple polyhedra, auxiliary planes, circles, and segments, with sliders controlling rotation angles for the animation.
The prompt above was compiled from the original drawing commands and is for reference only.