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Three-Piece Spinner

Three-Piece Spinner
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2020-07-17 00:30
PolygonVectorTrigonometrySolid GeometryGeometric Construction
In collections Action Modular Origami
This example demonstrates an animated three-dimensional model folded from paper, sourced from the book 'Action Modular Origami'. Core mathematical concepts include intersection, area, quadrilaterals, rectangles, and statistical characteristics. Construction highlights involve commands/objects like Vector, WaterbombPreliminaryFold, Line, Point, Rotate, Plane, Intersect, Midpoint, Translate, Circle, Polygon, OrthogonalLine, Sphere, LineBisector, Sequence, Mirror, and PointIn in GeoGebra. Suitable for high school students and math enthusiasts studying spatial geometry and modular origami.

GeoGebra Commands & Parameters

  1. Vector((0, 0, offset)) → vOffset
  2. WaterbombPreliminaryFold(xOyPlane, H, I, α) → list2, J, K, L, O
  3. Line(O, J) → p
  4. Line(J, L) → n
  5. Point(n) → P
  6. Rotate(P, 90°, p) → P'_1
  7. Rotate(P, (-(90°)), p) → P'
  8. Plane(P'_1, P, P') → c
  9. WaterbombPreliminaryFold(c, P, P', β) → list3, R, S, T, U
  10. Plane(U, J, L) → o
  11. Line(J, H) → f_1
  12. Rotate(f_1, (90 / 4)°, J) → g_1
  13. Vector(L, K) → v
  14. Line(J, K) → j_1
  15. Line(I, H) → d
  16. Intersect(g_1, d) → B_1
  17. Midpoint(I, H) → C_1
  18. Translate(L, Vector[(frac * v)]) → L'
  19. Plane(J, K, L') → w
  20. Circle(K, L', w) → d_1
  21. Intersect(j_1, d_1) → H_1, I_1
  22. Polygon(I_1, L', H_1) → poly5, h_2, i_2, l'_1
  23. OrthogonalLine(I_1, n, w) → k_1
  24. Intersect(n, k_1) → J_1
  25. Polygon(J, J_1, I_1) → poly6, i_3, j_2, j_3
  26. Midpoint(J, I_1) → M_1
  27. Polygon(A, B, 4, xOyPlane) → poly1, f, g, h, i, C, D
  28. Midpoint(A, C) → E
  29. Sphere(E, A) → b
  30. OrthogonalLine(E, xOyPlane) → j
  31. Intersect(j, b) → M, N
  32. Line(D, B) → m
  33. Midpoint(C, B) → G
  34. LineBisector(D, C, xOyPlane) → k
  35. LineBisector(D, B, xOyPlane) → l
  36. Rotate(G, If[α < 0°, α, (2 * α)], k) → G'
  37. Rotate(C, If[α ≥ 0°, α, (2 * α)], m) → C'
  38. Polygon(E, G', C') → poly2, c', e, g'
  39. Plane(E, C, F) → a
  40. Sequence(Rotate[{poly2, Mirror[poly2, a]}, (nn * 90°), j], nn, 0, 3, 1) → list1
  41. Rotate(m, (-(90°)), p) → m'
  42. Point(list2) → Q
  43. Point(p) → V
  44. Point(p) → W
  45. Vector(V, W) → u
  46. Polygon(H, I, 4, xOyPlane) → poly3, q, r, s, t, Z, A_1
  47. Rotate(poly3, (90 / 2)°, J) → poly3'
  48. Mirror(L', o) → L''
  49. Line(L', L'') → h_1
  50. Intersect(h_1, o) → D_1
  51. Circle(D_1, L, o) → c_1
  52. Line(D_1, L) → i_1
  53. Intersect(i_1, c_1) → E_1, F_1
  54. PointIn(o) → G_1
  55. Polygon(L, L', G_1) → poly4, g_2, l_1, l'
  56. Rotate(Translate[Sequence[Rotate[{poly5, poly6, Mirror[{poly5, poly6}, o]}, (nn * 90°), p], nn, 0, 3, 1], vOffset], γ, p) → list5
  57. Translate(list5, Vector[(-2 * vOffset)]) → list5'
  58. Plane(J, H_1, O) → a_1
  59. Circle(H_1, I_1, a_1) → e_1
  60. Point(e_1) → K_1
  61. Plane(J, W, I) → b_1
  62. Mirror(Q, b_1) → Q'
  63. Polygon(Q', H_1, K_1) → poly7, k_2, q', h_3
  64. OrthogonalLine(G_1, xOyPlane) → m_1
  65. Line(J, P) → n_1
  66. Intersect(n_1, m_1) → L_1
  67. Plane(O, K_1, J) → o_1
  68. Mirror(Q', o_1) → Q''
  69. Polygon(J_1, M_1, J) → poly9, s_1, t_1, f_2
  70. Polygon(Q'', Translate[P, vOffset], J_1) → poly8, p_1, q_1, r_1
  71. PointIn(o) → N_1
  72. Rotate(Translate[Sequence[Rotate[{poly8, poly9, Mirror[{poly8, poly9}, o]}, (nn * 90°), p], nn, 0, 3, 1], Vector[(-vOffset)]], γ, p) → list7
  73. Rotate(Translate[Sequence[Rotate[{poly7, Mirror[{poly7}, o]}, (nn * 90°), p], nn, 0, 3, 1], vOffset], γ, p) → list6
  74. Rotate(Sequence[Rotate[{poly4, Mirror[poly4, o]}, (nn * 90°), p], nn, 0, 3, 1], γ, p) → list4
  75. Rotate(Translate[list3, u], γ, p) → list3'
  76. Rotate(list2, γ, p) → list2'