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Double Skew Tetrahedra Model

Double Skew Tetrahedra Model
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2020-07-17 00:24
Solid GeometryTransformationGeometric Construction
In collections Action Modular Origami
This is an animation of a modular origami model from the book 'Action Modular Origami'. The model demonstrates the folding process of double skew tetrahedra. Key mathematical concepts include solid geometry and geometric construction. Using various GeoGebra commands such as rotation, mirroring, and plane intersection, a complex geometric structure can be constructed. Suitable for high school students and origami enthusiasts interested in geometry.

GeoGebra Commands & Parameters

  1. OrthogonalLine(A, xOyPlane) → f
  2. Point(f) → D_1
  3. Line(A, B) → g
  4. Point(g) → V
  5. Plane(D_1, V, A) → d_1
  6. Point(g) → C
  7. Midpoint(C, V) → C_1
  8. Point(Circle[C, Distance[A, C], Segment[A, C]]) → F
  9. Rotate(C, 45°, F) → C'
  10. Point(Circle[C', Distance[F, C'], Segment[F, C']]) → P
  11. Cube(F, C', P) → n, Q, facePQUT, faceRSTU, edgeFQ, edgeC'F, edgeC'P, edgePQ, edgeFR, edgeQU, edgeRU, edgeRS, R, edgeC'S, edgeST, edgePT, edgeTU, S, T, U, faceC'FQP, faceFQUR, faceC'FRS, faceC'PTS
  12. Cube(A, C, F) → b, G, faceFGKJ, faceHIJK, edgeAG, edgeAC, edgeCF, edgeFG, edgeAH, edgeGK, edgeHK, edgeHI, H, edgeCI, edgeIJ, edgeFJ, edgeJK, I, J, K, faceACFG, faceAGKH, faceACIH, faceCFJI
  13. Plane(A, F, J) → d
  14. Line(A, F) → q
  15. Point(q) → W
  16. Midpoint(W, F) → F_1
  17. Rotate(C, (2 * α), C_1, d_1) → C'_1
  18. Sphere(C'_1, 1) → u_1
  19. Rotate(R, (-α), F, d) → R'
  20. Line(F, L) → t
  21. OrthogonalLine(W, t, d) → g_1
  22. Intersect(t, g_1) → A_1
  23. Circle(A_1, 1, d) → q_1
  24. Intersect(t, q_1) → I_1, J_1
  25. Rotate(F, (-2 * α), F_1, d) → F'_1
  26. Plane(C'_1, I_1, F'') → o_1
  27. OrthogonalLine(C'_1, o_1) → t_1
  28. Intersect(t_1, u_1) → K_1, L_1
  29. OrthogonalLine(L_1, d_1) → g_2
  30. Intersect(g_2, d_1) → M_1
  31. OrthogonalLine(C, g, xOyPlane) → i
  32. Circle(C, A, xOyPlane) → c
  33. Intersect(i, c) → D, E
  34. Polygon(A, C, E) → poly1, e, a, c_1
  35. OrthogonalLine(F, q, xOyPlane) → h_1
  36. Vector(E, A_1) → v
  37. Rotate(Translate[Rotate[F, (-α), h_1], v], β, q) → F'
  38. Plane(F'', F', F) → b_1
  39. Angle(F'', F, F', b_1) → δ
  40. Polygon(C, F, J) → poly2, j, c_2, f_1
  41. Polygon(A, C, J) → poly3, j_1, a_1, c_3
  42. Polygon(A, F, J) → poly4, j_2, a_2, f_2
  43. Translate({poly1, poly2, poly3, poly4}, Vector[A, A]) → SkewTet
  44. Translate(Rotate[SkewTet, (-α), h_1], v) → SkewTet'_1
  45. OrthogonalLine(F, b_1) → k_1
  46. OrthogonalLine(A_1, q, d) → i_1
  47. Intersect(q, i_1) → B_1
  48. Polygon(F, S, C') → poly5, c'_1, f_3, s
  49. Intersect(g, c) → M, N
  50. OrthogonalLine(M, g, xOyPlane) → l
  51. OrthogonalLine(A, g, xOyPlane) → h
  52. OrthogonalLine(F, h, xOyPlane) → m
  53. Intersect(l, m) → O
  54. Rotate(O, 45°, F) → O'
  55. Polygon(S, F, O') → poly8, o'_2, s_2, f_5
  56. Polygon(S, O', C') → poly7, c'_2, s_1, o'_1
  57. Polygon(C', F, O') → poly6, o', c', f_4
  58. Mirror(SkewTet, d) → SkewTet'
  59. Sequence(Rotate[{SkewTet, SkewTet'}, (n * 90°), f], n, 0, 3, 1) → pos1
  60. Circle(i, C) → k
  61. Vector(E, W) → u
  62. Translate({poly5, poly6, poly7, poly8}, u) → SkewTet2
  63. Mirror(SkewTet2, d) → SkewTet2'
  64. Sequence(Rotate[{SkewTet2, SkewTet2'}, (n * 90°), f], n, 0, 3, 1) → pos2
  65. Translate(Q, u) → Q'
  66. Plane(V, faceCFJI) → o
  67. OrthogonalLine(Q', g, xOyPlane) → r
  68. Intersect(r, g) → Z
  69. Rotate(SkewTet2, 180°, r) → SkewTet3_1
  70. Mirror(SkewTet3_1, d) → SkewTet3'_1
  71. Rotate(SkewTet, (-(180°)), i) → SkewTet3'
  72. Rotate(SkewTet', (-(180°)), m) → SkewTet3
  73. Sequence(Rotate[{SkewTet3, SkewTet3'}, (n * 90°), f], n, 0, 3, 1) → pos3
  74. Mirror(pos1, xOyPlane) → pos4
  75. Circle(F, R, d) → p
  76. Rotate(Rotate[SkewTet'_1, β, q], γ, k_1) → SkewTet01
  77. Mirror(SkewTet01, d) → SkewTet01'
  78. Vector(W, W) → w
  79. Translate(SkewTet01, w) → SkewTet01'_1
  80. Sequence(Rotate[{SkewTet01'_1, Mirror[SkewTet01'_1, d]}, (n * 90°), f], n, 0, 3, 1) → pos01
  81. Rotate(Translate[Rotate[poly4, (-α), h_1], v], β, q) → poly9
  82. Mirror(poly9, d) → poly9'
  83. Point(pos01) → E_1
  84. Circle(C_1, C, d_1) → e_1
  85. Circle(F_1, F, d) → p_1
  86. Intersect(h_1, m) → G_1
  87. Segment(F, J) → l_1
  88. Point(xAxis) → H_1
  89. Segment(F'', I_1) → m_1
  90. Segment(C'_1, F'') → n_1
  91. Polygon(ZZZ, C'_1, F'') → poly10, h_2, k_2, l_2
  92. Segment(C'_1, I_1) → r_1
  93. Polygon(ZZZ, I_1, F'') → poly11, i_2, m_2, n_2
  94. Polygon(F'', I_1, C'_1) → poly12, c'_4, f''_2, i_3
  95. Polygon(ZZZ, C'_1, I_1) → poly13, p_2, q_2, r_2
  96. Sequence(Rotate[{{poly10, poly11, poly12, poly13}, Mirror[{poly10, poly11, poly12, poly13}, d]}, (n * 90°), f], n, 0, 3, 1) → pos10
  97. Distance(C'_1, F'_1) → distanceC'1F'1

AI Drawing Prompt Reference

Draw a double skew tetrahedron model, including four tetrahedra n, b, SkewTet, and SkewTet'_1. First, construct tetrahedra n and b, then use rotation and mirroring to create SkewTet. Next, apply a series of rotations and translations to create SkewTet'_1. Ensure all edges and faces of the tetrahedra are correctly connected to form a complex geometric structure.
The prompt above was compiled from the original drawing commands and is for reference only.