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Shape Based on an Octahedron

Shape Based on an Octahedron
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2017-01-21 18:33
PolygonVectorSolid GeometryGeometric Construction
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This example demonstrates the construction of a complex geometric shape based on an octahedron. It involves core mathematical concepts such as solid geometry, polygon construction, vector operations, and geometric transformations. Key GeoGebra commands used include Angle, Rotate, Ray, Sequence, Point, IndexOf, Element, Flatten, Intersect, AffineRatio, Midpoint, Polygon, and Vector. It is suitable for high school and above students to explore methods of solid geometry construction.

GeoGebra Commands & Parameters

  1. Angle(A, O, B) → α
  2. Rotate(FF, α / 2, O_1) → F'
  3. Ray(O_1, F') → f
  4. Ray(O_1, FF) → g
  5. Ray(M, Vector[O_1, FF]) → h
  6. Ray(F', Vector[O_1, F']) → i
  7. Sequence(O_1 + (i * Vector[O_1, M]), i, 1.5, 5, 0.5) → B_1, {(0, 1.0606601717798214), (0, 1.4142135623730951), (0, 1.7677669529663689), (0, 2.121320343559643), (0, 2.4748737341529163), (0, 2.8284271247461903), (0, 3.1819805153394642), (0, 3.5355339059327378)}
  8. Point(g) → A_1
  9. Sequence(Distance[A_1, Element[B_1, i]], i, 1, Length[B_1]) → C_1, {1.414213562373095, 1.0606601717798212, 0.7071067811865475, 0.3535533905932735, 0, 0.35355339059327395, 0.7071067811865479, 1.0606601717798214}
  10. IndexOf(Min[C_1], C_1) → D_1
  11. Element(B_1, D_1) → E_1
  12. Sequence(O_1 + (i * Vector[O_1, F']), i, 1.5, 5, 0.5) → B_6, {(-1.0606601717798212, 1.0606601717798214), (-1.414213562373095, 1.4142135623730951), (-1.7677669529663687, 1.7677669529663689), (-2.1213203435596424, 2.121320343559643), (-2.474873734152916, 2.4748737341529163), (-2.82842712474619, 2.8284271247461903), (-3.181980515339464, 3.1819805153394642), (-3.5355339059327373, 3.5355339059327378)}
  13. Point(f) → A_6
  14. Sequence(Distance[A_6, Element[B_6, i]], i, 1, Length[B_6]) → C_6, {3.4999999999999996, 3, 2.5, 1.9999999999999998, 1.5000000000000002, 0.9999999999999999, 0.4999999999999996, 0}
  15. IndexOf(Min[C_6], C_6) → D_6
  16. Element(B_6, D_6) → E_6
  17. Flatten(Unique[Sequence[Sequence[O_1 + (ii * Vector[O_1, M]) + (jj * Vector[O_1, F']), ii, 1, 5, 0.5], jj, 0.5, 5, 0.5]]) → LL
  18. Sequence(Distance[A_2, Element[LL, i]], i, 1, Length[LL]) → C_2, {0.49999999999999994, 0.35355339059327373, 0.49999999999999994, 0.790569415042095, 1.1180339887498951, 1.4577379737113252, 1.8027756377319948, 2.150581316760657, 2.5000000000000004, 0, 0.35355339059327373, 0.7071067811865477, 1.0606601717798212, 1.4142135623730951, 1.7677669529663687, 2.121320343559643, 2.474873734152917, 2.8284271247461907, 0.49999999999999994, 0.790569415042095, 1.1180339887498951, 1.4577379737113252, 1.8027756377319948, 2.150581316760657, 2.5000000000000004, 2.8504385627478457, 3.2015621187164243, 1, 1.2747548783981963, 1.5811388300841898, 1.903943276465977, 2.23606797749979, 2.5739075352467506, 2.915475947422651, 3.259601202601325, 3.6055512754639896, 1.4999999999999998, 1.7677669529663689, 2.06155281280883, 2.3717082451262845, 2.6925824035672523, 3.0207614933986435, 3.3541019662496847, 3.691205765058351, 4.031128874149275, 2, 2.2638462845343543, 2.5495097567963922, 2.8504385627478452, 3.16227766016838, 3.48209706929603, 3.807886552931955, 4.138236339311712, 4.47213595499958, 2.4999999999999996, 2.7613402542968153, 3.0413812651491097, 3.3354160160315836, 3.640054944640259, 3.952847075210474, 4.272001872658765, 4.59619407771256, 4.924428900898053, 3, 3.2596012026013246, 3.535533905932738, 3.824264635194589, 4.1231056256176615, 4.430011286667337, 4.743416490252569, 5.062114182829148, 5.385164807134504, 3.5000000000000004, 3.7583240945932275, 4.0311288741492755, 4.315669125408018, 4.6097722286464435, 4.911720676097126, 5.220153254455275, 5.5339859052946645, 5.852349955359814, 4, 4.257346591481601, 4.527692569068709, 4.808846015417836, 5.0990195135927845, 5.396758286230726, 5.70087712549569, 6.0104076400856545, 6.324555320336759}
  19. Sequence(Distance[A_3, Element[LL, i]], i, 1, Length[LL]) → C_3, {1.5, 1.2747548783981961, 1.118033988749895, 1.0606601717798212, 1.118033988749895, 1.2747548783981961, 1.5, 1.767766952966369, 2.0615528128088303, 1, 0.7905694150420949, 0.7071067811865475, 0.7905694150420947, 0.9999999999999999, 1.274754878398196, 1.5811388300841895, 1.9039432764659772, 2.23606797749979, 0.5000000000000001, 0.35355339059327373, 0.5000000000000001, 0.7905694150420948, 1.118033988749895, 1.457737973711325, 1.8027756377319948, 2.150581316760657, 2.4999999999999996, 0, 0.35355339059327395, 0.7071067811865475, 1.060660171779821, 1.414213562373095, 1.7677669529663689, 2.121320343559643, 2.474873734152917, 2.82842712474619, 0.4999999999999998, 0.7905694150420948, 1.1180339887498945, 1.457737973711325, 1.8027756377319948, 2.150581316760657, 2.4999999999999996, 2.8504385627478457, 3.2015621187164243, 0.9999999999999999, 1.2747548783981961, 1.5811388300841895, 1.9039432764659772, 2.23606797749979, 2.5739075352467498, 2.915475947422651, 3.2596012026013246, 3.6055512754639896, 1.4999999999999996, 1.7677669529663687, 2.0615528128088303, 2.3717082451262845, 2.6925824035672514, 3.0207614933986426, 3.3541019662496847, 3.6912057650583505, 4.031128874149275, 1.9999999999999998, 2.2638462845343543, 2.5495097567963922, 2.850438562747845, 3.16227766016838, 3.48209706929603, 3.8078865529319543, 4.138236339311712, 4.472135954999579, 2.5, 2.7613402542968157, 3.04138126514911, 3.335416016031584, 3.640054944640259, 3.9528470752104745, 4.272001872658766, 4.59619407771256, 4.924428900898053, 3, 3.2596012026013237, 3.5355339059327373, 3.824264635194589, 4.123105625617661, 4.430011286667337, 4.743416490252569, 5.062114182829148, 5.385164807134504}
  20. Sequence(Distance[A_4, Element[LL, i]], i, 1, Length[LL]) → C_4, {2.1505813167606567, 1.8027756377319946, 1.457737973711325, 1.1180339887498945, 0.7905694150420943, 0.4999999999999998, 0.35355339059327373, 0.5000000000000004, 0.7905694150420952, 1.7677669529663687, 1.414213562373095, 1.060660171779821, 0.7071067811865475, 0.3535533905932735, 0, 0.35355339059327395, 0.7071067811865479, 1.0606601717798219, 1.457737973711325, 1.1180339887498945, 0.7905694150420943, 0.4999999999999998, 0.35355339059327373, 0.5000000000000001, 0.7905694150420952, 1.1180339887498953, 1.457737973711325, 1.274754878398196, 0.9999999999999996, 0.7905694150420947, 0.7071067811865475, 0.7905694150420949, 1.0000000000000002, 1.2747548783981966, 1.5811388300841904, 1.9039432764659772, 1.2747548783981961, 1.1180339887498947, 1.0606601717798212, 1.118033988749895, 1.2747548783981963, 1.5000000000000002, 1.7677669529663687, 2.061552812808831, 2.3717082451262845, 1.457737973711325, 1.414213562373095, 1.457737973711325, 1.5811388300841898, 1.767766952966369, 1.9999999999999998, 2.2638462845343548, 2.5495097567963922, 2.8504385627478452, 1.7677669529663684, 1.8027756377319943, 1.903943276465977, 2.0615528128088303, 2.263846284534354, 2.5, 2.7613402542968153, 3.0413812651491097, 3.3354160160315836, 2.1505813167606567, 2.23606797749979, 2.3717082451262845, 2.5495097567963922, 2.7613402542968157, 3, 3.2596012026013246, 3.535533905932738, 3.824264635194589, 2.57390753524675, 2.6925824035672523, 2.8504385627478452, 3.0413812651491106, 3.2596012026013246, 3.5000000000000004, 3.7583240945932275, 4.0311288741492755, 4.315669125408018, 3.020761493398643, 3.162277660168379, 3.335416016031583, 3.5355339059327378, 3.758324094593227, 4, 4.257346591481601, 4.5276925690687095, 4.808846015417836}
  21. Sequence(Distance[A_5, Element[LL, i]], i, 1, Length[LL]) → C_5, {2.5, 2.2638462845343543, 2.0615528128088303, 1.903943276465977, 1.8027756377319946, 1.7677669529663687, 1.8027756377319946, 1.9039432764659772, 2.0615528128088303, 2, 1.7677669529663689, 1.5811388300841895, 1.457737973711325, 1.414213562373095, 1.457737973711325, 1.5811388300841895, 1.7677669529663689, 2, 1.5, 1.2747548783981961, 1.1180339887498947, 1.0606601717798212, 1.118033988749895, 1.2747548783981961, 1.5, 1.767766952966369, 2.06155281280883, 0.9999999999999999, 0.7905694150420947, 0.7071067811865475, 0.7905694150420947, 0.9999999999999999, 1.2747548783981961, 1.58113883008419, 1.9039432764659776, 2.2360679774997894, 0.5000000000000001, 0.35355339059327373, 0.4999999999999998, 0.7905694150420948, 1.118033988749895, 1.4577379737113254, 1.8027756377319943, 2.150581316760657, 2.5, 0, 0.35355339059327395, 0.7071067811865475, 1.0606601717798214, 1.4142135623730954, 1.7677669529663684, 2.1213203435596433, 2.4748737341529163, 2.8284271247461903, 0.4999999999999996, 0.7905694150420947, 1.118033988749895, 1.4577379737113254, 1.8027756377319943, 2.1505813167606567, 2.5, 2.8504385627478452, 3.2015621187164247, 0.9999999999999999, 1.2747548783981961, 1.58113883008419, 1.9039432764659767, 2.2360679774997902, 2.57390753524675, 2.9154759474226504, 3.2596012026013246, 3.605551275463989, 1.5000000000000002, 1.767766952966369, 2.0615528128088307, 2.371708245126285, 2.692582403567252, 3.0207614933986435, 3.3541019662496847, 3.6912057650583505, 4.0311288741492755, 2, 2.263846284534354, 2.5495097567963922, 2.850438562747845, 3.1622776601683795, 3.4820970692960302, 3.807886552931954, 4.138236339311713, 4.47213595499958}
  22. IndexOf(Min[C_2], C_2) → D_2
  23. IndexOf(Min[C_3], C_3) → D_3
  24. IndexOf(Min[C_4], C_4) → D_4
  25. IndexOf(Min[C_5], C_5) → D_5
  26. Element(LL, D_2) → E_2
  27. Element(LL, D_3) → E_3
  28. Element(LL, D_4) → E_4
  29. Element(LL, D_5) → E_5
  30. Intersect(g, Line[A_2, f]) → G_2
  31. Intersect(g, Line[A_3, f]) → G_3
  32. Intersect(g, Line[A_4, f]) → G_4
  33. Intersect(g, Line[A_5, f]) → G_5
  34. Intersect(f, Line[A_2, g]) → H_2
  35. Intersect(f, Line[A_3, g]) → H_3
  36. Intersect(f, Line[A_4, g]) → H_4
  37. Intersect(f, Line[A_5, g]) → H_5
  38. AffineRatio(O_1, M, A_1) → I_1
  39. AffineRatio(O_1, M, G_2) → I_2
  40. AffineRatio(O_1, M, G_3) → I_3
  41. AffineRatio(O_1, M, G_4) → I_4
  42. AffineRatio(O_1, M, G_5) → I_5
  43. AffineRatio(O_1, F', A_6) → J_6
  44. AffineRatio(O_1, F', H_5) → J_5
  45. AffineRatio(O_1, F', H_4) → J_4
  46. AffineRatio(O_1, F', H_3) → J_3
  47. AffineRatio(O_1, F', H_2) → J_2
  48. Midpoint(A1, B1) → C1
  49. Polygon(F', M, A_1, A_2, A_3, A_4, A_5, A_6) → wielokąt1, f', m, a_1, a_2, a_3, a_4, a_5, a_6
  50. Vector(O, A1) → D1
  51. Vector(O, B1) → E1
  52. Vector(O, C1) → F1
  53. Polygon(B1, A1, O1, P1, Q1, G1, H1, I1, J1, K1, L1, M1, N1) → R1, S1, AB1, AC1, AD1, AE1, T1, U1, V1, W1, X1, Y1, Z1, AA1
  54. Midpoint(A2, B2) → C2
  55. Midpoint(A3, B3) → C3
  56. Midpoint(A4, B4) → C4
  57. Midpoint(A5, B5) → C5
  58. Midpoint(A6, B6) → C6
  59. Midpoint(A7, B7) → C7
  60. Midpoint(A8, B8) → C8
  61. Midpoint(A9, B9) → C9
  62. Midpoint(A10, B10) → C10
  63. Midpoint(A11, B11) → C11
  64. Midpoint(A12, B12) → C12
  65. Vector(O, A2) → D2
  66. Vector(O, B2) → E2
  67. Vector(O, C2) → F2
  68. Vector(O, A3) → D3
  69. Vector(O, B3) → E3
  70. Vector(O, C3) → F3
  71. Vector(O, A4) → D4
  72. Vector(O, B4) → E4
  73. Vector(O, C4) → F4
  74. Vector(O, A5) → D5
  75. Vector(O, B5) → E5
  76. Vector(O, C5) → F5
  77. Vector(O, A6) → D6
  78. Vector(O, B6) → E6
  79. Vector(O, C6) → F6
  80. Vector(O, A7) → D7
  81. Vector(O, B7) → E7
  82. Vector(O, C7) → F7
  83. Vector(O, A8) → D8
  84. Vector(O, B8) → E8
  85. Vector(O, C8) → F8
  86. Vector(O, A9) → D9
  87. Vector(O, B9) → E9
  88. Vector(O, C9) → F9
  89. Vector(O, A10) → D10
  90. Vector(O, B10) → E10
  91. Vector(O, C10) → F10
  92. Vector(O, A11) → D11
  93. Vector(O, B11) → E11
  94. Vector(O, C11) → F11
  95. Vector(O, A12) → D12
  96. Vector(O, B12) → E12
  97. Vector(O, C12) → F12
  98. Polygon(B2, A2, O2, P2, Q2, G2, H2, I2, J2, K2, L2, M2, N2) → R2, S2, AB2, AC2, AD2, AE2, T2, U2, V2, W2, X2, Y2, Z2, AA2
  99. Polygon(B3, A3, O3, P3, Q3, G3, H3, I3, J3, K3, L3, M3, N3) → R3, S3, AB3, AC3, AD3, AE3, T3, U3, V3, W3, X3, Y3, Z3, AA3
  100. Polygon(B4, A4, O4, P4, Q4, G4, H4, I4, J4, K4, L4, M4, N4) → R4, S4, AB4, AC4, AD4, AE4, T4, U4, V4, W4, X4, Y4, Z4, AA4
  101. Polygon(B5, A5, O5, P5, Q5, G5, H5, I5, J5, K5, L5, M5, N5) → R5, S5, AB5, AC5, AD5, AE5, T5, U5, V5, W5, X5, Y5, Z5, AA5
  102. Polygon(B6, A6, O6, P6, Q6, G6, H6, I6, J6, K6, L6, M6, N6) → R6, S6, AB6, AC6, AD6, AE6, T6, U6, V6, W6, X6, Y6, Z6, AA6
  103. Polygon(B7, A7, O7, P7, Q7, G7, H7, I7, J7, K7, L7, M7, N7) → R7, S7, AB7, AC7, AD7, AE7, T7, U7, V7, W7, X7, Y7, Z7, AA7
  104. Polygon(B8, A8, O8, P8, Q8, G8, H8, I8, J8, K8, L8, M8, N8) → R8, S8, AB8, AC8, AD8, AE8, T8, U8, V8, W8, X8, Y8, Z8, AA8
  105. Polygon(B9, A9, O9, P9, Q9, G9, H9, I9, J9, K9, L9, M9, N9) → R9, S9, AB9, AC9, AD9, AE9, T9, U9, V9, W9, X9, Y9, Z9, AA9
  106. Polygon(B10, A10, O10, P10, Q10, G10, H10, I10, J10, K10, L10, M10, N10) → R10, S10, AB10, AC10, AD10, AE10, T10, U10, V10, W10, X10, Y10, Z10, AA10
  107. Polygon(B11, A11, O11, P11, Q11, G11, H11, I11, J11, K11, L11, M11, N11) → R11, S11, AB11, AC11, AD11, AE11, T11, U11, V11, W11, X11, Y11, Z11, AA11
  108. Polygon(B12, A12, O12, P12, Q12, G12, H12, I12, J12, K12, L12, M12, N12) → R12, S12, AB12, AC12, AD12, AE12, T12, U12, V12, W12, X12, Y12, Z12, AA12

AI Drawing Prompt Reference

Draw a complex geometric shape based on an octahedron, involving multiple points, lines, rays, and faces. Start by establishing a central point and the basic structure of an octahedron, then generate several sets of points through rotation and vector operations. Next, use these point sets and geometric transformations to construct the boundaries of the shape, calculating distances and ratio relationships between key points. Finally, combine these elements to form a complex geometric body with octahedral characteristics.
The prompt above was compiled from the original drawing commands and is for reference only.