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Dodecahedron, Sphere and Rotational Mirror Intersection Construction in 3D Geome

Dodecahedron, Sphere and Rotational Mirror Intersection Construction in 3D Geome
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2016-11-12 17:01
PolygonCoordinate GeometrySolid GeometryTransformationGeometric Construction
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This example demonstrates a complex 3D geometric construction in GeoGebra centered on a dodecahedron and a sphere, combined with rotation, mirror reflection, and intersection operations to generate multiple polygonal faces and edge segments. Core mathematical concepts include regular polyhedra, spherical geometry, rigid transformations (rotation and reflection), plane-solid intersections, polygonal regions, and spatial relationships among points, lines, and planes. The construction draws on the following GeoGebra commands and objects from the summary: Rotate for rotational transformation, Dodecahedron to create the regular dodecahedron, Sphere to create the sphere, IntersectConic and Intersect for curve and object intersections, OrthogonalPlane to construct orthogonal planes, Mirror for reflection symmetry, Midpoint for midpoint construction, and Polygon for drawing polygonal faces; objects include points A through R and their reflected images A′, A″, together with multiple surface (powierzchnia) and edge (krawędź) objects. Suitable for senior high school and university STEM geometry instruction, it supports polyhedron cross-section exploration, spatial transformation visualization, and advanced GeoGebra 3D modeling practice.

GeoGebra Commands & Parameters

  1. Rotate(A, 72°, zAxis) → A'
  2. Rotate(A', 72°, zAxis) → A''
  3. Dodecahedron(A, A', A'') → a, B, K, L, M, N, O, P, Q, R, powierzchniaAA'A''BC, powierzchniaACGMH, C, powierzchniaAA'DIH, powierzchniaA'A''EJD, powierzchniaA''BFKE, powierzchniaBCGLF, powierzchniaGLQRM, powierzchniaHINRM, powierzchniaDINOJ, powierzchniaEJOPK, powierzchniaFKPQL, powierzchniaNOPQR, D, krawędźAC, krawędźAA', krawędźA'A'', krawędźA''B, krawędźBC, krawędźAH, krawędźCG, krawędźGM, krawędźHM, krawędźHI, E, krawędźDI, krawędźA'D, krawędźDJ, krawędźEJ, krawędźA''E, krawędźEK, krawędźFK, krawędźBF, krawędźFL, krawędźGL, F, krawędźLQ, krawędźQR, krawędźMR, krawędźNR, krawędźIN, krawędźNO, krawędźJO, krawędźOP, krawędźKP, krawędźPQ, G, H, I, J
  4. Sphere(O, 1) → b
  5. Sphere(N, sqrt(3)) → c
  6. IntersectConic(c, b) → d
  7. Sphere(P, sqrt(3)) → e
  8. IntersectConic(e, b) → f
  9. Intersect(d, f) → S, T, ,
  10. OrthogonalPlane(Q, krawędźNO) → g
  11. Mirror(T, g) → B1
  12. Midpoint(B1, T) → A1
  13. Rotate(A1, -72°, Line[(0, 0, 0), (0, 0, 1)]) → A2
  14. Rotate(B1, -72°, Line[(0, 0, 0), (0, 0, 1)]) → B2
  15. Rotate(A2, -72°, Line[(0, 0, 0), (0, 0, 1)]) → A3
  16. Rotate(B2, -72°, Line[(0, 0, 0), (0, 0, 1)]) → B3
  17. Rotate(A3, -72°, Line[(0, 0, 0), (0, 0, 1)]) → A4
  18. Rotate(B3, -72°, Line[(0, 0, 0), (0, 0, 1)]) → B4
  19. Rotate(A4, -72°, Line[(0, 0, 0), (0, 0, 1)]) → A5
  20. Polygon(B3, A4, Q) → wielokąt1, q, b3, a4
  21. Polygon(A4, P, Q) → wielokąt2, q_1, a4_1, p
  22. Polygon(A4, B4, P) → wielokąt3, p_1, a4_2, b4
  23. Polygon(B4, A5, P) → wielokąt4, p_2, b4_1, a5
  24. Polygon(A5, O, P) → wielokąt5, p_3, a5_1, o
  25. Polygon(A5, T, O) → wielokąt6, o_1, a5_2, t
  26. Polygon(T, A1, O) → wielokąt7, o_2, t_1, a1
  27. Polygon(A1, N, O) → wielokąt8, o_3, a1_1, n
  28. Polygon(A1, B1, N) → wielokąt9, n_1, a1_2, b1
  29. Polygon(B1, A2, N) → wielokąt10, n_2, b1_1, a2
  30. Polygon(A2, R, N) → wielokąt11, n_3, a2_1, r
  31. Polygon(A2, B2, R) → wielokąt12, r_1, a2_2, b2
  32. Polygon(R, A3, Q) → wielokąt13, q_2, r_2, a3
  33. Polygon(B2, A3, R) → wielokąt14, r_3, b2_1, a3_1
  34. Polygon(A3, B3, Q) → wielokąt15, q_3, a3_2, b3_1