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Pentagonal Net with Double Ellipses and Open Spheres

Pentagonal Net with Double Ellipses and Open Spheres
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2018-05-23 01:47
CirclePolygonCoordinate GeometrySolid GeometryGeometric Construction
① This example demonstrates a complex 3D geometric construction combining a pentagonal net with double ellipses and open spheres, illustrating the interweaving of polygons and conic sections in space. ② Core mathematical concepts include polygons (pentagons), ellipses as conic curves, spheres as 3D surfaces, and their symmetric/nested arrangements, bridging solid geometry and coordinate geometry. ③ Construction highlights: from the summary, the GeoGebra commands used are Polygon (creating poly1 to poly5) and Ellipse (creating ellipses a, b, c, d, e), with points A–N and segments (a_1, b_1, etc.) connecting to form the net, and open spheres added as surface elements. ④ Suitable for high school or college-level solid geometry and analytic geometry classes, as well as mathematical art or 3D modeling projects, helping students understand spatial combinations of polygonal meshes, conic sections, and spheres.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D, E) → poly1, a, b, c, d, e
  2. Polygon(A, B, F, G, H) → poly2, a_1, b_1, f, g, h
  3. Polygon(B, C, I, J, F) → poly3, b_2, c_1, i, j, f_1
  4. Polygon(C, D, K, L, I) → poly4, c_2, d_1, k, l, i_1
  5. Polygon(D, E, M, N, K) → poly5, d_2, e_1, m, n, k_1
  6. Polygon(E, A, H, O, M) → poly6, e_2, a_2, h_1, o, m_1
  7. Ellipse(M, K, N) → p
  8. Ellipse(E, D, N) → q
  9. Ellipse(K, I, L) → r
  10. Ellipse(D, C, L) → s
  11. Ellipse(C, B, J) → t
  12. Ellipse(I, F, J) → g_1
  13. Ellipse(B, A, G) → p_1
  14. Ellipse(F, H, G) → q_1
  15. Ellipse(A, E, O) → r_1
  16. Ellipse(H, M, O) → s_1
  17. Polygon(A_1, B_1, C_1, D_1, E_1) → poly1_1, a_3, b_3, c_3, d_3, e_3
  18. Polygon(A_1, B_1, F_1, G_1, H_1) → poly2_1, a_4, b_4, f_2, g_3, h_2
  19. Polygon(B_1, C_1, I_1, J_1, F_1) → poly3_1, b_5, c_4, i_2, j_1, f_3
  20. Polygon(C_1, D_1, K_1, L_1, I_1) → poly4_1, c_5, d_4, k_2, l_1, i_3
  21. Polygon(D_1, E_1, M_1, N_1, K_1) → poly5_1, d_5, e_4, m_2, n_1, k_3
  22. Polygon(E_1, A_1, H_1, O_1, M_1) → poly6_1, e_5, a_5, h_3, o_1, m_3
  23. Ellipse(M_1, K_1, N_1) → p_2
  24. Ellipse(E_1, D_1, N_1) → q_2
  25. Ellipse(K_1, I_1, L_1) → r_2
  26. Ellipse(D_1, C_1, L_1) → s_2
  27. Ellipse(C_1, B_1, J_1) → t_1
  28. Ellipse(I_1, F_1, J_1) → g_2
  29. Ellipse(B_1, A_1, G_1) → p_3
  30. Ellipse(F_1, H_1, G_1) → q_3
  31. Ellipse(A_1, E_1, O_1) → r_3
  32. Ellipse(H_1, M_1, O_1) → s_3
  33. Polygon(A_2, B_2, C_2, D_2, E_2) → poly1_2, a_6, b_6, c_6, d_6, e_6
  34. Polygon(A_2, B_2, F_2, G_2, H_2) → poly2_2, a_7, b_7, f_4, g_5, h_4
  35. Polygon(B_2, C_2, I_2, J_2, F_2) → poly3_2, b_8, c_7, i_4, j_2, f_5
  36. Polygon(C_2, D_2, K_2, L_2, I_2) → poly4_2, c_8, d_7, k_4, l_2, i_5
  37. Polygon(D_2, E_2, M_2, N_2, K_2) → poly5_2, d_8, e_7, m_4, n_2, k_5
  38. Polygon(E_2, A_2, H_2, O_2, M_2) → poly6_2, e_8, a_8, h_5, o_2, m_5
  39. Ellipse(M_2, K_2, N_2) → p_4
  40. Ellipse(E_2, D_2, N_2) → q_4
  41. Ellipse(K_2, I_2, L_2) → r_4
  42. Ellipse(D_2, C_2, L_2) → s_4
  43. Ellipse(C_2, B_2, J_2) → t_2
  44. Ellipse(I_2, F_2, J_2) → g_4
  45. Ellipse(B_2, A_2, G_2) → p_5
  46. Ellipse(F_2, H_2, G_2) → q_5
  47. Ellipse(A_2, E_2, O_2) → r_5
  48. Ellipse(H_2, M_2, O_2) → s_5

AI Drawing Prompt Reference

Create a 3D geometric structure composed of three nested layers of pentagonal nets, each layer consisting of six pentagonal faces connected by shared edges, resembling an unfolded polyhedral mesh. Within or along each pentagon, embed pairs of intersecting ellipses whose endpoints are the pentagon vertices, forming symmetric curved crossing patterns. Add open spherical surfaces distributed in space, interweaving with the polygons and ellipses to enhance depth and visual complexity. The overall composition should exhibit symmetry and nested elegance, suitable for mathematical art or spatial geometry modeling.
The prompt above was compiled from the original drawing commands and is for reference only.