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Pyramid Dissection

Pyramid Dissection
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2019-04-25 18:23
FunctionSolid GeometryOptimizationParametric EquationGeometric Construction
This example demonstrates the geometric dissection and artistic representation of a pyramid. It involves core mathematical concepts such as solid geometry, parametric equations, function graphing, and geometric construction. Key GeoGebra commands and objects used in the construction include Polygon, Translate, Plane, Centroid, Pyramid, and IntersectPath. Suitable for high school and above, it is ideal for learning solid geometry constructions, parametric equations, and optimization problems.

GeoGebra Commands & Parameters

  1. Polygon(A, B, D, C) → p0, a, b, d, c
  2. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (1 - 1)) - 1, 0)]) → P1
  3. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (2 - 1)) - 1, 0)]) → P2
  4. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (3 - 1)) - 1, 0)]) → P3
  5. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (4 - 1)) - 1, 0)]) → P4
  6. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (5 - 1)) - 1, 0)]) → P5
  7. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (6 - 1)) - 1, 0)]) → P6
  8. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (7 - 1)) - 1, 0)]) → P7
  9. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (8 - 1)) - 1, 0)]) → P8
  10. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (9 - 1)) - 1, 0)]) → P9
  11. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (10 - 1)) - 1, 0)]) → P10
  12. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (11 - 1)) - 1, 0)]) → P11
  13. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (12 - 1)) - 1, 0)]) → P12
  14. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (13 - 1)) - 1, 0)]) → P13
  15. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (14 - 1)) - 1, 0)]) → P14
  16. Translate(p0, Vector[(0, -1, 0), (0, (2 / n * (15 - 1)) - 1, 0)]) → P15
  17. Plane(P1) → PO1
  18. Plane(P2) → PO2
  19. Plane(P3) → PO3
  20. Plane(P4) → PO4
  21. Plane(P5) → PO5
  22. Plane(P6) → PO6
  23. Plane(P7) → PO7
  24. Plane(P8) → PO8
  25. Plane(P9) → PO9
  26. Plane(P10) → PO10
  27. Plane(P11) → PO11
  28. Plane(P12) → PO12
  29. Plane(P13) → PO13
  30. Plane(P14) → PO14
  31. Plane(P15) → PO15
  32. Polygon((o, (-o) + t, (-o)), ((-o), (-o) + t, (-o)), ((-o), o + t, (-o)), (o, o + t, (-o))) → base, f, g, h, i
  33. Centroid(base) → T
  34. Pyramid(base, (x(T), y(T), z(T) + 0.9)) → s, face1, face3, face4, face2, edge1, edge2, edge3, edge4
  35. IntersectPath(PO1, s) → S1_1, F_2, G_2, H_2, I_2, h_3, i_3, j_3, k_3
  36. IntersectPath(PO2, s) → S2_1, B_2, C_2, D_2, E_2, d_2, e_2, f_3, g_3
  37. IntersectPath(PO3, s) → S3_1, V_1, W_1, Z_1, A_2, t_2, a_2, b_2, c_2
  38. IntersectPath(PO4, s) → S4_1, E, F, T_1, U_1, j, k, r_2, s_2
  39. IntersectPath(PO5, s) → S5_1, G, H, I, J, l, m, p, q
  40. IntersectPath(PO6, s) → S6_1, K, L, M, N, r, e, f_1, g_1
  41. IntersectPath(PO7, s) → S7_1, O, P, Q, R, h_1, i_1, j_1, k_1
  42. IntersectPath(PO8, s) → S8_1, S, U, V, W, l_1, m_1, n_1, p_1
  43. IntersectPath(PO9, s) → S9_1, Z, A_1, B_1, C_1, q_1, r_1, s_1, t_1
  44. IntersectPath(PO10, s) → S10_1, D_1, E_1, F_1, G_1, a_1, b_1, c_1, d_1
  45. IntersectPath(PO11, s) → S11_1, H_1, I_1, J_1, K_1, e_1, f_2, g_2, h_2
  46. IntersectPath(PO12, s) → S12_1, L_1, M_1, N_1, O_1, i_2, j_2, k_2, l_2
  47. IntersectPath(PO13, s) → S13_1, P_1, Q_1, R_1, S_1, m_2, n_2, p_2, q_2
  48. IntersectPath(PO14, s) → S14_1, J_2, K_2, L_2, M_2, l_3, m_3, n_3, p_3
  49. IntersectPath(PO15, s) → S15_1, N_2, O_2, P_2, Q_2, q_3, r_3, s_3, t_3