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Cube Division Demonstration

Cube Division Demonstration
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2020-11-10 21:48
Solid GeometryCalculusOptimizationGeometric Construction
This example demonstrates the geometric construction and relationships of a cube's division. It involves core mathematical concepts such as Definite Integral, Natural Numbers, Constructions, Linear Programming/Optimization, GCD, and LCM. GeoGebra commands and objects like Intersect, Point, Polygon, Pyramid, Mirror, Plane, IntersectPath, and Midpoint are used in the construction. Suitable for high school students and math enthusiasts learning solid geometry, calculus, and optimization methods.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 4) → pol1, f, g, h, i, C, D
  4. Polygon(A, B, 4) → pol2, j, k, l, m, E, F
  5. Pyramid(pol2, f / 2) → a_{3}, G, faceABG, faceBEG, faceEFG, faceAFG, arestaAG, arestaBG, arestaEG, arestaFG
  6. Mirror(a_{3}, faceBEG) → a'
  7. Mirror(a_{3}, G) → a'_{1}
  8. Plane(G, A, B) → p
  9. IntersectPath(p, a'_{1}) → pol3, H, I, J, n, q, r
  10. Polygon(H, E, J) → t2, j_2, h_2, e
  11. Polygon(J, E, F) → t3, f_1, j_3, e_1
  12. Polygon(J, F, A) → t4, a_1, j_4, f_2
  13. Polygon(J, A, B) → t5, b, j_5, a_2
  14. Polygon(H, B, E) → t6, e_2, h_3, b_1
  15. Polygon(H, J, B) → t7, b_2, h_4, j_6
  16. Polygon(P_7, P_5, P_6) → t1, p_6, p_7, p_5
  17. Polygon(P_3, P_6, P_5, P_4) → q1, p_3, s, t, p_4
  18. Polygon(P_7, P_4, P_5) → t8, c, d, g_1
  19. Polygon(P_2, P_4, P_3) → t9, h_1, p_2, i_1
  20. Polygon(P_2, P_7, P_4) → t10, j_1, k_1, l_1
  21. Polygon(P_3, P_2, P_7) → t11, m_1, n_1, p_1
  22. Polygon(P_3, P_7, P_6) → t12, q_1, r_1, s_1
  23. Midpoint(R_4, R_1) → K
  24. Polygon(R_4, R_3, R_2, R_5) → q2, r_4, r_3, r_2, r_5
  25. Polygon(R_5, R_2, R_1) → t13, t_1, c_1, d_1
  26. Polygon(K, R_5, R_4) → t14, g_2, k_2, i_2
  27. Polygon(K, R_3, R_4) → t15, l_2, k_3, m_2
  28. Polygon(R_1, K, R_5) → t16, n_2, q_2, k_4
  29. Polygon(K, R_1, R_2) → t17, s_2, k_5, t_2
  30. Polygon(K, R_3, R_2) → t18, c_2, k_6, d_2
  31. Midpoint(S_5, S_2) → L
  32. Polygon(S_3, L, S_4) → t19, s_4, s_3, l_3
  33. Polygon(S_3, S_5, S_1) → t20, f_3, g_3, s_5
  34. Polygon(S_3, L, S_5) → t21, i_3, m_3, l_4
  35. Polygon(S_3, S_4, S_2, S_1) → q3, n_3, q_3, t_3, b_3
  36. Polygon(S_2, L, S_1) → t22, c_3, d_3, l_5
  37. Polygon(S_1, L, S_5) → t23, e_3, f_4, l_6
  38. Polygon(S_4, S_2, L) → t24, l_7, g_4, i_4

AI Drawing Prompt Reference

In 3D space, start from two perpendicular coordinate axes and their intersection as the main reference point. Using points on these axes, construct two corresponding quadrilateral base faces (each sharing the same pair of axis-derived points, with other points completing the four corners). Lift one quadrilateral face into a pyramid defined by a height parameter, then mirror the pyramid’s key vertex across a specific face and mirror it again across the pyramid’s apex line so that a new mirrored vertex and its associated mirrored direction are created. Use a plane through the mirrored setup to intersect the mirrored direction path, obtaining several intersection points. Build multiple triangular and quadrilateral surface patches by connecting those points to form a divided cube-like solid, and take midpoints on selected edges to create further internal vertices, then connect them to finalize the subdivision faces.
The prompt above was compiled from the original drawing commands and is for reference only.