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PP Dissection Method Demonstration

PP Dissection Method Demonstration
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2019-03-21 00:44
PolygonTransformationGeometric Construction
In collections Cutting up squares
This example demonstrates the geometric construction process of the PP dissection method through dynamic visualization of shape transformations and combinations. It involves core mathematical concepts such as Similarity Transformation, Combinatorics, Indefinite Integral, Function Graph, and Multiplication. The construction highlights GeoGebra commands like Circle, Line, Intersect, Polygon, OrthogonalLine, Point, Vector, Translate, Segment, Midpoint, and Rotate. Suitable for high school and above, it provides an intuitive representation of complex geometric transformations and integral operations.

GeoGebra Commands & Parameters

  1. Circle(A, 1) → c
  2. Line(A, B) → a
  3. Intersect(c, a, 2) → C
  4. Polygon(A, C, 3) → poly1, d, a_1, c_1, D
  5. Polygon(C, D, 6) → poly2, c_2, H, d_1, b, e, f, g, E, F, G
  6. Circle(I, 0.6580370064762462) → h
  7. OrthogonalLine(I, d_1) → i
  8. OrthogonalLine(I, i) → j
  9. Intersect(h, i, 1) → J
  10. Polygon(J, I, 4) → poly3, j_1, i_1, k, l, K, L
  11. Polygon(K, I, 4) → poly4, k_1, i_2, m, n, M, N
  12. Polygon(N, M, 4) → poly5, n_1, m_1, p, q, O, P
  13. Polygon(P, O, 4) → poly6, p_1, o, r, s, Q, R
  14. Polygon(M, I, 4) → poly7, m_2, i_3, t, b_1, S, T
  15. Polygon(K, N, 4) → poly8, k_2, n_2, e_1, f_1, U, V
  16. Polygon(J, L, S, I, K, V, U, N, R, Q, M, T) → poly9, j_2, t_1, s_1, i_4, l_1, k_3, v, u, n_3, r_1, q_1, m_3
  17. Line(W, Z) → g_1
  18. Circle(W, (2 * side)) → h_1
  19. OrthogonalLine(W, g_1) → a_2
  20. Intersect(h_1, a_2, 1) → A_1
  21. OrthogonalLine(A_1, a_2) → b_2
  22. Point(b_2) → B_1
  23. Polygon(T, U, V, K, L, J, I, S) → poly10, t_2, u_1, v_1, k_4, l_2, j_3, i_5, s_2
  24. Vector(T, B_1) → w
  25. Translate(T, w) → T'
  26. Translate(U, w) → U'
  27. Translate(V, w) → V'
  28. Translate(K, w) → K'
  29. Translate(L, w) → L'
  30. Translate(J, w) → J'
  31. Translate(I, w) → I'
  32. Translate(S, w) → S'
  33. Polygon(T', U', V', K', L', J', I', S') → poly10', t', u', v', k', l', j', i', s'
  34. Vector(B_1, U') → z
  35. Translate(T', z) → T''
  36. Translate(U', z) → U''
  37. Translate(V', z) → V''
  38. Translate(K', z) → K''
  39. Translate(L', z) → L''
  40. Translate(J', z) → J''
  41. Translate(I', z) → I''
  42. Translate(S', z) → S''
  43. Polygon(T'', U'', V'', K'', L'', J'', I'', S'') → poly10'', t'', u'', v'', k'', l'', j'', i'', s''
  44. Translate(T'', z) → T'''
  45. Translate(U'', z) → U'''
  46. Translate(V'', z) → V'''
  47. Translate(K'', z) → K'''
  48. Translate(L'', z) → L'''
  49. Translate(J'', z) → J'''
  50. Translate(I'', z) → I'''
  51. Translate(S'', z) → S'''
  52. Polygon(T''', U''', V''', K''', L''', J''', I''', S''') → poly10''', t''', u''', v''', k''', l''', j''', i''', s'''
  53. Translate(T''', z) → C_1
  54. Translate(U''', z) → D_1
  55. Translate(V''', z) → E_1
  56. Translate(K''', z) → F_1
  57. Translate(L''', z) → G_1
  58. Translate(J''', z) → H_1
  59. Translate(I''', z) → I_1
  60. Translate(S''', z) → J_1
  61. Polygon(C_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1) → poly11, c_3, d_2, e_2, f_2, g_2, h_2, i_6, j_4
  62. Segment(J', G_1) → p_2
  63. Segment(V''', I_1) → q_2
  64. Segment(I_1, H_1) → r_2
  65. Segment(J', H_1) → a_3
  66. Point(b_2) → K_1
  67. Circle(K_1, 3) → d_3
  68. Intersect(d_3, g_1, 2) → L_1
  69. Line(K_1, L_1) → b_3
  70. Circle(K_1, 0.8660254037844386) → e_3
  71. OrthogonalLine(K_1, b_3) → f_3
  72. Intersect(e_3, f_3, 2) → M_1
  73. Line(M_1, b_3) → g_3
  74. Circle(K_1, 1) → h_3
  75. Circle(K_1, 2) → p_3
  76. Point(b_3) → N_1
  77. Circle(N_1, 1) → q_3
  78. Circle(N_1, 2) → r_3
  79. Intersect(r_3, b_3, 2) → O_1
  80. Intersect(q_3, g_3, 2) → P_1
  81. Circle(O_1, 1) → s_3
  82. Intersect(s_3, g_3, 1) → Q_1
  83. Polygon(N_1, P_1, Q_1, O_1) → poly12, n_4, p_4, q_4, o_1
  84. Midpoint(q_4) → R_1
  85. Rotate(N_1, 180°, R_1) → S_1
  86. Rotate(P_1, 180°, R_1) → T_1
  87. Rotate(Q_1, 180°, R_1) → U_1
  88. Rotate(O_1, 180°, R_1) → V_1
  89. Polygon(S_1, T_1, U_1, V_1) → poly12', s_4, t_3, u_2, v_2
  90. Midpoint(n_4) → W_1
  91. Midpoint(s_4) → Z_1
  92. Rotate(N_1, 180°, W_1) → A_2
  93. Rotate(P_1, 180°, W_1) → B_2
  94. Rotate(Q_1, 180°, W_1) → C_2
  95. Rotate(O_1, 180°, W_1) → D_2
  96. Polygon(A_2, B_2, C_2, D_2) → poly12'_1, a_4, b_4, c_4, d_4
  97. Rotate(S_1, 180°, Z_1) → E_2
  98. Rotate(T_1, 180°, Z_1) → F_2
  99. Rotate(U_1, 180°, Z_1) → G_2
  100. Rotate(V_1, 180°, Z_1) → H_2
  101. Polygon(E_2, F_2, G_2, H_2) → poly12'', e_4, f_4, g_4, h_4
  102. Midpoint(c_4) → I_2
  103. Midpoint(g_4) → J_2
  104. Rotate(E_2, 180°, J_2) → K_2
  105. Rotate(F_2, 180°, J_2) → L_2
  106. Rotate(G_2, 180°, J_2) → M_2
  107. Rotate(H_2, 180°, J_2) → N_2
  108. Polygon(K_2, L_2, M_2, N_2) → poly12''', k_5, l_3, m_4, n_5
  109. Rotate(A_2, 180°, I_2) → O_2
  110. Rotate(B_2, 180°, I_2) → P_2
  111. Rotate(C_2, 180°, I_2) → Q_2
  112. Rotate(D_2, 180°, I_2) → R_2
  113. Polygon(O_2, P_2, Q_2, R_2) → poly13, o_2, p_5, q_5, r_4
  114. Intersect(b_2, g_3) → S_2
  115. Intersect(g_1, g_3) → T_2
  116. Intersect(b_3, g_1) → U_2
  117. Polygon(K_1, S_2, T_2, U_2) → q1, l_4, t_4, f_5, g_5

AI Drawing Prompt Reference

Visualize the dynamic transformation process of a geometric figure, starting with a circle and a line segment, constructing two polygons through their intersection points, then using similarity transformations and orthogonal lines to build multiple small quadrilaterals, combining these to form a complex shape, and finally demonstrating the symmetry and combination relationships of the figure through translation operations, embodying concepts of geometric construction, transformation, and integral operations.
The prompt above was compiled from the original drawing commands and is for reference only.