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使用Varignon图案着色的立方体

使用Varignon图案着色的立方体
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2020-01-09 00:08
多边形立体几何
所属专题 Color 3D surfaces
①这个范例演示了一个使用Varignon图案着色的立方体；②涉及的核心数学概念包括立体几何、多边形、积分和几何作图；③作图要点使用了GeoGebra命令如点、立方体、线、交点、多边形、中点和质心等；④适合高中及以上学生学习和理解复杂几何构造。

GeoGebra 命令及参数

  1. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  2. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, arêteAD, arêteAB, arêteBC, arêteCD, arêteAE, arêteDH, arêteEH, arêteEF, E, arêteBF, arêteFG, arêteCG, arêteGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  3. PointIn(faceABFE) → I
  4. Line(I, arêteEF) → f
  5. Line(I, arêteAE) → g
  6. Intersect(g, arêteEF) → J
  7. Intersect(arêteAB, g) → K
  8. Intersect(f, arêteAE) → L
  9. Intersect(arêteBF, f) → M
  10. Polygon(E, J, I, L) → q1, e, j, i, l
  11. Midpoint(E, J) → N
  12. Midpoint(I, L) → O
  13. Midpoint(J, I) → P
  14. Midpoint(L, E) → Q
  15. Polygon(N, Q, O, P) → q2, n, q, o, p
  16. Midpoint(n) → R
  17. Midpoint(p) → S
  18. Midpoint(o) → T
  19. Midpoint(q) → U
  20. Polygon(R, S, T, U) → q3, r, s, t, u
  21. Midpoint(r) → V
  22. Midpoint(u) → W
  23. Midpoint(s) → Z
  24. Midpoint(t) → A_1
  25. Polygon(V, W, A_1, Z) → q4, v, w, a_1, z_1
  26. Midpoint(N, E) → B_1
  27. Midpoint(N, J) → C_1
  28. Midpoint(J, P) → D_1
  29. Midpoint(P, I) → E_1
  30. Midpoint(I, O) → F_1
  31. Midpoint(O, L) → G_1
  32. Midpoint(L, Q) → H_1
  33. Midpoint(Q, E) → I_1
  34. Polygon(B_1, I_1, H_1, G_1, F_1, E_1, D_1, C_1) → poly1, b_1, i_1, h_1, g_1, f_1, e_1, d_1, c_1
  35. Polygon(J, F, M, I) → q5, j_1, f_2, m, i_2
  36. Midpoint(J, F) → J_1
  37. Midpoint(F, M) → K_1
  38. Midpoint(M, I) → L_1
  39. Midpoint(J_1, J) → M_1
  40. Midpoint(J_1, F) → N_1
  41. Midpoint(F, K_1) → O_1
  42. Midpoint(K_1, M) → P_1
  43. Midpoint(L_1, M) → Q_1
  44. Midpoint(L_1, I) → R_1
  45. Polygon(N_1, O_1, P_1, Q_1, R_1, E_1, D_1, M_1) → poly2, n_1, o_1, p_1, q_1, r_1, h, k, m_1
  46. Polygon(J_1, P, L_1, K_1) → q6, b, p_2, l_1, k_1
  47. Centroid(q6) → S_1
  48. Midpoint(b) → T_1
  49. Midpoint(k_1) → U_1
  50. Midpoint(l_1) → V_1
  51. Midpoint(p_2) → W_1
  52. Polygon(T_1, U_1, V_1, W_1) → q7, t_1, u_1, v_1, w_1
  53. Midpoint(t_1) → Z_1
  54. Midpoint(u_1) → A_2
  55. Midpoint(v_1) → B_2
  56. Midpoint(w_1) → C_2
  57. Polygon(Z_1, C_2, B_2, A_2) → q8, c, c_2, b_2, a_2
  58. Polygon(L, I, K, A) → q9, l_2, i_3, k_2, a_3
  59. Midpoint(L, A) → D_2
  60. Midpoint(I, K) → E_2
  61. Midpoint(A, K) → F_2
  62. Midpoint(L, D_2) → G_2
  63. Midpoint(D_2, A) → H_2
  64. Midpoint(A, F_2) → I_2
  65. Midpoint(F_2, K) → J_2
  66. Midpoint(E_2, I) → K_2
  67. Midpoint(E_2, K) → L_2
  68. Polygon(G_2, H_2, I_2, J_2, L_2, K_2, F_1, G_1) → poly3, g_2, h_2, d, j_2, s_1, m_2, n_2, q_2
  69. Polygon(O, D_2, F_2, E_2) → q10, o_2, d_2, r_2, e_2
  70. Midpoint(e_2) → M_2
  71. Midpoint(o_2) → N_2
  72. Midpoint(d_2) → O_2
  73. Midpoint(r_2) → P_2
  74. Polygon(N_2, M_2, P_2, O_2) → q11, s_2, t_2, f_3, g_3
  75. Midpoint(s_2) → Q_2
  76. Midpoint(g_3) → R_2
  77. Midpoint(f_3) → S_2
  78. Midpoint(t_2) → T_2
  79. Polygon(Q_2, R_2, S_2, T_2) → q12, h_3, j_3, k_3, l_3
  80. Polygon(I, M, B, K) → q13, i_4, m_3, b_3, k_4
  81. Midpoint(M, B) → U_2
  82. Midpoint(K, B) → V_2
  83. Midpoint(M, U_2) → W_2
  84. Midpoint(B, U_2) → Z_2
  85. Midpoint(V_2, B) → A_3
  86. Midpoint(V_2, K) → B_3
  87. Polygon(R_1, K_2, L_2, B_3, A_3, Z_2, W_2, Q_1) → poly4, n_3, p_3, q_3, r_3, s_3, z_2, w_2, t_3
  88. Polygon(L_1, E_2, V_2, U_2) → q14, c_3, d_3, v_2, u_2
  89. Midpoint(u_2) → C_3
  90. Midpoint(c_3) → D_3
  91. Midpoint(d_3) → E_3
  92. Midpoint(v_2) → F_3
  93. Polygon(C_3, F_3, E_3, D_3) → q15, e_3, f_4, g_4, h_4
  94. Midpoint(h_4) → G_3
  95. Midpoint(e_3) → H_3
  96. Midpoint(f_4) → I_3
  97. Midpoint(g_4) → J_3
  98. Polygon(G_3, J_3, I_3, H_3) → q16, j_4, l_4, m_4, n_4
  99. Midpoint(C, A) → K_3
  100. Line(K_3, zAxis) → p_4
  101. Sequence(Rotate[q1, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l1
  102. Sequence(Rotate[q5, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l2
  103. Sequence(Rotate[q9, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l3
  104. Midpoint(E, C) → L_3
  105. Midpoint(E, D) → M_3
  106. Line(M_3, L_3) → q_4
  107. Sequence(Rotate[q13, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l4
  108. Sequence(Rotate[poly1, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l5
  109. Sequence(Rotate[poly2, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l6
  110. Sequence(Rotate[poly3, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l7
  111. Sequence(Rotate[poly4, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l8
  112. Sequence(Rotate[q2, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l9
  113. Sequence(Rotate[q3, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l10
  114. Sequence(Rotate[q4, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l11
  115. Sequence(Rotate[q6, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l12
  116. Sequence(Rotate[q7, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l13
  117. Sequence(Rotate[q8, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l14
  118. Sequence(Rotate[q10, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l15
  119. Sequence(Rotate[q11, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l16
  120. Sequence(Rotate[q12, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l17
  121. Sequence(Rotate[q14, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l18
  122. Sequence(Rotate[q15, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l19
  123. Sequence(Rotate[q16, i, p_4], i, 0, (2 * π), (2 * π / 4)) → l20
  124. Midpoint(H, C) → N_3
  125. Sequence(Rotate[q1, i, q_4], i, π / 2, (3 * π / 2), (4 * π / 4)) → l21
  126. Sequence(Rotate[q5, i, q_4], i, π / 2, (3 * π / 2), (4 * π / 4)) → l22
  127. Sequence(Rotate[q9, i, q_4], i, π / 2, (3 * π / 2), (4 * π / 4)) → l23
  128. Sequence(Rotate[q13, i, q_4], i, π / 2, (3 * π / 2), (4 * π / 4)) → l24
  129. Sequence(Rotate[poly1, i, q_4], i, π / 2, (3 * π / 2), (4 * π / 4)) → l25
  130. Sequence(Rotate[poly2, i, q_4], i, π / 2, (3 * π / 2), (4 * π / 4)) → l26
  131. Sequence(Rotate[poly3, i, q_4], i, π / 2, (3 * π / 2), (4 * π / 4)) → l27
  132. Sequence(Rotate[poly4, i, q_4], i, π / 2, (3 * π / 2), (4 * π / 4)) → l28
  133. Sequence(Rotate[q2, i, q_4], i, π / 2, (3 * π / 2), π) → l29
  134. Sequence(Rotate[q3, i, q_4], i, π / 2, (3 * π / 2), π) → l30
  135. Sequence(Rotate[q4, i, q_4], i, π / 2, (3 * π / 2), π) → l31
  136. Sequence(Rotate[q6, i, q_4], i, π / 2, (3 * π / 2), π) → l32
  137. Sequence(Rotate[q7, i, q_4], i, π / 2, (3 * π / 2), π) → l33
  138. Sequence(Rotate[q8, i, q_4], i, π / 2, (3 * π / 2), π) → l34
  139. Sequence(Rotate[q10, i, q_4], i, π / 2, (3 * π / 2), π) → l35
  140. Sequence(Rotate[q11, i, q_4], i, π / 2, (3 * π / 2), π) → l36
  141. Sequence(Rotate[q12, i, q_4], i, π / 2, (3 * π / 2), π) → l37
  142. Sequence(Rotate[q14, i, q_4], i, π / 2, (3 * π / 2), π) → l38
  143. Sequence(Rotate[q15, i, q_4], i, π / 2, (3 * π / 2), π) → l39
  144. Sequence(Rotate[q16, i, q_4], i, π / 2, (3 * π / 2), π) → l40
  145. Polygon(D, C, B, A) → q17, d_4, c_4, b_4, a_4