三角形能否密铺？——中点旋转法探索

GeoGebra 命令及参数

1. Polygon(A, B, C) → poly1, c, a, b
2. Midpoint(B, C) → D
3. Midpoint(C, A) → E
4. Midpoint(A, B) → F
5. Rotate(A, rot, F) → A'
6. Rotate(B, rot, F) → B'
7. Rotate(C, rot, F) → C'
8. Polygon(A', B', C') → poly1', c', a', b'
9. Rotate(A, rot, D) → A'_1
10. Rotate(B, rot, D) → B'_1
11. Rotate(C, rot, D) → C'_1
12. Polygon(A'_1, B'_1, C'_1) → poly1'_1, c'_1, a'_1, b'_1
13. Rotate(A, rot, E) → A'_2
14. Rotate(B, rot, E) → B'_2
15. Rotate(C, rot, E) → C'_2
16. Polygon(A'_2, B'_2, C'_2) → poly1'_2, c'_2, a'_2, b'_2
17. Rotate(A, 180*°, F) → A'_3
18. Rotate(B, 180*°, F) → B'_3
19. Rotate(C, 180*°, F) → C'_3
20. Polygon(A'_3, B'_3, C'_3) → poly1'_3, c'_3, a'_3, b'_3
21. Rotate(A, 180*°, D) → A'_4
22. Rotate(B, 180*°, D) → B'_4
23. Rotate(C, 180*°, D) → C'_4
24. Polygon(A'_4, B'_4, C'_4) → poly1'_4, c'_4, a'_4, b'_4
25. Rotate(A, 180*°, E) → A'_5
26. Rotate(B, 180*°, E) → B'_5
27. Rotate(C, 180*°, E) → C'_5
28. Polygon(A'_5, B'_5, C'_5) → poly1'_5, c'_5, a'_5, b'_5
29. Midpoint(C'_3, B) → G
30. Midpoint(B, A'_4) → H
31. Midpoint(A'_4, C) → I
32. Midpoint(C, B'_5) → J
33. Midpoint(B'_5, A) → K
34. Midpoint(C'_3, A) → L
35. Rotate(A'_3, rot, G) → M
36. Rotate(B'_3, rot, G) → N
37. Rotate(C'_3, rot, G) → O
38. Polygon(M, N, O) → poly2, o, m, n
39. Rotate(A'_3, rot, L) → P
40. Rotate(B'_3, rot, L) → Q
41. Rotate(C'_3, rot, L) → R
42. Polygon(P, Q, R) → poly3, r, p, q
43. Rotate(A'_4, rot, H) → S
44. Rotate(B'_4, rot, H) → T
45. Rotate(C'_4, rot, H) → U
46. Polygon(S, T, U) → poly4, u, s, t
47. Rotate(A'_4, rot, I) → V
48. Rotate(B'_4, rot, I) → W
49. Rotate(C'_4, rot, I) → Z
50. Polygon(V, W, Z) → poly5, z, v, w
51. Rotate(A'_5, rot, K) → A_1
52. Rotate(B'_5, rot, K) → B_1
53. Rotate(C'_5, rot, K) → C_1
54. Polygon(A_1, B_1, C_1) → poly6, c_1, a_1, b_1
55. Rotate(A'_5, rot, J) → D_1
56. Rotate(B'_5, rot, J) → E_1
57. Rotate(C'_5, rot, J) → F_1
58. Polygon(D_1, E_1, F_1) → poly7, f_1, d_1, e_1
59. Angle(C, A, B) → α
60. Angle(A, B, C) → β
61. Angle(B, C, A) → γ
62. Midpoint(a'_3) → G_1
63. Midpoint(b'_3) → H_1
64. Midpoint(b'_4) → I_1
65. Midpoint(c'_4) → J_1
66. Midpoint(c'_5) → K_1
67. Midpoint(a'_5) → L_1
68. Rotate(A'_3, 180*°, H_1) → M_1
69. Rotate(B'_3, 180*°, H_1) → N_1
70. Rotate(C'_3, 180*°, H_1) → O_1
71. Polygon(M_1, N_1, O_1) → poly8, o_1, m_1, n_1
72. Rotate(A'_4, 180*°, I_1) → P_1
73. Rotate(B'_4, 180*°, I_1) → Q_1
74. Rotate(C'_4, 180*°, I_1) → R_1
75. Polygon(P_1, Q_1, R_1) → poly9, r_1, p_1, q_1
76. Rotate(A'_4, 180*°, J_1) → S_1
77. Rotate(B'_4, 180*°, J_1) → T_1
78. Rotate(C'_4, 180*°, J_1) → U_1
79. Polygon(S_1, T_1, U_1) → poly10, u_1, s_1, t_1
80. Rotate(A'_5, 180*°, K_1) → V_1
81. Rotate(B'_5, 180*°, K_1) → W_1
82. Rotate(C'_5, 180*°, K_1) → Z_1
83. Polygon(V_1, W_1, Z_1) → poly11, z_1, v_1, w_1
84. Rotate(A'_5, 180*°, L_1) → A_2
85. Rotate(B'_5, 180*°, L_1) → B_2
86. Rotate(C'_5, 180*°, L_1) → C_2
87. Polygon(A_2, B_2, C_2) → poly12, c_2, a_2, b_2
88. Rotate(A'_3, 180*°, G_1) → D_2
89. Rotate(B'_3, 180*°, G_1) → E_2
90. Rotate(C'_3, 180*°, G_1) → F_2
91. Polygon(D_2, E_2, F_2) → poly13, f_2, d_2, e_2
92. Midpoint(t_1) → G_2
93. Midpoint(r_1) → H_2
94. Rotate(P_1, 180*°, H_2) → I_2
95. Rotate(Q_1, 180*°, H_2) → J_2
96. Rotate(R_1, 180*°, H_2) → K_2
97. Polygon(I_2, J_2, K_2) → poly9', k_2, i_2, j_2
98. Rotate(S_1, 180*°, G_2) → L_2
99. Rotate(T_1, 180*°, G_2) → M_2
100. Rotate(U_1, 180*°, G_2) → N_2
101. Polygon(L_2, M_2, N_2) → poly10', n_2, l_2, m_2
102. Midpoint(a) → O_2
103. Midpoint(m_1) → P_2
104. Midpoint(c) → Q_2
105. Angle(I_1, B, H_2) → δ
106. Angle(H_1, B, Q_2) → ε
107. Angle(H_2, B, P_2) → ζ
108. Angle(P_2, B, H_1) → η
109. Angle(O_2, B, I_1) → θ