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Four Color Theorem Visualization in GeoGebra

Four Color Theorem Visualization in GeoGebra
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2018-07-03 03:58
PolygonTransformationOptimizationGeometric ConstructionGraphing
In collections The Best Math Book in the World
This GeoGebra example demonstrates the Four Color Theorem, which states that any planar map can be colored with at most four colors such that no adjacent regions share the same color. The core mathematical concepts involved include planar graph coloring, region adjacency relations, and the combinatorial optimization of the minimum number of colors needed. The construction uses GeoGebra commands such as Polygon to create map regions, Point and Line to define boundaries, Mirror and Intersect to generate symmetric or complex adjacency structures, and Sum, Min, KeepIf, and Length to handle color assignment and verification. The object list includes numerous vertices (A through S), edges (a through i_1), and auxiliary points (p01, p02, etc.), forming a detailed interactive model. This example is suitable for high school and above students or math enthusiasts, serving as a visual aid for understanding the geometric formulation of the Four Color Theorem, and can be used in introductory graph theory or discrete mathematics courses to explore how region adjacencies relate to color allocation strategies.

GeoGebra Commands & Parameters

  1. Polygon(A, D, C, B) → p01, a, d, c, b
  2. Polygon(I, H, E, F, G, A) → p02, i, h, e, f, g, a_1
  3. Point(i) → P
  4. Point(d) → S
  5. Polygon(P, Q, R, S, D, I) → p99, p, q, r, s, d_1, i_1
  6. Point(e) → N
  7. Polygon(N, O, M, L) → p05, n_{20}, o, m, l
  8. Polygon(N, O, J, K, H, E) → p04, n_1, o_1, j, k, h_1, e_1
  9. Point(k) → T
  10. Point(q) → W
  11. Polygon(P, T, U, V, W, Q) → p104, p_1, t, u, v, w, q_1
  12. Point(o) → Z
  13. Point(o_1) → B_1
  14. Polygon(Z, A_1, C_1, B_1) → p08, z_1, a_2, c_1, b_1
  15. Polygon(B_1, C_1, D_1, E_1, K, J) → p07, b_2, c_2, d_2, e_2, k_1, j_1
  16. Point(e_2) → F_1
  17. Point(u) → G_1
  18. Polygon(T, F_1, H_1, I_1, G_1, U) → p107, t_1, f_1, h_2, i_2, g_1, u_1
  19. Polygon(W, V, J_1, K_1) → p14, w_1, v_1, j_2, k_2
  20. Point(v_1) → M_1
  21. Polygon(G_1, I_1, L_1, M_1) → p15, g_2, i_3, l_1, m_1
  22. Point(a_2) → Q_1
  23. Polygon(N_1, Q_1, P_1, O_1) → p13, n_2, q_2, p_2, o_2
  24. Point(c_2) → R_1
  25. Polygon(R_1, S_1, P_1, Q_1) → p12, r_1, s_1, p_3, q_3
  26. Polygon(R_1, S_1, T_1, U_1, E_1, D_1) → p11, r_2, s_2, t_2, u_2, e_3, d_3
  27. Point(p_2) → V_1
  28. Polygon(V_1, W_1, Z_1, A_2) → p18, v_2, w_2, z_2, a_3
  29. Point(s_2) → B_2
  30. Polygon(B_2, C_2, Z_1, A_2) → p20, b_3, c_3, z_3, a_4
  31. Polygon(B_2, C_2, D_2, E_2, U_1, T_1) → p83, b_4, c_4, d_4, e_4, u_3, t_3
  32. Point(w_2) → F_2
  33. Polygon(I_2, H_2, G_2, F_2) → p22, i_4, h_3, g_3, f_2
  34. Point(c_3) → N_2
  35. Polygon(F_2, G_2, J_2, N_2) → p23, f_3, g_4, j_3, n_3
  36. Point(c_4) → O_2
  37. Polygon(N_2, J_2, K_2, O_2) → p25, n_4, j_4, k_3, o_3
  38. Polygon(O_2, K_2, L_2, M_2, E_2, D_2) → p85, o_4, k_4, l_2, m_2, e_5, d_5
  39. Point(h_3) → P_2
  40. Point(g_4) → S_2
  41. Polygon(P_2, Q_2, R_2, S_2) → p27, p_4, q_4, r_3, s_3
  42. Point(j_4) → U_2
  43. Polygon(S_2, R_2, T_2, U_2) → p29, s_4, r_4, t_4, u_4
  44. Point(k_4) → W_2
  45. Polygon(U_2, T_2, V_2, W_2) → p31, u_5, t_5, v_3, w_3
  46. Polygon(W_2, V_2, Z_2, A_3, M_2, L_2) → p87, w_4, v_4, z_4, a_5, m_3, l_3
  47. Point(q_4) → B_3
  48. Polygon(B_3, C_3, D_3, E_3) → p33, b_5, c_5, d_6, e_6
  49. Point(r_4) → F_3
  50. Polygon(F_3, G_3, C_3, B_3) → p34, f_4, g_5, c_6, b_6
  51. Point(t_5) → H_3
  52. Polygon(H_3, I_3, G_3, F_3) → p36, h_4, i_5, g_6, f_5
  53. Point(v_4) → J_3
  54. Polygon(J_3, K_3, I_3, H_3) → p38, j_5, k_5, i_6, h_5
  55. Polygon(J_3, K_3, L_3, M_3, A_3, Z_2) → p89, j_6, k_6, l_4, m_4, a_6, z_5
  56. Point(c_5) → N_3
  57. Point(g_5) → Q_3
  58. Polygon(N_3, O_3, P_3, Q_3) → p40, n_5, o_5, p_5, q_5
  59. Point(i_5) → S_3
  60. Polygon(Q_3, P_3, R_3, S_3) → p42, q_6, p_6, r_5, s_5
  61. Point(k_5) → U_3
  62. Polygon(S_3, R_3, T_3, U_3) → p44, s_6, r_6, t_6, u_6
  63. Point(k_6) → W_3
  64. Polygon(U_3, T_3, V_3, W_3) → p46, u_7, t_7, v_5, w_5
  65. Line(G, S) → f_6
  66. Mirror(W_3, f_6) → B_4
  67. Mirror(V_3, f_6) → C_4
  68. Mirror(Z_3, f_6) → D_4
  69. Mirror(M_3, f_6) → F_4
  70. Mirror(L_3, f_6) → G_4
  71. Mirror(U_3, f_6) → H_4
  72. Mirror(T_3, f_6) → I_4
  73. Mirror(V_3, f_6) → J_4
  74. Mirror(W_3, f_6) → K_4
  75. Polygon(H_4, I_4, J_4, K_4) → p47, h_6, i_7, j_7, k_7
  76. Mirror(S_3, f_6) → L_4
  77. Mirror(R_3, f_6) → M_4
  78. Mirror(T_3, f_6) → N_4
  79. Mirror(U_3, f_6) → O_4
  80. Polygon(L_4, M_4, N_4, O_4) → p45, l_6, m_6, n_6, o_6
  81. Mirror(Q_3, f_6) → P_4
  82. Mirror(P_3, f_6) → Q_4
  83. Mirror(R_3, f_6) → R_4
  84. Mirror(S_3, f_6) → S_4
  85. Polygon(P_4, Q_4, R_4, S_4) → p43, p_7, q_7, r_7, s_7
  86. Mirror(N_3, f_6) → T_4
  87. Mirror(O_3, f_6) → U_4
  88. Mirror(P_3, f_6) → V_4
  89. Mirror(Q_3, f_6) → W_4
  90. Polygon(T_4, U_4, V_4, W_4) → p41, t_8, u_8, v_7, w_7
  91. Mirror(J_3, f_6) → Z_4
  92. Mirror(K_3, f_6) → A_5
  93. Mirror(L_3, f_6) → B_5
  94. Mirror(M_3, f_6) → C_5
  95. Mirror(A_3, f_6) → D_5
  96. Mirror(Z_2, f_6) → E_5
  97. Polygon(Z_4, A_5, B_5, C_5, D_5, E_5) → p90, z_7, a_8, b_8, c_8, d_8, e_8
  98. Mirror(J_3, f_6) → F_5
  99. Mirror(K_3, f_6) → G_5
  100. Mirror(I_3, f_6) → H_5
  101. Mirror(H_3, f_6) → I_5
  102. Polygon(F_5, G_5, H_5, I_5) → p39, f_8, g_8, h_7, i_8
  103. Mirror(S_3, f_6) → J_5
  104. Mirror(H_3, f_6) → K_5
  105. Mirror(I_3, f_6) → L_5
  106. Mirror(G_3, f_6) → M_5
  107. Mirror(F_3, f_6) → N_5
  108. Polygon(K_5, L_5, M_5, N_5) → p37, k_8, l_7, m_7, n_7
  109. Mirror(F_3, f_6) → O_5
  110. Mirror(G_3, f_6) → P_5
  111. Mirror(C_3, f_6) → Q_5
  112. Mirror(B_3, f_6) → R_5
  113. Polygon(O_5, P_5, Q_5, R_5) → p35, o_7, p_8, q_8, r_8
  114. Mirror(W_2, f_6) → S_5
  115. Mirror(V_2, f_6) → T_5
  116. Mirror(Z_2, f_6) → U_5
  117. Mirror(A_3, f_6) → V_5
  118. Mirror(M_2, f_6) → W_5
  119. Mirror(L_2, f_6) → Z_5
  120. Polygon(S_5, T_5, U_5, V_5, W_5, Z_5) → p88, s_8, t_9, u_9, v_8, w_8, z_8
  121. Mirror(U_2, f_6) → A_6
  122. Mirror(T_2, f_6) → B_6
  123. Mirror(V_2, f_6) → C_6
  124. Mirror(W_2, f_6) → D_6
  125. Polygon(A_6, B_6, C_6, D_6) → p32, a_9, b_9, c_9, d_9
  126. Mirror(S_2, f_6) → E_6
  127. Mirror(R_2, f_6) → F_6
  128. Mirror(T_2, f_6) → G_6
  129. Mirror(U_2, f_6) → H_6
  130. Polygon(E_6, F_6, G_6, H_6) → p30, e_9, f_9, g_9, h_8
  131. Mirror(P_2, f_6) → I_6
  132. Mirror(Q_2, f_6) → J_6
  133. Mirror(R_2, f_6) → K_6
  134. Mirror(S_2, f_6) → L_6
  135. Polygon(I_6, J_6, K_6, L_6) → p28, i_9, j_8, k_9, l_8
  136. Mirror(O_2, f_6) → M_6
  137. Mirror(K_2, f_6) → N_6
  138. Mirror(L_2, f_6) → O_6
  139. Mirror(M_2, f_6) → P_6
  140. Mirror(E_2, f_6) → Q_6
  141. Mirror(D_2, f_6) → R_6
  142. Polygon(M_6, N_6, O_6, P_6, Q_6, R_6) → p86, m_8, n_8, o_8, p_9, q_9, r_9
  143. Mirror(N_2, f_6) → S_6
  144. Mirror(J_2, f_6) → T_6
  145. Mirror(K_2, f_6) → U_6
  146. Mirror(O_2, f_6) → V_6
  147. Polygon(S_6, T_6, U_6, V_6) → p26, s_9, t_{10}, u_{10}, v_9
  148. Mirror(F_2, f_6) → W_6
  149. Mirror(G_2, f_6) → Z_6
  150. Mirror(J_2, f_6) → A_7
  151. Mirror(N_2, f_6) → B_7
  152. Polygon(W_6, Z_6, A_7, B_7) → p24, w_9, z_9, a_{10}, b_{10}
  153. Mirror(B_2, f_6) → C_7
  154. Mirror(C_2, f_6) → D_7
  155. Mirror(D_2, f_6) → E_7
  156. Mirror(E_2, f_6) → F_7
  157. Mirror(U_1, f_6) → G_7
  158. Mirror(T_1, f_6) → H_7
  159. Polygon(C_7, D_7, E_7, F_7, G_7, H_7) → p84, c_{10}, d_{10}, e_{10}, f_{10}, g_{10}, h_9
  160. Mirror(B_2, f_6) → I_7
  161. Mirror(C_2, f_6) → J_7
  162. Mirror(Z_1, f_6) → K_7
  163. Mirror(A_2, f_6) → L_7
  164. Polygon(I_7, J_7, K_7, L_7) → p21, i_{10}, j_9, k_{10}, l_9
  165. Mirror(V_1, f_6) → M_7
  166. Mirror(W_1, f_6) → N_7
  167. Mirror(Z_1, f_6) → O_7
  168. Mirror(A_2, f_6) → P_7
  169. Polygon(M_7, N_7, O_7, P_7) → p19, m_9, n_9, o_9, p_{10}
  170. Mirror(R_1, f_6) → Q_7
  171. Mirror(S_1, f_6) → R_7
  172. Mirror(T_1, f_6) → S_7
  173. Mirror(U_1, f_6) → T_7
  174. Mirror(E_1, f_6) → U_7
  175. Mirror(D_1, f_6) → V_7
  176. Polygon(Q_7, R_7, S_7, T_7, U_7, V_7) → p82, q_{10}, r_{10}, s_{10}, t_{11}, u_{11}, v_{10}
  177. Mirror(R_1, f_6) → W_7
  178. Mirror(S_1, f_6) → Z_7
  179. Mirror(P_1, f_6) → A_8
  180. Mirror(Q_1, f_6) → B_8
  181. Polygon(W_7, Z_7, A_8, B_8) → p17, w_{10}, z_{10}, a_{11}, b_{11}
  182. Mirror(B_1, f_6) → C_8
  183. Mirror(C_1, f_6) → D_8
  184. Mirror(D_1, f_6) → E_8
  185. Mirror(E_1, f_6) → F_8
  186. Mirror(K, f_6) → K'
  187. Mirror(J, f_6) → J'
  188. Polygon(C_8, D_8, E_8, F_8, K', J') → p10, c_{11}, d_{11}, e_{11}, f_{11}, k', j'
  189. Mirror(Z, f_6) → Z'
  190. Mirror(A_1, f_6) → G_8
  191. Mirror(C_1, f_6) → H_8
  192. Mirror(B_1, f_6) → I_8
  193. Polygon(Z', G_8, H_8, I_8) → p09, z', g_{11}, h_{10}, i_{11}
  194. Mirror(N, f_6) → N'
  195. Mirror(O, f_6) → O'
  196. Mirror(J, f_6) → J'_1
  197. Mirror(K, f_6) → K'_1
  198. Mirror(H, f_6) → H'
  199. Mirror(E, f_6) → E'
  200. Polygon(N', O', J'_1, K'_1, H', E') → p06, n', o', j'_1, k'_1, h', e'
  201. Mirror(I, f_6) → I'
  202. Mirror(H, f_6) → H'_1
  203. Mirror(E, f_6) → E'_1
  204. Mirror(F, f_6) → F'
  205. Mirror(G, f_6) → G'
  206. Mirror(A, f_6) → A'
  207. Polygon(I', H'_1, E'_1, F', G', A') → p03, i', h'_1, e'_1, f', g', a'
  208. Point(u_2) → J_8
  209. Point(h_2) → M_8
  210. Polygon(J_8, K_8, L_8, M_8, H_1, F_1) → p91, j_{10}, k_{11}, l_{10}, m_{10}, h_{11}, f_{12}
  211. Point(i_3) → O_8
  212. Polygon(L_8, N_8, O_8, M_8) → p48, l_{11}, n_{10}, o_{10}, m_{11}
  213. Polygon(O_8, N_8, P_8, Q_8) → p50, o_{11}, n_{11}, p_{11}, q_{11}
  214. Point(e_4) → R_8
  215. Point(k_{11}) → U_8
  216. Polygon(R_8, S_8, T_8, U_8, K_8, J_8) → p93, r_{11}, s_{11}, t_{12}, u_{12}, k_{12}, j_{11}
  217. Point(l_{11}) → W_8
  218. Polygon(U_8, T_8, V_8, W_8) → p51, u_{13}, t_{13}, v_{11}, w_{11}
  219. Point(f_6) → Z_8
  220. Intersect(f_6, n_{11}) → A_9
  221. Polygon(W_8, V_8, Z_8, A_9) → p53, w_{12}, v_{12}, z_{11}, a_{12}
  222. Point(m_2) → B_9
  223. Point(s_{11}) → E_9
  224. Polygon(B_9, C_9, D_9, E_9, S_8, R_8) → p95, b_{12}, c_{12}, d_{12}, e_{12}, s_{12}, r_{12}
  225. Point(t_{13}) → G_9
  226. Polygon(E_9, D_9, F_9, G_9) → p55, e_{13}, d_{13}, f_{13}, g_{12}
  227. Point(v_{12}) → I_9
  228. Polygon(G_9, F_9, H_9, I_9) → p57, g_{13}, f_{14}, h_{12}, i_{12}
  229. Polygon(I_9, H_9, J_9, K_9) → p59, i_{13}, h_{13}, j_{12}, k_{13}
  230. Point(a_5) → L_9
  231. Point(c_{12}) → O_9
  232. Polygon(L_9, M_9, N_9, O_9, C_9, B_9) → p97, l_{12}, m_{12}, n_{12}, o_{12}, c_{13}, b_{13}
  233. Point(d_{13}) → Q_9
  234. Polygon(O_9, N_9, P_9, Q_9) → p60, o_{13}, n_{13}, p_{12}, q_{12}
  235. Point(f_{14}) → S_9
  236. Polygon(Q_9, P_9, R_9, S_9) → p62, q_{13}, p_{13}, r_{13}, s_{13}
  237. Point(f_6) → T_9
  238. Intersect(f_6, h_{13}) → U_9
  239. Polygon(S_9, R_9, T_9, U_9) → p64, s_{14}, r_{14}, t_{14}, u_{14}
  240. Point(m_4) → V_9
  241. Point(m_{12}) → A_{10}
  242. Polygon(V_9, W_9, Z_9, A_{10}, M_9, L_9) → p101, v_{13}, w_{13}, z_{12}, a_{13}, m_{13}, l_{13}
  243. Point(n_{13}) → C_{10}
  244. Polygon(A_{10}, Z_9, B_{10}, C_{10}) → p66, a_{14}, z_{13}, b_{14}, c_{14}
  245. Point(p_{13}) → E_{10}
  246. Polygon(C_{10}, B_{10}, D_{10}, E_{10}) → p68, c_{15}, b_{15}, d_{14}, e_{14}
  247. Point(r_{14}) → G_{10}
  248. Polygon(E_{10}, D_{10}, F_{10}, G_{10}) → p70, e_{15}, d_{15}, f_{15}, g_{14}
  249. Polygon(G_{10}, F_{10}, H_{10}, I_{10}) → p72, g_{15}, f_{16}, h_{14}, i_{14}
  250. Point(w_{13}) → L_{10}
  251. Polygon(M_3, J_{10}, K_{10}, L_{10}, W_9, V_9) → p103, m_{14}, j_{13}, k_{14}, l_{14}, w_{14}, v_{14}
  252. Point(z_{13}) → N_{10}
  253. Polygon(L_{10}, K_{10}, M_{10}, N_{10}) → p73, l_{15}, k_{15}, m_{15}, n_{14}
  254. Point(b_{15}) → P_{10}
  255. Polygon(N_{10}, M_{10}, O_{10}, P_{10}) → p75, n_{15}, m_{16}, o_{14}, p_{14}
  256. Point(d_{15}) → R_{10}
  257. Polygon(P_{10}, O_{10}, Q_{10}, R_{10}) → p77, p_{15}, o_{15}, q_{14}, r_{15}
  258. Point(f_6) → S_{10}
  259. Intersect(f_6, f_{16}) → T_{10}
  260. Polygon(R_{10}, Q_{10}, S_{10}, T_{10}) → p79, r_{16}, q_{15}, s_{15}, t_{15}
  261. Mirror(V_9, f_6) → U_{10}
  262. Mirror(W_9, f_6) → V_{10}
  263. Mirror(Z_9, f_6) → W_{10}
  264. Mirror(A_{10}, f_6) → Z_{10}
  265. Mirror(M_9, f_6) → A_{11}
  266. Mirror(L_9, f_6) → B_{11}
  267. Polygon(U_{10}, V_{10}, W_{10}, Z_{10}, A_{11}, B_{11}) → p102, u_{15}, v_{15}, w_{15}, z_{14}, a_{15}, b_{16}
  268. Mirror(A_{10}, f_6) → C_{11}
  269. Mirror(Z_9, f_6) → D_{11}
  270. Mirror(B_{10}, f_6) → E_{11}
  271. Mirror(C_{10}, f_6) → F_{11}
  272. Polygon(C_{11}, D_{11}, E_{11}, F_{11}) → p67, c_{16}, d_{16}, e_{16}, f_{17}
  273. Mirror(C_{10}, f_6) → G_{11}
  274. Mirror(B_{10}, f_6) → H_{11}
  275. Mirror(D_{10}, f_6) → I_{11}
  276. Mirror(E_{10}, f_6) → J_{11}
  277. Polygon(G_{11}, H_{11}, I_{11}, J_{11}) → p69, g_{16}, h_{15}, i_{15}, j_{14}
  278. Mirror(E_{10}, f_6) → K_{11}
  279. Mirror(D_{10}, f_6) → L_{11}
  280. Mirror(F_{10}, f_6) → M_{11}
  281. Mirror(G_{10}, f_6) → N_{11}
  282. Polygon(K_{11}, L_{11}, M_{11}, N_{11}) → p71, k_{16}, l_{16}, m_{17}, n_{16}
  283. Mirror(L_9, f_6) → O_{11}
  284. Mirror(M_9, f_6) → P_{11}
  285. Mirror(N_9, f_6) → Q_{11}
  286. Mirror(O_9, f_6) → R_{11}
  287. Mirror(C_9, f_6) → S_{11}
  288. Mirror(B_9, f_6) → T_{11}
  289. Polygon(O_{11}, P_{11}, Q_{11}, R_{11}, S_{11}, T_{11}) → p98, o_{16}, p_{16}, q_{16}, r_{17}, s_{16}, t_{16}
  290. Mirror(B_9, f_6) → U_{11}
  291. Mirror(C_9, f_6) → V_{11}
  292. Mirror(D_9, f_6) → W_{11}
  293. Mirror(E_9, f_6) → Z_{11}
  294. Mirror(S_8, f_6) → A_{12}
  295. Mirror(R_8, f_6) → B_{12}
  296. Polygon(U_{11}, V_{11}, W_{11}, Z_{11}, A_{12}, B_{12}) → p96, u_{16}, v_{16}, w_{16}, z_{15}, a_{16}, b_{17}
  297. Mirror(R_8, f_6) → C_{12}
  298. Mirror(S_8, f_6) → D_{12}
  299. Mirror(T_8, f_6) → E_{12}
  300. Mirror(U_8, f_6) → F_{12}
  301. Mirror(K_8, f_6) → G_{12}
  302. Mirror(J_8, f_6) → H_{12}
  303. Polygon(C_{12}, D_{12}, E_{12}, F_{12}, G_{12}, H_{12}) → p94, c_{17}, d_{17}, e_{17}, f_{18}, g_{17}, h_{16}
  304. Mirror(J_8, f_6) → I_{12}
  305. Mirror(K_8, f_6) → J_{12}
  306. Mirror(L_8, f_6) → K_{12}
  307. Mirror(M_8, f_6) → L_{12}
  308. Mirror(H_1, f_6) → M_{12}
  309. Mirror(F_1, f_6) → N_{12}
  310. Polygon(I_{12}, J_{12}, K_{12}, L_{12}, M_{12}, N_{12}) → p92, i_{16}, j_{15}, k_{17}, l_{17}, m_{18}, n_{17}
  311. Mirror(T, f_6) → T'
  312. Mirror(F_1, f_6) → O_{12}
  313. Mirror(H_1, f_6) → P_{12}
  314. Mirror(I_1, f_6) → Q_{12}
  315. Mirror(G_1, f_6) → R_{12}
  316. Mirror(U, f_6) → U'
  317. Polygon(T', O_{12}, P_{12}, Q_{12}, R_{12}, U') → p108, t', o_{17}, p_{17}, q_{17}, r_{18}, u'
  318. Mirror(P, f_6) → P'
  319. Mirror(T, f_6) → T'_1
  320. Mirror(U, f_6) → U'_1
  321. Mirror(V, f_6) → V'
  322. Mirror(W, f_6) → W'
  323. Mirror(Q, f_6) → Q'
  324. Polygon(P', T'_1, U'_1, V', W', Q') → p105, p', t'_1, u'_1, v', w', q'
  325. Mirror(P, f_6) → P'_1
  326. Mirror(Q, f_6) → Q'_1
  327. Mirror(R, f_6) → R'
  328. Mirror(S, f_6) → S'
  329. Mirror(D, f_6) → D'
  330. Mirror(I, f_6) → I'_1
  331. Polygon(P'_1, Q'_1, R', S', D', I'_1) → p100, p'_1, q'_1, r', s', d', i'_1
  332. Mirror(G_1, f_6) → S_{12}
  333. Mirror(I_1, f_6) → T_{12}
  334. Mirror(L_1, f_6) → U_{12}
  335. Mirror(M_1, f_6) → V_{12}
  336. Polygon(S_{12}, T_{12}, U_{12}, V_{12}) → p16, s_{17}, t_{17}, u_{17}, v_{17}
  337. Mirror(L_8, f_6) → W_{12}
  338. Mirror(N_8, f_6) → Z_{12}
  339. Mirror(O_8, f_6) → A_{13}
  340. Mirror(M_8, f_6) → B_{13}
  341. Polygon(W_{12}, Z_{12}, A_{13}, B_{13}) → p49, w_{17}, z_{16}, a_{17}, b_{18}
  342. Mirror(W_8, f_6) → C_{13}
  343. Mirror(V_8, f_6) → D_{13}
  344. Mirror(Z_8, f_6) → E_{13}
  345. Mirror(A_9, f_6) → F_{13}
  346. Polygon(C_{13}, D_{13}, E_{13}, F_{13}) → p54, c_{18}, d_{18}, e_{18}, f_{19}
  347. Mirror(U_8, f_6) → G_{13}
  348. Mirror(T_8, f_6) → H_{13}
  349. Mirror(V_8, f_6) → I_{13}
  350. Mirror(W_8, f_6) → J_{13}
  351. Polygon(G_{13}, H_{13}, I_{13}, J_{13}) → p52, g_{18}, h_{17}, i_{17}, j_{16}
  352. Mirror(G_9, f_6) → K_{13}
  353. Mirror(F_9, f_6) → L_{13}
  354. Mirror(H_9, f_6) → M_{13}
  355. Mirror(I_9, f_6) → N_{13}
  356. Polygon(K_{13}, L_{13}, M_{13}, N_{13}) → p58, k_{18}, l_{18}, m_{19}, n_{18}
  357. Mirror(E_9, f_6) → O_{13}
  358. Mirror(D_9, f_6) → P_{13}
  359. Mirror(F_9, f_6) → Q_{13}
  360. Mirror(G_9, f_6) → R_{13}
  361. Polygon(O_{13}, P_{13}, Q_{13}, R_{13}) → p56, o_{18}, p_{18}, q_{18}, r_{19}
  362. Mirror(S_9, f_6) → S_{13}
  363. Mirror(R_9, f_6) → T_{13}
  364. Mirror(T_9, f_6) → U_{13}
  365. Mirror(U_9, f_6) → V_{13}
  366. Polygon(S_{13}, T_{13}, U_{13}, V_{13}) → p65, s_{18}, t_{18}, u_{18}, v_{18}
  367. Mirror(Q_9, f_6) → W_{13}
  368. Mirror(P_9, f_6) → Z_{13}
  369. Mirror(R_9, f_6) → A_{14}
  370. Mirror(S_9, f_6) → B_{14}
  371. Polygon(W_{13}, Z_{13}, A_{14}, B_{14}) → p63, w_{18}, z_{17}, a_{18}, b_{19}
  372. Mirror(O_9, f_6) → C_{14}
  373. Mirror(N_9, f_6) → D_{14}
  374. Mirror(P_9, f_6) → E_{14}
  375. Mirror(Q_9, f_6) → F_{14}
  376. Polygon(C_{14}, D_{14}, E_{14}, F_{14}) → p61, c_{19}, d_{19}, e_{19}, f_{20}
  377. Mirror(R_{10}, f_6) → G_{14}
  378. Mirror(Q_{10}, f_6) → H_{14}
  379. Mirror(S_{10}, f_6) → I_{14}
  380. Mirror(T_{10}, f_6) → J_{14}
  381. Polygon(G_{14}, H_{14}, I_{14}, J_{14}) → p80, g_{19}, h_{18}, i_{18}, j_{17}
  382. Mirror(P_{10}, f_6) → K_{14}
  383. Mirror(O_{10}, f_6) → L_{14}
  384. Mirror(Q_{10}, f_6) → M_{14}
  385. Mirror(R_{10}, f_6) → N_{14}
  386. Polygon(K_{14}, L_{14}, M_{14}, N_{14}) → p78, k_{19}, l_{19}, m_{20}, n_{19}
  387. Mirror(N_{10}, f_6) → O_{14}
  388. Mirror(M_{10}, f_6) → P_{14}
  389. Mirror(O_{10}, f_6) → Q_{14}
  390. Mirror(P_{10}, f_6) → R_{14}
  391. Polygon(O_{14}, P_{14}, Q_{14}, R_{14}) → p76, o_{19}, p_{19}, q_{19}, r_{20}
  392. Mirror(L_{10}, f_6) → S_{14}
  393. Mirror(K_{10}, f_6) → T_{14}
  394. Mirror(M_{10}, f_6) → U_{14}
  395. Mirror(N_{10}, f_6) → V_{14}
  396. Polygon(S_{14}, T_{14}, U_{14}, V_{14}) → p74, s_{19}, t_{19}, u_{19}, v_{19}
  397. Point(h_{18}) → W_{14}
  398. Polygon(W_3, V_3, Z_3, A_4, E_4, W_{14}, J_{10}, L_3) → p109, w_6, v_6, z_6, a_7, e_7, w_{19}, j_{18}, l_5
  399. Polygon(B_4, C_4, D_4, Z_{14}, F_4, G_4) → p106, b_7, c_7, d_7, z_{18}, f_7, g_7
  400. Polygon(Z_3, A_{15}, B_{15}, C_{15}, T_{14}, S_{14}, V_{10}, U_{10}, Z_{14}, D_4) → p110, z_{19}, d_{20}, a_{19}, b_{20}, c_{20}, t_{20}, s_{20}, v_{20}, u_{20}, z_{20}
  401. Polygon(W_{14}, E_4, D_{15}, C_{15}) → p81, w_{20}, e_{20}, d_{21}, c_{21}
  402. Sum(regiones) → sumaRegiones
  403. Min(regiones) → mínimoRegiones
  404. KeepIf(x ≟ 0, diferencias) → ceros
  405. Length(ceros) → coincidencias

AI Drawing Prompt Reference

Draw a planar map with multiple adjacent regions, colored with four colors such that no adjacent regions share the same color. Start by creating several basic polygons to represent regions, then define boundaries using points and lines, generate complex adjacency structures through mirroring and intersection, and finally use a coloring tool to ensure adjacent regions have different colors, demonstrating the Four Color Theorem.
The prompt above was compiled from the original drawing commands and is for reference only.