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自然数、平方和立方的和的视觉证明

自然数、平方和立方的和的视觉证明
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2015-08-13 08:02
数列解析几何几何作图
①这个范例通过动态图形演示了自然数、平方数和立方数的和的几何解释；②涉及的核心数学概念包括数列求和、几何证明、解析几何；③作图要点——从下方『构造摘要』提炼用到了Segment（线段）、Sequence（序列）、Point（点）、Polygon（多边形）、Intersect（交点）、OrthogonalLine（垂线）、Translate（平移）等GeoGebra命令/对象；④适合的学段或用途——特别适用于学习Edexcel FP1课程中的数列和级数部分。

GeoGebra 命令及参数

  1. Segment(C, A) → a
  2. Segment(B, D) → b
  3. Segment(D, C) → c
  4. Segment(A, B) → d
  5. Sequence(Segment[(k, 0), (k, 15)], k, 0, 11) → list2
  6. Sequence(Segment[(0, k), (11, k)], k, 0, 14) → list3
  7. Point(list2) → E
  8. Point(list2) → F
  9. Point(list2) → G
  10. Point(list2) → H
  11. Point(list2) → I
  12. Point(list2) → J
  13. Point(list2) → K
  14. Point(list2) → L
  15. Point(list2) → M
  16. Point(list2) → N
  17. Point(list2) → O
  18. Point(list2) → P
  19. Point(list2) → Q
  20. Point(list2) → R
  21. Point(list2) → S
  22. Point(list2) → T
  23. Point(c) → U
  24. Point(c) → V
  25. Polygon(U, S, T, V) → poly1, u, s, t, v
  26. Polygon(Q, P, L, R) → poly2, q, p, l, r
  27. Polygon(O, N, J, K) → poly3, o, n, j, k
  28. Polygon(M, G, H, I) → poly4, m, g, h, i
  29. Point(d) → W
  30. Point(d) → Z
  31. Polygon(E, W, Z, F) → poly5, e, w, z_1, f
  32. Polygon(D_1, C_1, B_1, E_1) → poly6, d_1, c_1, b_1, e_1
  33. Point(xAxis) → U_2
  34. Point(xAxis) → A_3
  35. Point(xAxis) → E_3
  36. Polygon(N_1, M_1, J_1, I_1) → poly7, n_1, m_1, j_1, i_1
  37. Polygon(L_1, O_1, P_1, K_1) → poly8, l_1, o_1, p_1, k_1
  38. Polygon(R_1, T_1, S_1, Q_1) → poly9, r_1, t_1, s_1, q_1
  39. Polygon(H_1, G_1, F_1, A_1) → poly10, h_1, g_1, f_1, a_1
  40. Polygon(D_1, C_1, B_1, E_1) → poly11, d_2, c_2, b_2, e_2
  41. Polygon(U_1, V_1, W_1, A_1) → poly12, u_1, v_1, w_1, a_2
  42. Polygon(Z_1, A_2, B_2, I_1) → poly13, z_2, a_3, b_3, i_2
  43. Polygon(C_2, D_2, E_2, K_1) → poly14, c_3, d_3, e_3, k_2
  44. Polygon(R_2, W_2, S_2, Q_1) → poly15, r_2, w_2, s_2, q_2
  45. Polygon(Q_2, Z_2, T_2, S_2, W_2, R_2) → poly16, q_3, z_3, t_2, s_3, w_3, r_3
  46. Polygon(F_2, G_2, H_2, E_2, D_2, C_2) → poly17, f_2, g_2, h_2, e_4, d_4, c_4
  47. Polygon(L_2, M_2, N_2, B_2, A_2, Z_1) → poly18, l_2, m_2, n_2, b_4, a_4, z_4
  48. Polygon(H_1, G_1, F_1, W_1, V_1, U_1) → poly19, h_3, g_3, f_3, w_4, v_2, u_2
  49. Polygon(N_1, M_1, J_1, N_2, M_2, L_2) → poly20, n_3, m_3, j_2, n_4, m_4, l_3
  50. Polygon(K_2, J_2, I_2, H_2, G_2, F_2) → poly21, k_3, j_3, i_3, h_4, g_4, f_4
  51. Polygon(P_2, A_3, U_2, T_2, Z_2, Q_2) → poly22, p_2, a_5, u_3, t_3, z_5, q_4
  52. Polygon(L_1, K_2, J_2, I_2, P_1, O_1) → poly23, l_4, k_4, j_4, i_4, p_3, o_2
  53. Polygon(O_2, B_3, V_2, U_2, A_3, P_2) → poly24, o_3, b_5, v_3, u_4, a_6, p_4
  54. Polygon(R_1, O_2, B_3, V_2, S_1, T_1) → poly25, r_4, o_4, b_6, v_4, s_4, t_4
  55. Segment(U_1, Q_3) → c_5
  56. Segment(R_3, W_1) → d_5
  57. Segment(Z_1, V_3) → e_5
  58. Segment(L_2, U_3) → f_5
  59. Segment(T_3, N_2) → g_5
  60. Segment(B_2, S_3) → h_5
  61. Segment(K_2, N_3) → i_5
  62. Segment(F_2, O_3) → j_5
  63. Segment(C_2, P_3) → k_5
  64. Segment(K_3, E_2) → l_5
  65. Segment(L_3, H_2) → m_5
  66. Segment(M_3, I_2) → n_5
  67. Segment(O_2, G_3) → p_5
  68. Segment(C_3, S_2) → q_5
  69. Segment(D_3, T_2) → r_5
  70. Segment(U_2, E_3) → s_5
  71. Segment(F_3, V_2) → t_5
  72. Segment(H_3, P_2) → c_6
  73. Segment(Q_2, I_3) → d_6
  74. Segment(R_2, J_3) → e_6
  75. Point(a) → W_3
  76. Point(a) → Z_3
  77. Point(b) → A_4
  78. Point(b) → B_4
  79. Intersect(s, t) → C_4
  80. Intersect(u, s) → D_4
  81. Intersect(q, p) → E_4
  82. Intersect(o, n) → F_4
  83. Intersect(p, l) → G_4
  84. Intersect(n, j) → H_4
  85. Intersect(g, h) → I_4
  86. Intersect(m, g) → J_4
  87. Point(a) → K_4
  88. Point(a) → L_4
  89. Point(b) → M_4
  90. Point(b) → N_4
  91. Intersect(a, d) → O_4
  92. Intersect(d, e) → P_4
  93. Point(d) → Q_4
  94. Intersect(b, d) → R_4
  95. Segment(Z_3, E_4) → f_6
  96. Segment(A_4, G_4) → g_6
  97. Segment(H_4, M_4) → h_6
  98. Segment(I_4, N_4) → i_6
  99. Segment(Q_4, R_4) → j_6
  100. Segment(O_4, P_4) → k_6
  101. Segment(K_4, J_4) → l_6
  102. Segment(L_4, F_4) → m_6
  103. Segment(W_3, D_4) → n_6
  104. Segment(C_4, B_4) → p_6
  105. Intersect(a, c) → S_4
  106. Intersect(c, u) → T_4
  107. Intersect(c, t) → U_4
  108. Intersect(b, c) → V_4
  109. Segment(S_4, V_4) → q_6
  110. Segment(S_4, O_4) → r_6
  111. Segment(V_4, R_4) → s_6
  112. Segment(R_4, O_4) → t_6
  113. Segment(Q_6, Z_6) → a_7
  114. Segment(Z_6, N_7) → b_7
  115. Segment(N_7, M_7) → c_7
  116. Segment(M_7, W_6) → d_7
  117. Segment(Q_6, A_7) → e_7
  118. Segment(A_7, B_7) → f_7
  119. Segment(B_7, C_7) → g_7
  120. Segment(C_7, D_7) → h_7
  121. Segment(D_7, E_7) → i_7
  122. Segment(E_7, F_7) → j_7
  123. Segment(F_7, G_7) → k_7
  124. Segment(G_7, H_7) → l_7
  125. Segment(H_7, I_7) → m_7
  126. Segment(I_7, J_7) → n_7
  127. Segment(J_7, K_7) → p_7
  128. Segment(K_7, L_7) → q_7
  129. Segment(V_6, J_7) → r_7
  130. Segment(H_7, U_6) → s_7
  131. Segment(T_6, F_7) → t_7
  132. Segment(D_7, S_6) → a_8
  133. Segment(R_6, B_7) → b_8
  134. Segment(A_7, O_7) → c_8
  135. Segment(D_7, P_7) → d_8
  136. Segment(F_7, Q_7) → e_8
  137. Segment(H_7, R_7) → f_8
  138. Segment(J_7, S_7) → g_8
  139. Segment(L_7, T_7) → h_8
  140. Segment(Q_6, N_7) → i_8
  141. Polygon(Q_6, A_7, J_7, K_7, L_7, M_7, N_7, B_7, C_7, D_7, E_7, F_7, G_7, H_7, I_7) → poly26, q_8, i_9, j_8, k_8, l_8, m_8, n_8, a_9, b_9, c_9, d_9, e_9, f_9, g_9, h_9
  142. Segment(V_7, U_7) → p_8
  143. Segment(U_7, Z_7) → r_8
  144. Segment(Z_7, W_7) → s_8
  145. Segment(W_7, V_7) → t_8
  146. Segment(K_8, A_8) → j_9
  147. Segment(F_8, P_8) → k_9
  148. Segment(J_8, T_8) → l_9
  149. Segment(O_8, E_8) → m_9
  150. Intersect(j_9, l_9) → U_8
  151. Intersect(j_9, k_9) → V_8
  152. Intersect(k_9, m_9) → W_8
  153. Intersect(l_9, m_9) → Z_8
  154. OrthogonalLine(L_8, l_9) → n_9
  155. OrthogonalLine(M_8, l_9) → p_9
  156. OrthogonalLine(N_8, l_9) → q_9
  157. OrthogonalLine(I_8, j_9) → r_9
  158. OrthogonalLine(H_8, j_9) → s_9
  159. OrthogonalLine(G_8, j_9) → t_9
  160. Intersect(r_9, j_9) → A_9
  161. Intersect(s_9, j_9) → B_9
  162. Intersect(t_9, j_9) → C_9
  163. Intersect(n_9, k_9) → D_9
  164. Intersect(p_9, k_9) → E_9
  165. Intersect(q_9, k_9) → F_9
  166. Intersect(t_9, m_9) → G_9
  167. Intersect(s_9, m_9) → H_9
  168. Intersect(r_9, m_9) → I_9
  169. Intersect(n_9, l_9) → J_9
  170. Intersect(p_9, l_9) → K_9
  171. Intersect(q_9, l_9) → L_9
  172. Segment(L_8, J_9) → a_{10}
  173. Segment(M_8, K_9) → b_{10}
  174. Segment(N_8, L_9) → c_{10}
  175. Segment(I_9, S_8) → d_{10}
  176. Segment(R_8, H_9) → e_{10}
  177. Segment(G_9, Q_8) → f_{10}
  178. Segment(I_8, A_9) → g_{10}
  179. Segment(H_8, B_9) → h_{10}
  180. Segment(G_8, C_9) → i_{10}
  181. Segment(D_9, B_8) → j_{10}
  182. Segment(E_9, C_8) → k_{10}
  183. Segment(F_9, D_8) → l_{10}
  184. Translate(V_8, (0, 2.4)) → C_{11}
  185. Translate(M_9, (0, 2.4)) → H_{10}
  186. Translate(N_9, (0, 2.4)) → I_{10}
  187. Translate(O_9, (0, 2.4)) → J_{10}
  188. Translate(P_9, (0, 2.4)) → K_{10}
  189. Translate(W_9, (0, 2.4)) → R_{10}
  190. Translate(U_9, (0, 2.4)) → P_{10}
  191. Translate(Q_9, (0, 2.4)) → L_{10}
  192. Translate(R_9, (0, 2.4)) → M_{10}
  193. Translate(S_9, (0, 2.4)) → N_{10}
  194. Translate(T_9, (0, 2.4)) → O_{10}
  195. Translate(V_9, (0, 2.4)) → Q_{10}
  196. Translate(A_{10}, (0, 2.4)) → T_{10}
  197. Translate(Z_9, (0, 2.4)) → S_{10}
  198. Translate(D_{10}, (0, 2.4)) → W_{10}
  199. Translate(C_{10}, (0, 2.4)) → V_{10}
  200. Translate(B_{10}, (0, 2.4)) → U_{10}
  201. Translate(W_8, (0, 2.4)) → D_{11}
  202. Segment(C_{11}, K_{10}) → m_{10}
  203. Segment(K_{10}, P_9) → n_{10}
  204. Segment(K_{10}, O_{10}) → p_{10}
  205. Segment(O_{10}, T_9) → q_{10}
  206. Segment(O_{10}, S_{10}) → r_{10}
  207. Segment(S_{10}, Z_9) → s_{10}
  208. Segment(S_{10}, W_{10}) → t_{10}
  209. Segment(W_{10}, D_{10}) → a_{11}
  210. Segment(D_{11}, W_{10}) → b_{11}
  211. Segment(H_{10}, M_9) → c_{11}
  212. Segment(I_{10}, N_9) → d_{11}
  213. Segment(J_{10}, O_9) → e_{11}
  214. Segment(L_{10}, Q_9) → f_{11}
  215. Segment(M_{10}, R_9) → g_{11}
  216. Segment(N_{10}, S_9) → h_{11}
  217. Segment(P_{10}, U_9) → i_{11}
  218. Segment(Q_{10}, V_9) → j_{11}
  219. Segment(R_{10}, W_9) → k_{11}
  220. Segment(T_{10}, A_{10}) → l_{11}
  221. Segment(U_{10}, B_{10}) → m_{11}
  222. Segment(V_{10}, C_{10}) → n_{11}
  223. Segment(G_{10}, B_{11}) → p_{11}
  224. Segment(A_{11}, F_{10}) → q_{11}
  225. Segment(E_{10}, Z_{10}) → r_{11}
  226. Translate(D_{10}, (0, -3)) → E_{11}
  227. Translate(C_{10}, (0, -3)) → F_{11}
  228. Translate(B_{10}, (0, -3)) → G_{11}
  229. Translate(A_{10}, (0, -3)) → H_{11}
  230. Translate(M_9, (0, -3)) → I_{11}
  231. Translate(N_9, (0, -3)) → J_{11}
  232. Translate(O_9, (0, -3)) → K_{11}
  233. Translate(P_9, (0, -3)) → L_{11}
  234. Translate(Q_9, (0, -3)) → M_{11}
  235. Translate(R_9, (0, -3)) → N_{11}
  236. Translate(S_9, (0, -3)) → O_{11}
  237. Translate(T_9, (0, -3)) → P_{11}
  238. Translate(U_9, (0, -3)) → Q_{11}
  239. Translate(V_9, (0, -3)) → R_{11}
  240. Translate(W_9, (0, -3)) → S_{11}
  241. Translate(Z_9, (0, -3)) → T_{11}
  242. Segment(M_9, I_{11}) → s_{11}
  243. Segment(N_9, J_{11}) → t_{11}
  244. Segment(O_9, K_{11}) → a_{12}
  245. Segment(P_9, L_{11}) → b_{12}
  246. Segment(R_9, N_{11}) → c_{12}
  247. Segment(S_9, O_{11}) → d_{12}
  248. Segment(T_9, P_{11}) → e_{12}
  249. Segment(U_9, Q_{11}) → f_{12}
  250. Segment(W_9, S_{11}) → g_{12}
  251. Segment(Z_9, T_{11}) → h_{12}
  252. Segment(A_{10}, H_{11}) → i_{12}
  253. Segment(B_{10}, G_{11}) → j_{12}
  254. Segment(D_{10}, E_{11}) → k_{12}
  255. Translate(D_{12}, (0, 3)) → E_{12}
  256. Translate(C_{12}, (0, 3)) → F_{12}
  257. Translate(B_{12}, (0, 3)) → G_{12}
  258. Translate(A_{12}, (0, 3)) → H_{12}
  259. Translate(U_{11}, (0, 3)) → I_{12}
  260. Translate(V_{11}, (0, 3)) → J_{12}
  261. Translate(W_{11}, (0, 3)) → K_{12}
  262. Translate(Z_{11}, (0, 3)) → L_{12}
  263. Segment(W_8, Z_{11}) → l_{12}
  264. Segment(K_{12}, W_{11}) → m_{12}
  265. Segment(J_{12}, V_{11}) → n_{12}
  266. Segment(I_{12}, U_{11}) → p_{12}
  267. Segment(H_{12}, A_{12}) → q_{12}
  268. Segment(G_{12}, B_{12}) → r_{12}
  269. Segment(F_{12}, C_{12}) → s_{12}
  270. Segment(E_{12}, D_{12}) → t_{12}
  271. Segment(P_{12}, T_{12}) → a_{13}
  272. Segment(S_{12}, O_{12}) → b_{13}
  273. Segment(N_{12}, R_{12}) → c_{13}
  274. Segment(Q_{12}, M_{12}) → d_{13}
  275. Segment(W_{12}, K_{10}) → e_{13}
  276. Segment(W_{12}, Z_{12}) → f_{13}
  277. Segment(Z_{12}, T_{10}) → g_{13}
  278. Translate(L_{10}, (0, 1.8)) → A_{13}
  279. Translate(M_{10}, (0, 1.8)) → B_{13}
  280. Translate(N_{10}, (0, 1.8)) → C_{13}
  281. Translate(O_{10}, (0, 1.8)) → D_{13}
  282. Translate(P_{10}, (0, 1.8)) → E_{13}
  283. Translate(Q_{10}, (0, 1.8)) → F_{13}
  284. Translate(R_{10}, (0, 1.8)) → G_{13}
  285. Translate(S_{10}, (0, 1.8)) → H_{13}
  286. Segment(V_{12}, J_{13}) → h_{13}
  287. Segment(I_{13}, U_{12}) → i_{13}
  288. Segment(A_{13}, L_{10}) → j_{13}
  289. Segment(B_{13}, M_{10}) → k_{13}
  290. Segment(C_{13}, N_{10}) → l_{13}
  291. Segment(D_{13}, O_{10}) → m_{13}
  292. Segment(E_{13}, P_{10}) → n_{13}
  293. Segment(F_{13}, Q_{10}) → p_{13}
  294. Segment(G_{13}, R_{10}) → q_{13}
  295. Segment(H_{13}, S_{10}) → r_{13}
  296. Translate(C_{13}, (0, 1.2)) → K_{13}
  297. Translate(D_{13}, (0, 1.2)) → L_{13}
  298. Translate(E_{13}, (0, 1.2)) → M_{13}
  299. Translate(F_{13}, (0, 1.2)) → N_{13}
  300. Translate(G_{13}, (0, 1.2)) → O_{13}
  301. Segment(K_{13}, O_{13}) → s_{13}
  302. Segment(O_{13}, G_{13}) → t_{13}
  303. Segment(N_{13}, F_{13}) → a_{14}
  304. Segment(M_{13}, E_{13}) → b_{14}
  305. Segment(L_{13}, D_{13}) → c_{14}
  306. Segment(K_{13}, C_{13}) → d_{14}
  307. Translate(C_{13}, (0, 0.6)) → P_{13}
  308. Translate(D_{13}, (0, 0.6)) → Q_{13}
  309. Translate(E_{13}, (0, 0.6)) → R_{13}
  310. Translate(F_{13}, (0, 0.6)) → S_{13}
  311. Translate(G_{13}, (0, 0.6)) → T_{13}
  312. Segment(P_{13}, T_{13}) → e_{14}
  313. Translate(M_{13}, (0, 0.6)) → U_{13}
  314. Translate(N_{13}, (0, 0.6)) → V_{13}
  315. Segment(U_{13}, V_{13}) → f_{14}
  316. Segment(V_{13}, N_{13}) → g_{14}
  317. Segment(U_{13}, M_{13}) → h_{14}
  318. Segment(C_{14}, B_{14}) → i_{14}
  319. Segment(E_{14}, C_{14}) → j_{14}
  320. Segment(W_{13}, C_{14}) → k_{14}
  321. Segment(D_{14}, Z_{13}) → l_{14}
  322. Segment(E_{14}, A_{14}) → m_{14}
  323. Segment(W_{12}, E_{14}) → n_{14}
  324. Translate(C_{13}, (0, 1.8)) → F_{14}
  325. Translate(F_{14}, (-1.8, 0)) → G_{14}
  326. Segment(G_{14}, F_{14}) → p_{14}
  327. Segment(F_{14}, C_{13}) → q_{14}
  328. Translate(G_{14}, (0, 1.8)) → H_{14}
  329. Translate(C_{14}, (1.8, 0)) → I_{14}
  330. Translate(H_{14}, (1.8, 0)) → J_{14}
  331. Segment(I_{14}, C_{14}) → r_{14}
  332. Segment(I_{14}, F_{14}) → s_{14}
  333. Segment(H_{14}, J_{14}) → t_{14}
  334. Translate(B_{14}, (2.4, 0)) → K_{14}
  335. Translate(K_{14}, (2.4, 0)) → L_{14}
  336. Translate(L_{14}, (2.4, 0)) → M_{14}
  337. Translate(M_{14}, (0, -2.4)) → N_{14}
  338. Translate(L_{14}, (0, -2.4)) → O_{14}
  339. Translate(K_{14}, (0, -2.4)) → P_{14}
  340. Segment(I_{14}, N_{14}) → a_{15}
  341. Segment(N_{14}, M_{14}) → b_{15}
  342. Segment(L_{14}, O_{14}) → c_{15}
  343. Segment(K_{14}, P_{14}) → d_{15}
  344. Translate(N_{14}, (2.4, 0)) → Q_{14}
  345. Segment(N_{14}, Q_{14}) → e_{15}
  346. Translate(N_{14}, (0, -2.4)) → R_{14}
  347. Translate(R_{14}, (0, -2.4)) → S_{14}
  348. Translate(R_{14}, (2.4, 0)) → T_{14}
  349. Translate(S_{14}, (2.4, 0)) → U_{14}
  350. Segment(N_{14}, W_{10}) → f_{15}
  351. Segment(S_{14}, U_{14}) → g_{15}
  352. Segment(R_{14}, T_{14}) → h_{15}
  353. Translate(J_{14}, (1.8, 0)) → V_{14}
  354. Translate(V_{14}, (1.8, 0)) → W_{14}
  355. Translate(W_{14}, (1.8, 0)) → Z_{14}
  356. Segment(J_{14}, Z_{14}) → i_{15}
  357. Translate(W_{14}, (0, 1.8)) → A_{15}
  358. Translate(V_{14}, (0, 1.8)) → B_{15}
  359. Segment(B_{15}, V_{14}) → j_{15}
  360. Segment(A_{15}, W_{14}) → k_{15}
  361. Translate(Z_{12}, (1.8, 0)) → C_{15}
  362. Segment(Z_{12}, C_{15}) → l_{15}
  363. Translate(K_{13}, (0, 1.2)) → D_{15}
  364. Translate(J_{14}, (1.2, 0)) → E_{15}
  365. Translate(D_{15}, (1.2, 0)) → F_{15}
  366. Segment(U_{13}, E_{15}) → m_{15}
  367. Segment(F_{15}, D_{15}) → n_{15}
  368. Translate(F_{15}, (1.2, 0)) → G_{15}
  369. Translate(G_{15}, (1.2, 0)) → H_{15}
  370. Segment(H_{15}, F_{15}) → p_{15}
  371. Segment(H_{15}, W_{14}) → q_{15}
  372. Translate(G_{15}, (0, 1.2)) → I_{15}
  373. Segment(I_{15}, G_{15}) → r_{15}
  374. Segment(G_{15}, O_{13}) → s_{15}
  375. Translate(V_{13}, (0, 0.6)) → J_{15}
  376. Segment(V_{13}, J_{15}) → t_{15}
  377. Translate(V_{13}, (0.6, 0)) → K_{15}
  378. Segment(V_{13}, K_{15}) → a_{16}
  379. Translate(W_{14}, (0, -1.8)) → L_{15}
  380. Translate(L_{15}, (1.8, 0)) → M_{15}
  381. Segment(W_{14}, L_{15}) → b_{16}
  382. Segment(L_{15}, M_{15}) → c_{16}
  383. Segment(L_{15}, Z_{12}) → d_{16}
  384. Translate(O_{13}, (1.2, 0)) → N_{15}
  385. Segment(N_{15}, O_{13}) → e_{16}
  386. Polygon(F_8, V_8, J_{15}, F_{15}, E_{15}, J_{14}, I_{14}, C_{14}, B_{14}, U_8, K_8, V_7, C_{11}, K_{10}, W_{12}, C_{13}, K_{13}, M_{13}, U_{13}, V_{13}) → poly27, f_{16}, v_{13}, j_{16}, f_{17}, e_{17}, j_{17}, i_{16}, c_{19}, b_{17}, u_8, k_{18}, v_8, v_7, c_{17}, k_{16}, w_{12}, c_{18}, k_{17}, m_{16}, u_{13}
  387. Polygon(K_8, U_8, K_{15}, G_{15}, H_{15}, W_{14}, Z_{14}, N_{14}, Q_{14}, Z_8, T_8, W_7, B_{14}, C_{14}, I_{14}, J_{14}, E_{15}, F_{15}, J_{15}, V_{13}) → poly28, k_{19}, v_6, k_{20}, g_{16}, h_{16}, w_{14}, z_{14}, n_{16}, q_{16}, z_8, t_{16}, u_6, w_7, b_{18}, c_{20}, i_{17}, j_{18}, e_{18}, f_{18}, j_{19}
  388. Polygon(Z_7, Z_{11}, V_{13}, K_{15}, G_{15}, H_{15}, W_{14}, Z_{14}, N_{14}, Q_{14}, Z_8, T_8, W_8, D_{10}, W_{10}, T_{10}, Z_{12}, G_{13}, O_{13}, N_{13}) → poly29, z_7, n_{17}, v_9, k_{21}, g_{18}, h_{17}, w_6, z_6, n_{18}, q_{17}, z_9, z_{11}, t_{18}, w_8, d_{17}, w_{10}, t_{17}, z_{12}, g_{17}, o_{13}

AI 绘图提示语参考

画一个 12×15 的正方形网格，垂直线从 x=0 到 11，水平线从 y=0 到 14；在每条垂直线上放置一系列点，并在一条从左上到右下的斜线段上也放置点；然后依次用这些点构建多个多边形，分别围成三角形、梯形和更复杂的多边形，直观展示自然数和、平方和与立方和的几何证明。
以上提示语为根据原始绘图命令整理而来，仅供参考。