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Visual proofs of sums of natural numbers, squares and cubes

Visual proofs of sums of natural numbers, squares and cubes
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2015-08-13 08:02
SequenceCoordinate GeometryGeometric Construction
This example demonstrates the geometric interpretations of sums of natural numbers, squares, and cubes through dynamic visuals. It involves core mathematical concepts such as series summation, geometric proofs, and analytical geometry. The construction utilizes GeoGebra commands like Segment, Sequence, Point, Polygon, Intersect, OrthogonalLine, and Translate. Ideal for students studying sequences and series in Edexcel FP1.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a 12 by 15 rectangular grid of unit squares with vertical lines at x = 0 through 11 and horizontal lines at y = 0 through 14. Place a series of points on each vertical line and also on a diagonal segment running from the top‑left to the bottom‑right corner. Then connect these points sequentially to form a collection of polygons — triangles, trapezoids, and more complex shapes — that visually demonstrate the geometric proofs of the sums of natural numbers, squares, and cubes.
The prompt above was compiled from the original drawing commands and is for reference only.