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Filling a Rectangle with Small Units

Filling a Rectangle with Small Units
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2015-03-26 14:36
CirclePolygonFunctionSequenceGeometric Construction
This example demonstrates a geometric task of placing several small units (such as small circles or polygons) into a given rectangle, forming a filling or arrangement pattern. The core mathematical concepts involve rectangles, polygons, circles, exponential functions, cosine functions, and statistical characteristics (like sums and midpoints), combining geometric construction with functional calculations to explore spatial allocation and arrangement rules. Construction highlights: from the construction summary, commands such as Intersect, Point, Segment, OrthogonalLine, Circle, Polygon, Sequence, Sum, and Midpoint are used, along with multiple points, segments, circles, polygons, text, and buttons for dynamic interaction. This is suitable for middle school or early college geometry and function teaching, useful for exploring optimization problems or statistical distributions of unit arrangements within a rectangle, and also as a dynamic geometry demonstration exercise.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Point(yAxis) → C
  4. Segment(A, B) → a
  5. Segment(A, C) → b
  6. Point(a) → D
  7. Point(b) → E
  8. OrthogonalLine(D, a) → c
  9. OrthogonalLine(E, b) → d
  10. Intersect(c, d) → F
  11. Segment(E, F) → e
  12. Segment(F, D) → f
  13. Segment(E, A) → g
  14. Segment(A, D) → h
  15. Circle(G, n) → k
  16. Point(k) → H
  17. Segment(G, H) → j
  18. Circle(G, i) → p
  19. OrthogonalLine(G, j) → l
  20. Intersect(p, l, 2) → I
  21. OrthogonalLine(I, j) → m
  22. OrthogonalLine(I, m) → q
  23. OrthogonalLine(H, j) → r
  24. Intersect(q, r) → J
  25. Segment(I, G) → s
  26. Segment(I, J) → t
  27. Segment(J, H) → a_1
  28. Polygon(I, G, H, J) → poly1, i_1, g_1, h_1, j_1
  29. Circle(G_1, n) → k_1
  30. Point(k_1) → H_1
  31. Segment(G_1, H_1) → j_2
  32. Circle(G_1, i) → p_1
  33. OrthogonalLine(G_1, j_2) → l_1
  34. Intersect(p_1, l_1, 2) → I_1
  35. OrthogonalLine(I_1, j_2) → m_1
  36. OrthogonalLine(I_1, m_1) → q_1
  37. OrthogonalLine(H_1, j_2) → r_1
  38. Intersect(q_1, r_1) → J_1
  39. Segment(I_1, G_1) → s_1
  40. Segment(I_1, J_1) → t_1
  41. Segment(J_1, H_1) → a_2
  42. Polygon(I_1, G_1, H_1, J_1) → poly1_1, i_2, g_2, h_2, j_3
  43. Circle(G_2, n) → k_2
  44. Point(k_2) → H_2
  45. Segment(G_2, H_2) → j_4
  46. Circle(G_2, i) → p_2
  47. OrthogonalLine(G_2, j_4) → l_2
  48. Intersect(p_2, l_2, 2) → I_2
  49. OrthogonalLine(I_2, j_4) → m_2
  50. OrthogonalLine(I_2, m_2) → q_2
  51. OrthogonalLine(H_2, j_4) → r_2
  52. Intersect(q_2, r_2) → J_2
  53. Segment(I_2, G_2) → s_2
  54. Segment(I_2, J_2) → t_2
  55. Segment(J_2, H_2) → a_3
  56. Polygon(I_2, G_2, H_2, J_2) → poly1_2, i_3, g_3, h_3, j_5
  57. Circle(G_3, n) → k_3
  58. Point(k_3) → H_3
  59. Segment(G_3, H_3) → j_6
  60. Circle(G_3, i) → p_3
  61. OrthogonalLine(G_3, j_6) → l_3
  62. Intersect(p_3, l_3, 2) → I_3
  63. OrthogonalLine(I_3, j_6) → m_3
  64. OrthogonalLine(I_3, m_3) → q_3
  65. OrthogonalLine(H_3, j_6) → r_3
  66. Intersect(q_3, r_3) → J_3
  67. Segment(I_3, G_3) → s_3
  68. Segment(I_3, J_3) → t_3
  69. Segment(J_3, H_3) → a_4
  70. Polygon(I_3, G_3, H_3, J_3) → poly1_3, i_4, g_4, h_4, j_7
  71. Circle(G_4, n) → k_4
  72. Point(k_4) → H_4
  73. Segment(G_4, H_4) → j_8
  74. Circle(G_4, i) → p_4
  75. OrthogonalLine(G_4, j_8) → l_4
  76. Intersect(p_4, l_4, 2) → I_4
  77. OrthogonalLine(I_4, j_8) → m_4
  78. OrthogonalLine(I_4, m_4) → q_4
  79. OrthogonalLine(H_4, j_8) → r_4
  80. Intersect(q_4, r_4) → J_4
  81. Segment(I_4, G_4) → s_4
  82. Segment(I_4, J_4) → t_4
  83. Segment(J_4, H_4) → a_5
  84. Polygon(I_4, G_4, H_4, J_4) → poly1_4, i_5, g_5, h_5, j_9
  85. Circle(G_5, n) → k_5
  86. Point(k_5) → H_5
  87. Segment(G_5, H_5) → j_{10}
  88. Circle(G_5, i) → p_5
  89. OrthogonalLine(G_5, j_{10}) → l_5
  90. Intersect(p_5, l_5, 2) → I_5
  91. OrthogonalLine(I_5, j_{10}) → m_5
  92. OrthogonalLine(I_5, m_5) → q_5
  93. OrthogonalLine(H_5, j_{10}) → r_5
  94. Intersect(q_5, r_5) → J_5
  95. Segment(I_5, G_5) → s_5
  96. Segment(I_5, J_5) → t_5
  97. Segment(J_5, H_5) → a_6
  98. Polygon(I_5, G_5, H_5, J_5) → poly1_5, i_6, g_6, h_6, j_{11}
  99. Circle(G_6, n) → k_6
  100. Point(k_6) → H_6
  101. Segment(G_6, H_6) → j_{12}
  102. Circle(G_6, i) → p_6
  103. OrthogonalLine(G_6, j_{12}) → l_6
  104. Intersect(p_6, l_6, 2) → I_6
  105. OrthogonalLine(I_6, j_{12}) → m_6
  106. OrthogonalLine(I_6, m_6) → q_6
  107. OrthogonalLine(H_6, j_{12}) → r_6
  108. Intersect(q_6, r_6) → J_6
  109. Segment(I_6, G_6) → s_6
  110. Segment(I_6, J_6) → t_6
  111. Segment(J_6, H_6) → a_7
  112. Polygon(I_6, G_6, H_6, J_6) → poly1_6, i_7, g_7, h_7, j_{13}
  113. Circle(G_7, n) → k_7
  114. Point(k_7) → H_7
  115. Segment(G_7, H_7) → j_{14}
  116. Circle(G_7, i) → p_7
  117. OrthogonalLine(G_7, j_{14}) → l_7
  118. Intersect(p_7, l_7, 2) → I_7
  119. OrthogonalLine(I_7, j_{14}) → m_7
  120. OrthogonalLine(I_7, m_7) → q_7
  121. OrthogonalLine(H_7, j_{14}) → r_7
  122. Intersect(q_7, r_7) → J_7
  123. Segment(I_7, G_7) → s_7
  124. Segment(I_7, J_7) → t_7
  125. Segment(J_7, H_7) → a_8
  126. Polygon(I_7, G_7, H_7, J_7) → poly1_7, i_8, g_8, h_8, j_{15}
  127. Circle(G_8, n) → k_8
  128. Point(k_8) → H_8
  129. Segment(G_8, H_8) → j_{16}
  130. Circle(G_8, i) → p_8
  131. OrthogonalLine(G_8, j_{16}) → l_8
  132. Intersect(p_8, l_8, 2) → I_8
  133. OrthogonalLine(I_8, j_{16}) → m_8
  134. OrthogonalLine(I_8, m_8) → q_8
  135. OrthogonalLine(H_8, j_{16}) → r_8
  136. Intersect(q_8, r_8) → J_8
  137. Segment(I_8, G_8) → s_8
  138. Segment(I_8, J_8) → t_8
  139. Segment(J_8, H_8) → a_9
  140. Polygon(I_8, G_8, H_8, J_8) → poly1_8, i_9, g_9, h_9, j_{17}
  141. Circle(G_9, n) → k_9
  142. Point(k_9) → H_9
  143. Segment(G_9, H_9) → j_{18}
  144. Circle(G_9, i) → p_9
  145. OrthogonalLine(G_9, j_{18}) → l_9
  146. Intersect(p_9, l_9, 2) → I_9
  147. OrthogonalLine(I_9, j_{18}) → m_9
  148. OrthogonalLine(I_9, m_9) → q_9
  149. OrthogonalLine(H_9, j_{18}) → r_9
  150. Intersect(q_9, r_9) → J_9
  151. Segment(I_9, G_9) → s_9
  152. Segment(I_9, J_9) → t_9
  153. Segment(J_9, H_9) → a_{10}
  154. Polygon(I_9, G_9, H_9, J_9) → poly1_9, i_{10}, g_{10}, h_{10}, j_{19}
  155. Circle(G_{10}, n) → k_{10}
  156. Point(k_{10}) → H_{10}
  157. Segment(G_{10}, H_{10}) → j_{20}
  158. Circle(G_{10}, i) → p_{10}
  159. OrthogonalLine(G_{10}, j_{20}) → l_{10}
  160. Intersect(p_{10}, l_{10}, 2) → I_{10}
  161. OrthogonalLine(I_{10}, j_{20}) → m_{10}
  162. OrthogonalLine(I_{10}, m_{10}) → q_{10}
  163. OrthogonalLine(H_{10}, j_{20}) → r_{10}
  164. Intersect(q_{10}, r_{10}) → J_{10}
  165. Segment(I_{10}, G_{10}) → s_{10}
  166. Segment(I_{10}, J_{10}) → t_{10}
  167. Segment(J_{10}, H_{10}) → a_{11}
  168. Polygon(I_{10}, G_{10}, H_{10}, J_{10}) → poly1_{10}, i_{11}, g_{11}, h_{11}, j_{21}
  169. Circle(G_{11}, n) → k_{11}
  170. Point(k_{11}) → H_{11}
  171. Segment(G_{11}, H_{11}) → j_{22}
  172. Circle(G_{11}, i) → p_{11}
  173. OrthogonalLine(G_{11}, j_{22}) → l_{11}
  174. Intersect(p_{11}, l_{11}, 2) → I_{11}
  175. OrthogonalLine(I_{11}, j_{22}) → m_{11}
  176. OrthogonalLine(I_{11}, m_{11}) → q_{11}
  177. OrthogonalLine(H_{11}, j_{22}) → r_{11}
  178. Intersect(q_{11}, r_{11}) → J_{11}
  179. Segment(I_{11}, G_{11}) → s_{11}
  180. Segment(I_{11}, J_{11}) → t_{11}
  181. Segment(J_{11}, H_{11}) → a_{12}
  182. Polygon(I_{11}, G_{11}, H_{11}, J_{11}) → poly1_{11}, i_{12}, g_{12}, h_{12}, j_{23}
  183. Circle(G_{12}, n) → k_{12}
  184. Point(k_{12}) → H_{12}
  185. Segment(G_{12}, H_{12}) → j_{24}
  186. Circle(G_{12}, i) → p_{12}
  187. OrthogonalLine(G_{12}, j_{24}) → l_{12}
  188. Intersect(p_{12}, l_{12}, 2) → I_{12}
  189. OrthogonalLine(I_{12}, j_{24}) → m_{12}
  190. OrthogonalLine(I_{12}, m_{12}) → q_{12}
  191. OrthogonalLine(H_{12}, j_{24}) → r_{12}
  192. Intersect(q_{12}, r_{12}) → J_{12}
  193. Segment(I_{12}, G_{12}) → s_{12}
  194. Segment(I_{12}, J_{12}) → t_{12}
  195. Segment(J_{12}, H_{12}) → a_{13}
  196. Polygon(I_{12}, G_{12}, H_{12}, J_{12}) → poly1_{12}, i_{13}, g_{13}, h_{13}, j_{25}
  197. Circle(G_{13}, n) → k_{13}
  198. Point(k_{13}) → H_{13}
  199. Segment(G_{13}, H_{13}) → j_{26}
  200. Circle(G_{13}, i) → p_{13}
  201. OrthogonalLine(G_{13}, j_{26}) → l_{13}
  202. Intersect(p_{13}, l_{13}, 2) → I_{13}
  203. OrthogonalLine(I_{13}, j_{26}) → m_{13}
  204. OrthogonalLine(I_{13}, m_{13}) → q_{13}
  205. OrthogonalLine(H_{13}, j_{26}) → r_{13}
  206. Intersect(q_{13}, r_{13}) → J_{13}
  207. Segment(I_{13}, G_{13}) → s_{13}
  208. Segment(I_{13}, J_{13}) → t_{13}
  209. Segment(J_{13}, H_{13}) → a_{14}
  210. Polygon(I_{13}, G_{13}, H_{13}, J_{13}) → poly1_{13}, i_{14}, g_{14}, h_{14}, j_{27}
  211. Circle(G_{14}, n) → k_{14}
  212. Point(k_{14}) → H_{14}
  213. Segment(G_{14}, H_{14}) → j_{28}
  214. Circle(G_{14}, i) → p_{14}
  215. OrthogonalLine(G_{14}, j_{28}) → l_{14}
  216. Intersect(p_{14}, l_{14}, 2) → I_{14}
  217. OrthogonalLine(I_{14}, j_{28}) → m_{14}
  218. OrthogonalLine(I_{14}, m_{14}) → q_{14}
  219. OrthogonalLine(H_{14}, j_{28}) → r_{14}
  220. Intersect(q_{14}, r_{14}) → J_{14}
  221. Segment(I_{14}, G_{14}) → s_{14}
  222. Segment(I_{14}, J_{14}) → t_{14}
  223. Segment(J_{14}, H_{14}) → a_{15}
  224. Polygon(I_{14}, G_{14}, H_{14}, J_{14}) → poly1_{14}, i_{15}, g_{15}, h_{15}, j_{29}
  225. Circle(G_{15}, n) → k_{15}
  226. Point(k_{15}) → H_{15}
  227. Segment(G_{15}, H_{15}) → j_{30}
  228. Circle(G_{15}, i) → p_{15}
  229. OrthogonalLine(G_{15}, j_{30}) → l_{15}
  230. Intersect(p_{15}, l_{15}, 2) → I_{15}
  231. OrthogonalLine(I_{15}, j_{30}) → m_{15}
  232. OrthogonalLine(I_{15}, m_{15}) → q_{15}
  233. OrthogonalLine(H_{15}, j_{30}) → r_{15}
  234. Intersect(q_{15}, r_{15}) → J_{15}
  235. Segment(I_{15}, G_{15}) → s_{15}
  236. Segment(I_{15}, J_{15}) → t_{15}
  237. Segment(J_{15}, H_{15}) → a_{16}
  238. Polygon(I_{15}, G_{15}, H_{15}, J_{15}) → poly1_{15}, i_{16}, g_{16}, h_{16}, j_{31}
  239. Circle(G_{16}, n) → k_{16}
  240. Point(k_{16}) → H_{16}
  241. Segment(G_{16}, H_{16}) → j_{32}
  242. Circle(G_{16}, i) → p_{16}
  243. OrthogonalLine(G_{16}, j_{32}) → l_{16}
  244. Intersect(p_{16}, l_{16}, 2) → I_{16}
  245. OrthogonalLine(I_{16}, j_{32}) → m_{16}
  246. OrthogonalLine(I_{16}, m_{16}) → q_{16}
  247. OrthogonalLine(H_{16}, j_{32}) → r_{16}
  248. Intersect(q_{16}, r_{16}) → J_{16}
  249. Segment(I_{16}, G_{16}) → s_{16}
  250. Segment(I_{16}, J_{16}) → t_{16}
  251. Segment(J_{16}, H_{16}) → a_{17}
  252. Polygon(I_{16}, G_{16}, H_{16}, J_{16}) → poly1_{16}, i_{17}, g_{17}, h_{17}, j_{33}
  253. Circle(G_{17}, n) → k_{17}
  254. Point(k_{17}) → H_{17}
  255. Segment(G_{17}, H_{17}) → j_{34}
  256. Circle(G_{17}, i) → p_{17}
  257. OrthogonalLine(G_{17}, j_{34}) → l_{17}
  258. Intersect(p_{17}, l_{17}, 2) → I_{17}
  259. OrthogonalLine(I_{17}, j_{34}) → m_{17}
  260. OrthogonalLine(I_{17}, m_{17}) → q_{17}
  261. OrthogonalLine(H_{17}, j_{34}) → r_{17}
  262. Intersect(q_{17}, r_{17}) → J_{17}
  263. Segment(I_{17}, G_{17}) → s_{17}
  264. Segment(I_{17}, J_{17}) → t_{17}
  265. Segment(J_{17}, H_{17}) → a_{18}
  266. Polygon(I_{17}, G_{17}, H_{17}, J_{17}) → poly1_{17}, i_{18}, g_{18}, h_{18}, j_{35}
  267. Circle(G_{18}, n) → k_{18}
  268. Point(k_{18}) → H_{18}
  269. Segment(G_{18}, H_{18}) → j_{36}
  270. Circle(G_{18}, i) → p_{18}
  271. OrthogonalLine(G_{18}, j_{36}) → l_{18}
  272. Intersect(p_{18}, l_{18}, 2) → I_{18}
  273. OrthogonalLine(I_{18}, j_{36}) → m_{18}
  274. OrthogonalLine(I_{18}, m_{18}) → q_{18}
  275. OrthogonalLine(H_{18}, j_{36}) → r_{18}
  276. Intersect(q_{18}, r_{18}) → J_{18}
  277. Segment(I_{18}, G_{18}) → s_{18}
  278. Segment(I_{18}, J_{18}) → t_{18}
  279. Segment(J_{18}, H_{18}) → a_{19}
  280. Polygon(I_{18}, G_{18}, H_{18}, J_{18}) → poly1_{18}, i_{19}, g_{19}, h_{19}, j_{37}
  281. Circle(G_{19}, n) → k_{19}
  282. Point(k_{19}) → H_{19}
  283. Segment(G_{19}, H_{19}) → j_{38}
  284. Circle(G_{19}, i) → p_{19}
  285. OrthogonalLine(G_{19}, j_{38}) → l_{19}
  286. Intersect(p_{19}, l_{19}, 2) → I_{19}
  287. OrthogonalLine(I_{19}, j_{38}) → m_{19}
  288. OrthogonalLine(I_{19}, m_{19}) → q_{19}
  289. OrthogonalLine(H_{19}, j_{38}) → r_{19}
  290. Intersect(q_{19}, r_{19}) → J_{19}
  291. Segment(I_{19}, G_{19}) → s_{19}
  292. Segment(I_{19}, J_{19}) → t_{19}
  293. Segment(J_{19}, H_{19}) → a_{20}
  294. Polygon(I_{19}, G_{19}, H_{19}, J_{19}) → poly1_{19}, i_{20}, g_{20}, h_{20}, j_{39}
  295. Circle(G_{20}, n) → k_{20}
  296. Point(k_{20}) → H_{20}
  297. Segment(G_{20}, H_{20}) → j_{40}
  298. Circle(G_{20}, i) → p_{20}
  299. OrthogonalLine(G_{20}, j_{40}) → l_{20}
  300. Intersect(p_{20}, l_{20}, 2) → I_{20}
  301. OrthogonalLine(I_{20}, j_{40}) → m_{20}
  302. OrthogonalLine(I_{20}, m_{20}) → q_{20}
  303. OrthogonalLine(H_{20}, j_{40}) → r_{20}
  304. Intersect(q_{20}, r_{20}) → J_{20}
  305. Segment(I_{20}, G_{20}) → s_{20}
  306. Segment(I_{20}, J_{20}) → t_{20}
  307. Segment(J_{20}, H_{20}) → a_{21}
  308. Polygon(I_{20}, G_{20}, H_{20}, J_{20}) → poly1_{20}, i_{21}, g_{21}, h_{21}, j_{41}
  309. Circle(G_{21}, n) → k_{21}
  310. Point(k_{21}) → H_{21}
  311. Segment(G_{21}, H_{21}) → j_{42}
  312. Circle(G_{21}, i) → p_{21}
  313. OrthogonalLine(G_{21}, j_{42}) → l_{21}
  314. Intersect(p_{21}, l_{21}, 2) → I_{21}
  315. OrthogonalLine(I_{21}, j_{42}) → m_{21}
  316. OrthogonalLine(I_{21}, m_{21}) → q_{21}
  317. OrthogonalLine(H_{21}, j_{42}) → r_{21}
  318. Intersect(q_{21}, r_{21}) → J_{21}
  319. Segment(I_{21}, G_{21}) → s_{21}
  320. Segment(I_{21}, J_{21}) → t_{21}
  321. Segment(J_{21}, H_{21}) → a_{22}
  322. Polygon(I_{21}, G_{21}, H_{21}, J_{21}) → poly1_{21}, i_{22}, g_{22}, h_{22}, j_{43}
  323. Circle(G_{22}, n) → k_{22}
  324. Point(k_{22}) → H_{22}
  325. Segment(G_{22}, H_{22}) → j_{44}
  326. Circle(G_{22}, i) → p_{22}
  327. OrthogonalLine(G_{22}, j_{44}) → l_{22}
  328. Intersect(p_{22}, l_{22}, 2) → I_{22}
  329. OrthogonalLine(I_{22}, j_{44}) → m_{22}
  330. OrthogonalLine(I_{22}, m_{22}) → q_{22}
  331. OrthogonalLine(H_{22}, j_{44}) → r_{22}
  332. Intersect(q_{22}, r_{22}) → J_{22}
  333. Segment(I_{22}, G_{22}) → s_{22}
  334. Segment(I_{22}, J_{22}) → t_{22}
  335. Segment(J_{22}, H_{22}) → a_{23}
  336. Polygon(I_{22}, G_{22}, H_{22}, J_{22}) → poly1_{22}, i_{23}, g_{23}, h_{23}, j_{45}
  337. Circle(G_{23}, n) → k_{23}
  338. Point(k_{23}) → H_{23}
  339. Segment(G_{23}, H_{23}) → j_{46}
  340. Circle(G_{23}, i) → p_{23}
  341. OrthogonalLine(G_{23}, j_{46}) → l_{23}
  342. Intersect(p_{23}, l_{23}, 2) → I_{23}
  343. OrthogonalLine(I_{23}, j_{46}) → m_{23}
  344. OrthogonalLine(I_{23}, m_{23}) → q_{23}
  345. OrthogonalLine(H_{23}, j_{46}) → r_{23}
  346. Intersect(q_{23}, r_{23}) → J_{23}
  347. Segment(I_{23}, G_{23}) → s_{23}
  348. Segment(I_{23}, J_{23}) → t_{23}
  349. Segment(J_{23}, H_{23}) → a_{24}
  350. Polygon(I_{23}, G_{23}, H_{23}, J_{23}) → poly1_{23}, i_{24}, g_{24}, h_{24}, j_{47}
  351. Circle(G_{24}, n) → k_{24}
  352. Point(k_{24}) → H_{24}
  353. Segment(G_{24}, H_{24}) → j_{48}
  354. Circle(G_{24}, i) → p_{24}
  355. OrthogonalLine(G_{24}, j_{48}) → l_{24}
  356. Intersect(p_{24}, l_{24}, 2) → I_{24}
  357. OrthogonalLine(I_{24}, j_{48}) → m_{24}
  358. OrthogonalLine(I_{24}, m_{24}) → q_{24}
  359. OrthogonalLine(H_{24}, j_{48}) → r_{24}
  360. Intersect(q_{24}, r_{24}) → J_{24}
  361. Segment(I_{24}, G_{24}) → s_{24}
  362. Segment(I_{24}, J_{24}) → t_{24}
  363. Segment(J_{24}, H_{24}) → a_{25}
  364. Polygon(I_{24}, G_{24}, H_{24}, J_{24}) → poly1_{24}, i_{25}, g_{25}, h_{25}, j_{49}
  365. Circle(G_{25}, n) → k_{25}
  366. Point(k_{25}) → H_{25}
  367. Segment(G_{25}, H_{25}) → j_{50}
  368. Circle(G_{25}, i) → p_{25}
  369. OrthogonalLine(G_{25}, j_{50}) → l_{25}
  370. Intersect(p_{25}, l_{25}, 2) → I_{25}
  371. OrthogonalLine(I_{25}, j_{50}) → m_{25}
  372. OrthogonalLine(I_{25}, m_{25}) → q_{25}
  373. OrthogonalLine(H_{25}, j_{50}) → r_{25}
  374. Intersect(q_{25}, r_{25}) → J_{25}
  375. Segment(I_{25}, G_{25}) → s_{25}
  376. Segment(I_{25}, J_{25}) → t_{25}
  377. Segment(J_{25}, H_{25}) → a_{26}
  378. Polygon(I_{25}, G_{25}, H_{25}, J_{25}) → poly1_{25}, i_{26}, g_{26}, h_{26}, j_{51}
  379. Circle(G_{26}, n) → k_{26}
  380. Point(k_{26}) → H_{26}
  381. Segment(G_{26}, H_{26}) → j_{52}
  382. Circle(G_{26}, i) → p_{26}
  383. OrthogonalLine(G_{26}, j_{52}) → l_{26}
  384. Intersect(p_{26}, l_{26}, 2) → I_{26}
  385. OrthogonalLine(I_{26}, j_{52}) → m_{26}
  386. OrthogonalLine(I_{26}, m_{26}) → q_{26}
  387. OrthogonalLine(H_{26}, j_{52}) → r_{26}
  388. Intersect(q_{26}, r_{26}) → J_{26}
  389. Segment(I_{26}, G_{26}) → s_{26}
  390. Segment(I_{26}, J_{26}) → t_{26}
  391. Segment(J_{26}, H_{26}) → a_{27}
  392. Polygon(I_{26}, G_{26}, H_{26}, J_{26}) → poly1_{26}, i_{27}, g_{27}, h_{27}, j_{53}
  393. Circle(G_{27}, n) → k_{27}
  394. Point(k_{27}) → H_{27}
  395. Segment(G_{27}, H_{27}) → j_{54}
  396. Circle(G_{27}, i) → p_{27}
  397. OrthogonalLine(G_{27}, j_{54}) → l_{27}
  398. Intersect(p_{27}, l_{27}, 2) → I_{27}
  399. OrthogonalLine(I_{27}, j_{54}) → m_{27}
  400. OrthogonalLine(I_{27}, m_{27}) → q_{27}
  401. OrthogonalLine(H_{27}, j_{54}) → r_{27}
  402. Intersect(q_{27}, r_{27}) → J_{27}
  403. Segment(I_{27}, G_{27}) → s_{27}
  404. Segment(I_{27}, J_{27}) → t_{27}
  405. Segment(J_{27}, H_{27}) → a_{28}
  406. Polygon(I_{27}, G_{27}, H_{27}, J_{27}) → poly1_{27}, i_{28}, g_{28}, h_{28}, j_{55}
  407. Circle(G_{28}, n) → k_{28}
  408. Point(k_{28}) → H_{28}
  409. Segment(G_{28}, H_{28}) → j_{56}
  410. Circle(G_{28}, i) → p_{28}
  411. OrthogonalLine(G_{28}, j_{56}) → l_{28}
  412. Intersect(p_{28}, l_{28}, 2) → I_{28}
  413. OrthogonalLine(I_{28}, j_{56}) → m_{28}
  414. OrthogonalLine(I_{28}, m_{28}) → q_{28}
  415. OrthogonalLine(H_{28}, j_{56}) → r_{28}
  416. Intersect(q_{28}, r_{28}) → J_{28}
  417. Segment(I_{28}, G_{28}) → s_{28}
  418. Segment(I_{28}, J_{28}) → t_{28}
  419. Segment(J_{28}, H_{28}) → a_{29}
  420. Polygon(I_{28}, G_{28}, H_{28}, J_{28}) → poly1_{28}, i_{29}, g_{29}, h_{29}, j_{57}
  421. Circle(G_{29}, n) → k_{29}
  422. Point(k_{29}) → H_{29}
  423. Segment(G_{29}, H_{29}) → j_{58}
  424. Circle(G_{29}, i) → p_{29}
  425. OrthogonalLine(G_{29}, j_{58}) → l_{29}
  426. Intersect(p_{29}, l_{29}, 2) → I_{29}
  427. OrthogonalLine(I_{29}, j_{58}) → m_{29}
  428. OrthogonalLine(I_{29}, m_{29}) → q_{29}
  429. OrthogonalLine(H_{29}, j_{58}) → r_{29}
  430. Intersect(q_{29}, r_{29}) → J_{29}
  431. Segment(I_{29}, G_{29}) → s_{29}
  432. Segment(I_{29}, J_{29}) → t_{29}
  433. Segment(J_{29}, H_{29}) → a_{30}
  434. Polygon(I_{29}, G_{29}, H_{29}, J_{29}) → poly1_{29}, i_{30}, g_{30}, h_{30}, j_{59}
  435. Circle(G_{30}, n) → k_{30}
  436. Point(k_{30}) → H_{30}
  437. Segment(G_{30}, H_{30}) → j_{60}
  438. Circle(G_{30}, i) → p_{30}
  439. OrthogonalLine(G_{30}, j_{60}) → l_{30}
  440. Intersect(p_{30}, l_{30}, 2) → I_{30}
  441. OrthogonalLine(I_{30}, j_{60}) → m_{30}
  442. OrthogonalLine(I_{30}, m_{30}) → q_{30}
  443. OrthogonalLine(H_{30}, j_{60}) → r_{30}
  444. Intersect(q_{30}, r_{30}) → J_{30}
  445. Segment(I_{30}, G_{30}) → s_{30}
  446. Segment(I_{30}, J_{30}) → t_{30}
  447. Segment(J_{30}, H_{30}) → a_{31}
  448. Polygon(I_{30}, G_{30}, H_{30}, J_{30}) → poly1_{30}, i_{31}, g_{31}, h_{31}, j_{61}
  449. Circle(G_{31}, n) → k_{31}
  450. Point(k_{31}) → H_{31}
  451. Segment(G_{31}, H_{31}) → j_{62}
  452. Circle(G_{31}, i) → p_{31}
  453. OrthogonalLine(G_{31}, j_{62}) → l_{31}
  454. Intersect(p_{31}, l_{31}, 2) → I_{31}
  455. OrthogonalLine(I_{31}, j_{62}) → m_{31}
  456. OrthogonalLine(I_{31}, m_{31}) → q_{31}
  457. OrthogonalLine(H_{31}, j_{62}) → r_{31}
  458. Intersect(q_{31}, r_{31}) → J_{31}
  459. Segment(I_{31}, G_{31}) → s_{31}
  460. Segment(I_{31}, J_{31}) → t_{31}
  461. Segment(J_{31}, H_{31}) → a_{32}
  462. Polygon(I_{31}, G_{31}, H_{31}, J_{31}) → poly1_{31}, i_{32}, g_{32}, h_{32}, j_{63}
  463. Circle(G_{32}, n) → k_{32}
  464. Point(k_{32}) → H_{32}
  465. Segment(G_{32}, H_{32}) → j_{64}
  466. Circle(G_{32}, i) → p_{32}
  467. OrthogonalLine(G_{32}, j_{64}) → l_{32}
  468. Intersect(p_{32}, l_{32}, 2) → I_{32}
  469. OrthogonalLine(I_{32}, j_{64}) → m_{32}
  470. OrthogonalLine(I_{32}, m_{32}) → q_{32}
  471. OrthogonalLine(H_{32}, j_{64}) → r_{32}
  472. Intersect(q_{32}, r_{32}) → J_{32}
  473. Segment(I_{32}, G_{32}) → s_{32}
  474. Segment(I_{32}, J_{32}) → t_{32}
  475. Segment(J_{32}, H_{32}) → a_{33}
  476. Polygon(I_{32}, G_{32}, H_{32}, J_{32}) → poly1_{32}, i_{33}, g_{33}, h_{33}, j_{65}
  477. Circle(G_{33}, n) → k_{33}
  478. Point(k_{33}) → H_{33}
  479. Segment(G_{33}, H_{33}) → j_{66}
  480. Circle(G_{33}, i) → p_{33}
  481. OrthogonalLine(G_{33}, j_{66}) → l_{33}
  482. Intersect(p_{33}, l_{33}, 2) → I_{33}
  483. OrthogonalLine(I_{33}, j_{66}) → m_{33}
  484. OrthogonalLine(I_{33}, m_{33}) → q_{33}
  485. OrthogonalLine(H_{33}, j_{66}) → r_{33}
  486. Intersect(q_{33}, r_{33}) → J_{33}
  487. Segment(I_{33}, G_{33}) → s_{33}
  488. Segment(I_{33}, J_{33}) → t_{33}
  489. Segment(J_{33}, H_{33}) → a_{34}
  490. Polygon(I_{33}, G_{33}, H_{33}, J_{33}) → poly1_{33}, i_{34}, g_{34}, h_{34}, j_{67}
  491. Circle(G_{34}, n) → k_{34}
  492. Point(k_{34}) → H_{34}
  493. Segment(G_{34}, H_{34}) → j_{68}
  494. Circle(G_{34}, i) → p_{34}
  495. OrthogonalLine(G_{34}, j_{68}) → l_{34}
  496. Intersect(p_{34}, l_{34}, 2) → I_{34}
  497. OrthogonalLine(I_{34}, j_{68}) → m_{34}
  498. OrthogonalLine(I_{34}, m_{34}) → q_{34}
  499. OrthogonalLine(H_{34}, j_{68}) → r_{34}
  500. Intersect(q_{34}, r_{34}) → J_{34}
  501. Segment(I_{34}, G_{34}) → s_{34}
  502. Segment(I_{34}, J_{34}) → t_{34}
  503. Segment(J_{34}, H_{34}) → a_{35}
  504. Polygon(I_{34}, G_{34}, H_{34}, J_{34}) → poly1_{34}, i_{35}, g_{35}, h_{35}, j_{69}
  505. Circle(G_{35}, n) → k_{35}
  506. Point(k_{35}) → H_{35}
  507. Segment(G_{35}, H_{35}) → j_{70}
  508. Circle(G_{35}, i) → p_{35}
  509. OrthogonalLine(G_{35}, j_{70}) → l_{35}
  510. Intersect(p_{35}, l_{35}, 2) → I_{35}
  511. OrthogonalLine(I_{35}, j_{70}) → m_{35}
  512. OrthogonalLine(I_{35}, m_{35}) → q_{35}
  513. OrthogonalLine(H_{35}, j_{70}) → r_{35}
  514. Intersect(q_{35}, r_{35}) → J_{35}
  515. Segment(I_{35}, G_{35}) → s_{35}
  516. Segment(I_{35}, J_{35}) → t_{35}
  517. Segment(J_{35}, H_{35}) → a_{36}
  518. Polygon(I_{35}, G_{35}, H_{35}, J_{35}) → poly1_{35}, i_{36}, g_{36}, h_{36}, j_{71}
  519. Circle(G_{36}, n) → k_{36}
  520. Point(k_{36}) → H_{36}
  521. Segment(G_{36}, H_{36}) → j_{72}
  522. Circle(G_{36}, i) → p_{36}
  523. OrthogonalLine(G_{36}, j_{72}) → l_{36}
  524. Intersect(p_{36}, l_{36}, 2) → I_{36}
  525. OrthogonalLine(I_{36}, j_{72}) → m_{36}
  526. OrthogonalLine(I_{36}, m_{36}) → q_{36}
  527. OrthogonalLine(H_{36}, j_{72}) → r_{36}
  528. Intersect(q_{36}, r_{36}) → J_{36}
  529. Segment(I_{36}, G_{36}) → s_{36}
  530. Segment(I_{36}, J_{36}) → t_{36}
  531. Segment(J_{36}, H_{36}) → a_{37}
  532. Polygon(I_{36}, G_{36}, H_{36}, J_{36}) → poly1_{36}, i_{37}, g_{37}, h_{37}, j_{73}
  533. Polygon(E, F, D, A) → poly2, e_1, f_1, d_1, a_{38}
  534. Sequence(If[IsInRegion[Element[list1, n], poly2] ∧ IsInRegion[Element[List2, n], poly2] ∧ IsInRegion[Element[List3, n], poly2] ∧ IsInRegion[Element[List4, n], poly2], 1, 0], n, 1, 36) → list5
  535. Sum(list5) → z
  536. Midpoint(A, D) → K
  537. Midpoint(F, D) → L
  538. Sequence(Segment[(q, 0), (q, y(E))], q, 1, h - 1) → list2
  539. Sequence(Segment[(0, q), (x(D), q)], q, 1, y(E) - 1) → list3

AI Drawing Prompt Reference

Draw a rectangle and fill it with a series of smaller squares. First, draw two sides of the rectangle, then construct a series of smaller squares inside the rectangle using the diagonals as sides, with each square's side length gradually decreasing until no more squares can be drawn.
The prompt above was compiled from the original drawing commands and is for reference only.