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Max Bill System: Mathematical Art with Rotation and Probability

Max Bill System: Mathematical Art with Rotation and Probability
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2020-12-14 10:59
SequenceSolid GeometryDerivativeProbability & Statistics
① This example demonstrates a mathematical artwork named 'Max Bill System', which combines rotation, probability, and 3D geometric elements to generate visually striking patterns. ② Core mathematical concepts include rotation, squares, probability, solids/3D shapes, confidence intervals, derivatives, and diagrams, highlighting the intersection of mathematics and art. ③ Construction highlights: Based on the construction summary, it uses GeoGebra commands such as Sequence, Flatten, Element, and Length, along with numerous objects (A1 to A23, B1 to B7, l1, l3, l4, a, b, M, N, etc.), to build complex structures via sequence generation and element extraction, possibly incorporating sliders or random numbers for probabilistic simulation. ④ Target audience: Suitable for high school to university levels, useful for mathematical art displays, probability/statistics visualization, 3D geometry exploration, or intuitive demonstrations of derivatives, and can serve as material for interdisciplinary projects (math + art).

GeoGebra Commands & Parameters

  1. Sequence(Polygon[A + (0, (i * (i - 1) / ((2 * N)))), A + (1, (i * (i - 1) / ((2 * N)))), A + (1, (i * (i + 1) / ((2 * N)))), A + (0, (i * (i + 1) / ((2 * N))))], i, 1, n) → l1
  2. Flatten(Sequence[Sequence[Translate[Rotate[Element[l1, 1], ((Mod[i, 2] + j) * 90°), A + (0.5, 0.5)], (i, j)], i, 0, m - 1], j, 0, m - 1]) → l3
  3. Element(l3, $A1) → B1
  4. Element(l3, $A2) → B2
  5. Element(l3, $A3) → B3
  6. Element(l3, $A4) → B4
  7. Element(l3, $A5) → B5
  8. Element(l3, $A6) → B6
  9. Element(l3, $A7) → B7
  10. Length(l3) → M
  11. Flatten(Sequence[Sequence[Translate[Rotate[Element[l1, 2], ((Mod[i, 2] + j) * 90°), A + (0.5, 0.5)], (i, j)], i, 0, m - 1], j, 0, m - 1]) → l4
  12. Element(l3, $A8) → B8
  13. Element(l3, $A9) → B9
  14. Element(l3, $A10) → B10
  15. Element(l3, $A11) → B11
  16. Element(l3, $A12) → B12
  17. Element(l3, $A13) → B13
  18. Element(l3, $A14) → B14
  19. Element(l3, $A15) → B15
  20. Element(l3, $A16) → B16
  21. Element(l3, $A17) → B17
  22. Element(l3, $A18) → B18
  23. Element(l3, $A19) → B19
  24. Element(l3, $A20) → B20
  25. Element(l3, $A21) → B21
  26. Element(l3, $A22) → B22
  27. Element(l3, $A23) → B23
  28. Element(l3, $A24) → B24
  29. Element(l3, $A25) → B25
  30. Element(l3, $A26) → B26
  31. Element(l3, $A27) → B27
  32. Element(l3, $A28) → B28
  33. Element(l3, $A29) → B29
  34. Element(l3, $A30) → B30
  35. Element(l3, $A31) → B31
  36. Element(l3, $A32) → B32
  37. Element(l3, $A33) → B33
  38. Element(l3, $A34) → B34
  39. Element(l3, $A35) → B35
  40. Element(l3, $A36) → B36
  41. Element(l3, $A37) → B37
  42. Element(l3, $A38) → B38
  43. Element(l3, $A39) → B39
  44. Element(l3, $A40) → B40
  45. Element(l3, $A41) → B41
  46. Element(l3, $A42) → B42
  47. Element(l3, $A43) → B43
  48. Element(l3, $A44) → B44
  49. Element(l3, $A45) → B45
  50. Element(l3, $A46) → B46
  51. Element(l3, $A47) → B47
  52. Element(l3, $A48) → B48
  53. Element(l3, $A49) → B49
  54. Element(l3, $A50) → B50
  55. Element(l3, $A51) → B51
  56. Element(l3, $A52) → B52
  57. Element(l3, $A53) → B53
  58. Element(l3, $A54) → B54
  59. Element(l3, $A55) → B55
  60. Element(l3, $A56) → B56
  61. Element(l3, $A57) → B57
  62. Element(l3, $A58) → B58
  63. Element(l3, $A59) → B59
  64. Element(l3, $A60) → B60
  65. Element(l3, $A61) → B61
  66. Element(l3, $A62) → B62
  67. Element(l3, $A63) → B63
  68. Element(l3, $A64) → B64
  69. Element(l4, $A1) → C1
  70. Flatten(Sequence[Sequence[Translate[Rotate[Element[l1, 3], ((Mod[i, 2] + j) * 90°), A + (0.5, 0.5)], (i, j)], i, 0, m - 1], j, 0, m - 1]) → l5
  71. Flatten(Sequence[Sequence[Translate[Rotate[Element[l1, 4], ((Mod[i, 2] + j) * 90°), A + (0.5, 0.5)], (i, j)], i, 0, m - 1], j, 0, m - 1]) → l6
  72. Flatten(Sequence[Sequence[Translate[Rotate[Element[l1, 5], ((Mod[i, 2] + j) * 90°), A + (0.5, 0.5)], (i, j)], i, 0, m - 1], j, 0, m - 1]) → l7
  73. Flatten(Sequence[Sequence[Translate[Rotate[Element[l1, 6], ((Mod[i, 2] + j) * 90°), A + (0.5, 0.5)], (i, j)], i, 0, m - 1], j, 0, m - 1]) → l8
  74. Flatten(Sequence[Sequence[Translate[Rotate[Element[l1, 7], ((Mod[i, 2] + j) * 90°), A + (0.5, 0.5)], (i, j)], i, 0, m - 1], j, 0, m - 1]) → l9
  75. Element(l4, $A2) → C2
  76. Element(l4, $A3) → C3
  77. Element(l4, $A4) → C4
  78. Element(l4, $A5) → C5
  79. Element(l4, $A6) → C6
  80. Element(l4, $A7) → C7
  81. Element(l4, $A8) → C8
  82. Element(l4, $A9) → C9
  83. Element(l4, $A10) → C10
  84. Element(l4, $A11) → C11
  85. Element(l4, $A12) → C12
  86. Element(l4, $A13) → C13
  87. Element(l4, $A14) → C14
  88. Element(l4, $A15) → C15
  89. Element(l4, $A16) → C16
  90. Element(l4, $A17) → C17
  91. Element(l4, $A18) → C18
  92. Element(l4, $A19) → C19
  93. Element(l4, $A20) → C20
  94. Element(l4, $A21) → C21
  95. Element(l4, $A22) → C22
  96. Element(l4, $A23) → C23
  97. Element(l4, $A24) → C24
  98. Element(l4, $A25) → C25
  99. Element(l4, $A26) → C26
  100. Element(l4, $A27) → C27
  101. Element(l4, $A28) → C28
  102. Element(l4, $A29) → C29
  103. Element(l4, $A30) → C30
  104. Element(l4, $A31) → C31
  105. Element(l4, $A32) → C32
  106. Element(l4, $A33) → C33
  107. Element(l4, $A34) → C34
  108. Element(l4, $A35) → C35
  109. Element(l4, $A36) → C36
  110. Element(l4, $A37) → C37
  111. Element(l4, $A38) → C38
  112. Element(l4, $A39) → C39
  113. Element(l4, $A40) → C40
  114. Element(l4, $A41) → C41
  115. Element(l4, $A42) → C42
  116. Element(l4, $A43) → C43
  117. Element(l4, $A44) → C44
  118. Element(l4, $A45) → C45
  119. Element(l4, $A46) → C46
  120. Element(l4, $A47) → C47
  121. Element(l4, $A48) → C48
  122. Element(l4, $A49) → C49
  123. Element(l4, $A50) → C50
  124. Element(l4, $A51) → C51
  125. Element(l4, $A52) → C52
  126. Element(l4, $A53) → C53
  127. Element(l4, $A54) → C54
  128. Element(l4, $A55) → C55
  129. Element(l4, $A56) → C56
  130. Element(l4, $A57) → C57
  131. Element(l4, $A58) → C58
  132. Element(l4, $A59) → C59
  133. Element(l4, $A60) → C60
  134. Element(l4, $A61) → C61
  135. Element(l4, $A62) → C62
  136. Element(l4, $A63) → C63
  137. Element(l4, $A64) → C64
  138. Element(l5, $A1) → D1
  139. Element(l5, $A2) → D2
  140. Element(l5, $A3) → D3
  141. Element(l5, $A4) → D4
  142. Element(l5, $A5) → D5
  143. Element(l5, $A6) → D6
  144. Element(l5, $A7) → D7
  145. Element(l5, $A8) → D8
  146. Element(l5, $A9) → D9
  147. Element(l5, $A10) → D10
  148. Element(l5, $A11) → D11
  149. Element(l5, $A12) → D12
  150. Element(l5, $A13) → D13
  151. Element(l5, $A14) → D14
  152. Element(l5, $A15) → D15
  153. Element(l5, $A16) → D16
  154. Element(l5, $A17) → D17
  155. Element(l5, $A18) → D18
  156. Element(l5, $A19) → D19
  157. Element(l5, $A20) → D20
  158. Element(l5, $A21) → D21
  159. Element(l5, $A22) → D22
  160. Element(l5, $A23) → D23
  161. Element(l5, $A24) → D24
  162. Element(l5, $A25) → D25
  163. Element(l5, $A26) → D26
  164. Element(l5, $A27) → D27
  165. Element(l5, $A28) → D28
  166. Element(l5, $A29) → D29
  167. Element(l5, $A30) → D30
  168. Element(l5, $A31) → D31
  169. Element(l5, $A32) → D32
  170. Element(l5, $A33) → D33
  171. Element(l5, $A34) → D34
  172. Element(l5, $A35) → D35
  173. Element(l5, $A36) → D36
  174. Element(l5, $A37) → D37
  175. Element(l5, $A38) → D38
  176. Element(l5, $A39) → D39
  177. Element(l5, $A40) → D40
  178. Element(l5, $A41) → D41
  179. Element(l5, $A42) → D42
  180. Element(l5, $A43) → D43
  181. Element(l5, $A44) → D44
  182. Element(l5, $A45) → D45
  183. Element(l5, $A46) → D46
  184. Element(l5, $A47) → D47
  185. Element(l5, $A48) → D48
  186. Element(l5, $A49) → D49
  187. Element(l5, $A50) → D50
  188. Element(l5, $A51) → D51
  189. Element(l5, $A52) → D52
  190. Element(l5, $A53) → D53
  191. Element(l5, $A54) → D54
  192. Element(l5, $A55) → D55
  193. Element(l5, $A56) → D56
  194. Element(l5, $A57) → D57
  195. Element(l5, $A58) → D58
  196. Element(l5, $A59) → D59
  197. Element(l5, $A60) → D60
  198. Element(l5, $A61) → D61
  199. Element(l5, $A62) → D62
  200. Element(l5, $A63) → D63
  201. Element(l5, $A64) → D64
  202. Element(l6, $A1) → E1
  203. Element(l6, $A2) → E2
  204. Element(l6, $A3) → E3
  205. Element(l6, $A4) → E4
  206. Element(l6, $A5) → E5
  207. Element(l6, $A6) → E6
  208. Element(l6, $A7) → E7
  209. Element(l6, $A8) → E8
  210. Element(l6, $A9) → E9
  211. Element(l6, $A10) → E10
  212. Element(l6, $A11) → E11
  213. Element(l6, $A12) → E12
  214. Element(l6, $A13) → E13
  215. Element(l6, $A14) → E14
  216. Element(l6, $A15) → E15
  217. Element(l6, $A16) → E16
  218. Element(l6, $A17) → E17
  219. Element(l6, $A18) → E18
  220. Element(l6, $A19) → E19
  221. Element(l6, $A20) → E20
  222. Element(l6, $A21) → E21
  223. Element(l6, $A22) → E22
  224. Element(l6, $A23) → E23
  225. Element(l6, $A24) → E24
  226. Element(l6, $A25) → E25
  227. Element(l6, $A26) → E26
  228. Element(l6, $A27) → E27
  229. Element(l6, $A28) → E28
  230. Element(l6, $A29) → E29
  231. Element(l6, $A30) → E30
  232. Element(l6, $A31) → E31
  233. Element(l6, $A32) → E32
  234. Element(l6, $A33) → E33
  235. Element(l6, $A34) → E34
  236. Element(l6, $A35) → E35
  237. Element(l6, $A36) → E36
  238. Element(l6, $A37) → E37
  239. Element(l6, $A38) → E38
  240. Element(l6, $A39) → E39
  241. Element(l6, $A40) → E40
  242. Element(l6, $A41) → E41
  243. Element(l6, $A42) → E42
  244. Element(l6, $A43) → E43
  245. Element(l6, $A44) → E44
  246. Element(l6, $A45) → E45
  247. Element(l6, $A46) → E46
  248. Element(l6, $A47) → E47
  249. Element(l6, $A48) → E48
  250. Element(l6, $A49) → E49
  251. Element(l6, $A50) → E50
  252. Element(l6, $A51) → E51
  253. Element(l6, $A52) → E52
  254. Element(l6, $A53) → E53
  255. Element(l6, $A54) → E54
  256. Element(l6, $A55) → E55
  257. Element(l6, $A56) → E56
  258. Element(l6, $A57) → E57
  259. Element(l6, $A58) → E58
  260. Element(l6, $A59) → E59
  261. Element(l6, $A60) → E60
  262. Element(l6, $A61) → E61
  263. Element(l6, $A62) → E62
  264. Element(l6, $A63) → E63
  265. Element(l6, $A64) → E64
  266. Element(l7, $A1) → F1
  267. Element(l7, $A2) → F2
  268. Element(l7, $A3) → F3
  269. Element(l7, $A4) → F4
  270. Element(l7, $A5) → F5
  271. Element(l7, $A6) → F6
  272. Element(l7, $A7) → F7
  273. Element(l7, $A8) → F8
  274. Element(l7, $A9) → F9
  275. Element(l7, $A10) → F10
  276. Element(l7, $A11) → F11
  277. Element(l7, $A12) → F12
  278. Element(l7, $A13) → F13
  279. Element(l7, $A14) → F14
  280. Element(l7, $A15) → F15
  281. Element(l7, $A16) → F16
  282. Element(l7, $A17) → F17
  283. Element(l7, $A18) → F18
  284. Element(l7, $A19) → F19
  285. Element(l7, $A20) → F20
  286. Element(l7, $A21) → F21
  287. Element(l7, $A22) → F22
  288. Element(l7, $A23) → F23
  289. Element(l7, $A24) → F24
  290. Element(l7, $A25) → F25
  291. Element(l7, $A26) → F26
  292. Element(l7, $A27) → F27
  293. Element(l7, $A28) → F28
  294. Element(l7, $A29) → F29
  295. Element(l7, $A30) → F30
  296. Element(l7, $A31) → F31
  297. Element(l7, $A32) → F32
  298. Element(l7, $A33) → F33
  299. Element(l7, $A34) → F34
  300. Element(l7, $A35) → F35
  301. Element(l7, $A36) → F36
  302. Element(l7, $A37) → F37
  303. Element(l7, $A38) → F38
  304. Element(l7, $A39) → F39
  305. Element(l7, $A40) → F40
  306. Element(l7, $A41) → F41
  307. Element(l7, $A42) → F42
  308. Element(l7, $A43) → F43
  309. Element(l7, $A44) → F44
  310. Element(l7, $A45) → F45
  311. Element(l7, $A46) → F46
  312. Element(l7, $A47) → F47
  313. Element(l7, $A48) → F48
  314. Element(l7, $A49) → F49
  315. Element(l7, $A50) → F50
  316. Element(l7, $A51) → F51
  317. Element(l7, $A52) → F52
  318. Element(l7, $A53) → F53
  319. Element(l7, $A54) → F54
  320. Element(l7, $A55) → F55
  321. Element(l7, $A56) → F56
  322. Element(l7, $A57) → F57
  323. Element(l7, $A58) → F58
  324. Element(l7, $A59) → F59
  325. Element(l7, $A60) → F60
  326. Element(l7, $A61) → F61
  327. Element(l7, $A62) → F62
  328. Element(l7, $A63) → F63
  329. Element(l7, $A64) → F64
  330. Element(l8, $A1) → G1
  331. Element(l8, $A2) → G2
  332. Element(l8, $A3) → G3
  333. Element(l8, $A4) → G4
  334. Element(l8, $A5) → G5
  335. Element(l8, $A6) → G6
  336. Element(l8, $A7) → G7
  337. Element(l8, $A8) → G8
  338. Element(l8, $A9) → G9
  339. Element(l8, $A10) → G10
  340. Element(l8, $A11) → G11
  341. Element(l8, $A12) → G12
  342. Element(l8, $A13) → G13
  343. Element(l8, $A14) → G14
  344. Element(l8, $A15) → G15
  345. Element(l8, $A16) → G16
  346. Element(l8, $A17) → G17
  347. Element(l8, $A18) → G18
  348. Element(l8, $A19) → G19
  349. Element(l8, $A20) → G20
  350. Element(l8, $A21) → G21
  351. Element(l8, $A22) → G22
  352. Element(l8, $A23) → G23
  353. Element(l8, $A24) → G24
  354. Element(l8, $A25) → G25
  355. Element(l8, $A26) → G26
  356. Element(l8, $A27) → G27
  357. Element(l8, $A28) → G28
  358. Element(l8, $A29) → G29
  359. Element(l8, $A30) → G30
  360. Element(l8, $A31) → G31
  361. Element(l8, $A32) → G32
  362. Element(l8, $A33) → G33
  363. Element(l8, $A34) → G34
  364. Element(l8, $A35) → G35
  365. Element(l8, $A36) → G36
  366. Element(l8, $A37) → G37
  367. Element(l8, $A38) → G38
  368. Element(l8, $A39) → G39
  369. Element(l8, $A40) → G40
  370. Element(l8, $A41) → G41
  371. Element(l8, $A42) → G42
  372. Element(l8, $A43) → G43
  373. Element(l8, $A44) → G44
  374. Element(l8, $A45) → G45
  375. Element(l8, $A46) → G46
  376. Element(l8, $A47) → G47
  377. Element(l8, $A48) → G48
  378. Element(l8, $A49) → G49
  379. Element(l8, $A50) → G50
  380. Element(l8, $A51) → G51
  381. Element(l8, $A52) → G52
  382. Element(l8, $A53) → G53
  383. Element(l8, $A54) → G54
  384. Element(l8, $A55) → G55
  385. Element(l8, $A56) → G56
  386. Element(l8, $A57) → G57
  387. Element(l8, $A58) → G58
  388. Element(l8, $A59) → G59
  389. Element(l8, $A60) → G60
  390. Element(l8, $A61) → G61
  391. Element(l8, $A62) → G62
  392. Element(l8, $A63) → G63
  393. Element(l8, $A64) → G64
  394. Element(l9, $A1) → H1
  395. Element(l9, $A2) → H2
  396. Element(l9, $A3) → H3
  397. Element(l9, $A4) → H4
  398. Element(l9, $A5) → H5
  399. Element(l9, $A6) → H6
  400. Element(l9, $A7) → H7
  401. Element(l9, $A8) → H8
  402. Element(l9, $A9) → H9
  403. Element(l9, $A10) → H10
  404. Element(l9, $A11) → H11
  405. Element(l9, $A12) → H12
  406. Element(l9, $A13) → H13
  407. Element(l9, $A14) → H14
  408. Element(l9, $A15) → H15
  409. Element(l9, $A16) → H16
  410. Element(l9, $A17) → H17
  411. Element(l9, $A18) → H18
  412. Element(l9, $A19) → H19
  413. Element(l9, $A20) → H20
  414. Element(l9, $A21) → H21
  415. Element(l9, $A22) → H22
  416. Element(l9, $A23) → H23
  417. Element(l9, $A24) → H24
  418. Element(l9, $A25) → H25
  419. Element(l9, $A26) → H26
  420. Element(l9, $A27) → H27
  421. Element(l9, $A28) → H28
  422. Element(l9, $A29) → H29
  423. Element(l9, $A30) → H30
  424. Element(l9, $A31) → H31
  425. Element(l9, $A32) → H32
  426. Element(l9, $A33) → H33
  427. Element(l9, $A34) → H34
  428. Element(l9, $A35) → H35
  429. Element(l9, $A36) → H36
  430. Element(l9, $A37) → H37
  431. Element(l9, $A38) → H38
  432. Element(l9, $A39) → H39
  433. Element(l9, $A40) → H40
  434. Element(l9, $A41) → H41
  435. Element(l9, $A42) → H42
  436. Element(l9, $A43) → H43
  437. Element(l9, $A44) → H44
  438. Element(l9, $A45) → H45
  439. Element(l9, $A46) → H46
  440. Element(l9, $A47) → H47
  441. Element(l9, $A48) → H48
  442. Element(l9, $A49) → H49
  443. Element(l9, $A50) → H50
  444. Element(l9, $A51) → H51
  445. Element(l9, $A52) → H52
  446. Element(l9, $A53) → H53
  447. Element(l9, $A54) → H54
  448. Element(l9, $A55) → H55
  449. Element(l9, $A56) → H56
  450. Element(l9, $A57) → H57
  451. Element(l9, $A58) → H58
  452. Element(l9, $A59) → H59
  453. Element(l9, $A60) → H60
  454. Element(l9, $A61) → H61
  455. Element(l9, $A62) → H62
  456. Element(l9, $A63) → H63
  457. Element(l9, $A64) → H64

AI Drawing Prompt Reference

Create a mathematical art piece combining rotation and probability, featuring complex 3D patterns formed by rotating and translating squares. Generate a sequence of squares, rotate and translate them at various angles, and highlight selected structures randomly to create a visually striking geometric pattern that showcases the intersection of mathematics and art.
The prompt above was compiled from the original drawing commands and is for reference only.