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Logarithmic/Golden Polygonal Spiral

Logarithmic/Golden Polygonal Spiral
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2020-06-13 21:09
PolygonFunctionTrigonometryTransformationGeometric Construction
This example demonstrates the construction of a logarithmic/golden polygonal spiral. It involves concepts of logarithmic growth, the golden ratio, and polygons. Using commands like Rotate, Intersect, If, and Circle, multiple polygons are constructed to illustrate the spiral generation process. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Rotate(A, α, P1) → P1'
  2. Line(P1, P1') → rt0
  3. Line(P0, A) → rp0
  4. Rotate(A, θ, P0) → A1
  5. Line(P0, A1) → rp1
  6. Intersect(rt0, rp1) → W1
  7. If(x(W1) ≥ 0, 1, 2) → INT1
  8. Circle(P0, W1) → c0
  9. Rotate(A, (2 * θ), P0) → A2
  10. Rotate(A, (3 * θ), P0) → A3
  11. Rotate(A, (4 * θ), P0) → A4
  12. Rotate(A, (5 * θ), P0) → A5
  13. Rotate(A, (6 * θ), P0) → A6
  14. Rotate(A, (7 * θ), P0) → A7
  15. Rotate(A, (8 * θ), P0) → A8
  16. Rotate(A, (9 * θ), P0) → A9
  17. Rotate(A, (10 * θ), P0) → A10
  18. Rotate(A, (11 * θ), P0) → A11
  19. Rotate(A, (12 * θ), P0) → A12
  20. Rotate(A, (13 * θ), P0) → A13
  21. Rotate(A, (14 * θ), P0) → A14
  22. Rotate(A, (15 * θ), P0) → A15
  23. Rotate(A, (16 * θ), P0) → A16
  24. Rotate(A, (17 * θ), P0) → A17
  25. Rotate(A, (18 * θ), P0) → A18
  26. Rotate(A, (19 * θ), P0) → A19
  27. Rotate(A, (20 * θ), P0) → A20
  28. Rotate(A, (21 * θ), P0) → A21
  29. Rotate(A, (22 * θ), P0) → A22
  30. Rotate(A, (23 * θ), P0) → A23
  31. Rotate(A, (24 * θ), P0) → A24
  32. Rotate(A, (25 * θ), P0) → A25
  33. Rotate(A, (26 * θ), P0) → A26
  34. Rotate(A, (27 * θ), P0) → A27
  35. Rotate(A, (28 * θ), P0) → A28
  36. Rotate(A, (29 * θ), P0) → A29
  37. Rotate(A, (30 * θ), P0) → A30
  38. Rotate(A, (31 * θ), P0) → A31
  39. Rotate(A, (32 * θ), P0) → A32
  40. Rotate(A, (33 * θ), P0) → A33
  41. Rotate(A, (34 * θ), P0) → A34
  42. Rotate(A, (35 * θ), P0) → A35
  43. Rotate(A, (36 * θ), P0) → A36
  44. Line(P0, A14) → rp14
  45. Line(P0, A33) → rp33
  46. Line(P0, A15) → rp15
  47. Line(P0, A34) → rp34
  48. Line(P0, A16) → rp16
  49. Line(P0, A35) → rp35
  50. Line(P0, A17) → rp17
  51. Line(P0, A36) → rp36
  52. Line(P0, A18) → rp18
  53. Line(P0, A19) → rp19
  54. Line(P0, A20) → rp20
  55. Line(P0, A2) → rp2
  56. Line(P0, A21) → rp21
  57. Line(P0, A3) → rp3
  58. Line(P0, A22) → rp22
  59. Line(P0, A4) → rp4
  60. Line(P0, A23) → rp23
  61. Line(P0, A5) → rp5
  62. Line(P0, A24) → rp24
  63. Line(P0, A6) → rp6
  64. Line(P0, A25) → rp25
  65. Line(P0, A7) → rp7
  66. Line(P0, A26) → rp26
  67. Line(P0, A8) → rp8
  68. Line(P0, A27) → rp27
  69. Line(P0, A9) → rp9
  70. Line(P0, A28) → rp28
  71. Line(P0, A10) → rp10
  72. Line(P0, A29) → rp29
  73. Line(P0, A11) → rp11
  74. Line(P0, A30) → rp30
  75. Line(P0, A12) → rp12
  76. Line(P0, A31) → rp31
  77. Line(P0, A13) → rp13
  78. Line(P0, A32) → rp32
  79. Intersect(c0, rp0, 1) → B1
  80. Line(B1, rt0) → rt1
  81. Intersect(rt1, rp1) → F1
  82. Circle(P0, F1) → c1
  83. Intersect(c1, rp0, 1) → B2
  84. Line(B2, rt0) → rt2
  85. Intersect(rt2, rp1) → F2
  86. Intersect(c1, rp2, 1) → W2
  87. Circle(P0, F2) → c2
  88. Intersect(c2, rp0, 1) → B3
  89. Line(B3, rt0) → rt3
  90. Intersect(rt3, rp1) → F3
  91. Circle(P0, F3) → c3
  92. Intersect(c3, rp0, 1) → B4
  93. Line(B4, rt0) → rt4
  94. Intersect(rt4, rp1) → F4
  95. Circle(P0, F4) → c4
  96. Intersect(c4, rp0, 1) → B5
  97. Line(B5, rt0) → rt5
  98. Intersect(rt5, rp1) → F5
  99. Circle(P0, F5) → c5
  100. Intersect(c5, rp0, 1) → B6
  101. Line(B6, rt0) → rt6
  102. Intersect(rt6, rp1) → F6
  103. Circle(P0, F6) → c6
  104. Intersect(c6, rp0, 1) → B7
  105. Line(B7, rt0) → rt7
  106. Intersect(rt7, rp1) → F7
  107. Circle(P0, F7) → c7
  108. Intersect(c7, rp0, 1) → B8
  109. Line(B8, rt0) → rt8
  110. Intersect(rt8, rp1) → F8
  111. Circle(P0, F8) → c8
  112. Intersect(c8, rp0, 1) → B9
  113. Line(B9, rt0) → rt9
  114. Intersect(rt9, rp1) → F9
  115. Circle(P0, F9) → c9
  116. Intersect(c9, rp0, 1) → B10
  117. Line(B10, rt0) → rt10
  118. Intersect(rt10, rp1) → F10
  119. Circle(P0, F10) → c10
  120. Intersect(c10, rp0, 1) → B11
  121. Line(B11, rt0) → rt11
  122. Intersect(rt11, rp1) → F11
  123. Circle(P0, F11) → c11
  124. Intersect(c11, rp0, 1) → B12
  125. Line(B12, rt0) → rt12
  126. Intersect(rt12, rp1) → F12
  127. Circle(P0, F12) → c12
  128. Intersect(c12, rp0, 1) → B13
  129. Line(B13, rt0) → rt13
  130. Intersect(rt13, rp1) → F13
  131. Circle(P0, F13) → c13
  132. Intersect(c13, rp0, 1) → B14
  133. Line(B14, rt0) → rt14
  134. Intersect(rt14, rp1) → F14
  135. Circle(P0, F14) → c14
  136. Intersect(c14, rp0, 1) → B15
  137. Line(B15, rt0) → rt15
  138. Intersect(rt15, rp1) → F15
  139. Circle(P0, F15) → c15
  140. Intersect(c15, rp0, 1) → B16
  141. Line(B16, rt0) → rt16
  142. Intersect(rt16, rp1) → F16
  143. Circle(P0, F16) → c16
  144. Intersect(c16, rp0, 1) → B17
  145. Line(B17, rt0) → rt17
  146. Intersect(rt17, rp1) → F17
  147. Circle(P0, F17) → c17
  148. Intersect(c17, rp0, 1) → B18
  149. Line(B18, rt0) → rt18
  150. Intersect(rt18, rp1) → F18
  151. Circle(P0, F18) → c18
  152. Intersect(c18, rp0, 1) → B19
  153. Line(B19, rt0) → rt19
  154. Intersect(rt19, rp1) → F19
  155. Circle(P0, F19) → c19
  156. Intersect(c19, rp0, 1) → B20
  157. Line(B20, rt0) → rt20
  158. Intersect(rt20, rp1) → F20
  159. Circle(P0, F20) → c20
  160. Intersect(c20, rp0, 1) → B21
  161. Line(B21, rt0) → rt21
  162. Intersect(rt21, rp1) → F21
  163. Circle(P0, F21) → c21
  164. Intersect(c21, rp0, 1) → B22
  165. Line(B22, rt0) → rt22
  166. Intersect(rt22, rp1) → F22
  167. Circle(P0, F22) → c22
  168. Intersect(c22, rp0, 1) → B23
  169. Line(B23, rt0) → rt23
  170. Intersect(rt23, rp1) → F23
  171. Circle(P0, F23) → c23
  172. Intersect(c23, rp0, 1) → B24
  173. Line(B24, rt0) → rt24
  174. Intersect(rt24, rp1) → F24
  175. Circle(P0, F24) → c24
  176. Intersect(c24, rp0, 1) → B25
  177. Line(B25, rt0) → rt25
  178. Intersect(rt25, rp1) → F25
  179. Circle(P0, F25) → c25
  180. Intersect(c25, rp0, 1) → B26
  181. Line(B26, rt0) → rt26
  182. Intersect(rt26, rp1) → F26
  183. Circle(P0, F26) → c26
  184. Intersect(c26, rp0, 1) → B27
  185. Line(B27, rt0) → rt27
  186. Intersect(rt27, rp1) → F27
  187. Circle(P0, F27) → c27
  188. Intersect(c27, rp0, 1) → B28
  189. Line(B28, rt0) → rt28
  190. Intersect(rt28, rp1) → F28
  191. Circle(P0, F28) → c28
  192. Intersect(c28, rp0, 1) → B29
  193. Line(B29, rt0) → rt29
  194. Intersect(rt29, rp1) → F29
  195. Circle(P0, F29) → c29
  196. Intersect(c29, rp0, 1) → B30
  197. Line(B30, rt0) → rt30
  198. Intersect(rt30, rp1) → F30
  199. Circle(P0, F30) → c30
  200. Intersect(c30, rp0, 1) → B31
  201. Line(B31, rt0) → rt31
  202. Intersect(rt31, rp1) → F31
  203. Circle(P0, F31) → c31
  204. Intersect(c31, rp0, 1) → B32
  205. Line(B32, rt0) → rt32
  206. Intersect(rt32, rp1) → F32
  207. Circle(P0, F32) → c32
  208. Intersect(c32, rp0, 1) → B33
  209. Line(B33, rt0) → rt33
  210. Intersect(rt33, rp1) → F33
  211. Circle(P0, F33) → c33
  212. Intersect(c33, rp0, 1) → B34
  213. Line(B34, rt0) → rt34
  214. Intersect(rt34, rp1) → F34
  215. Circle(P0, F34) → c34
  216. Intersect(c34, rp0, 1) → B35
  217. Line(B35, rt0) → rt35
  218. Intersect(rt35, rp1) → F35
  219. Circle(P0, F35) → c35
  220. Intersect(c2, rp3, 1) → W3
  221. Intersect(c3, rp4, 1) → W4
  222. Intersect(c4, rp5, 1) → W5
  223. Intersect(c5, rp6, 1) → W6
  224. Intersect(c6, rp7, 1) → W7
  225. Intersect(c7, rp8, 1) → W8
  226. Intersect(c8, rp9, 1) → W9
  227. Intersect(c9, rp10, 1) → W10
  228. Intersect(c10, rp11, 1) → W11
  229. Intersect(c11, rp12, 1) → W12
  230. Intersect(c12, rp13, 1) → W13
  231. Intersect(c13, rp14, 1) → W14
  232. Intersect(c14, rp15, 1) → W15
  233. Intersect(c15, rp16, 1) → W16
  234. Intersect(c16, rp17, 1) → W17
  235. Intersect(c17, rp18, 1) → W18
  236. Intersect(c18, rp19, 1) → W19
  237. Intersect(c19, rp20, 1) → W20
  238. Intersect(c20, rp21, 1) → W21
  239. Intersect(c21, rp22, 1) → W22
  240. Intersect(c22, rp23, 1) → W23
  241. Intersect(c23, rp24, 1) → W24
  242. Intersect(c24, rp25, 1) → W25
  243. Intersect(c25, rp26, 1) → W26
  244. Intersect(c26, rp27, 1) → W27
  245. Intersect(c27, rp28, 1) → W28
  246. Intersect(c28, rp29, 1) → W29
  247. Intersect(c29, rp30, 1) → W30
  248. Intersect(c30, rp31, 1) → W31
  249. Intersect(c31, rp32, 1) → W32
  250. Intersect(c32, rp33, 1) → W33
  251. Intersect(c33, rp34, 1) → W34
  252. Intersect(c34, rp35, 1) → W35
  253. Intersect(c35, rp36, 1) → W36
  254. Segment(P1, W1) → f
  255. Segment(W1, W2) → g
  256. Segment(W2, W3) → h
  257. Segment(W3, W4) → i
  258. Segment(W4, W5) → j
  259. Segment(W5, W6) → k
  260. Segment(W6, W7) → l
  261. Segment(W7, W8) → m
  262. Segment(W8, W9) → n
  263. Segment(W9, W10) → p
  264. Segment(W10, W11) → q
  265. Segment(W11, W12) → r
  266. Segment(W12, W13) → s
  267. Segment(W13, W14) → t
  268. Segment(W14, W15) → a
  269. Segment(W15, W16) → b
  270. Segment(W16, W17) → c
  271. Segment(W17, W18) → d
  272. Segment(W18, W19) → e
  273. Segment(W19, W20) → f_1
  274. Segment(W20, W21) → g_1
  275. Segment(W21, W22) → h_1
  276. Segment(W22, W23) → i_1
  277. Segment(W23, W24) → j_1
  278. Segment(W24, W25) → k_1
  279. Segment(W25, W26) → l_1
  280. Segment(W26, W27) → m_1
  281. Segment(W27, W28) → n_1
  282. Segment(W28, W29) → p_1
  283. Segment(W29, W30) → q_1
  284. Segment(W30, W31) → r_1
  285. Segment(W31, W32) → s_1
  286. Segment(W32, W33) → t_1
  287. Segment(W33, W34) → a_1
  288. Segment(W34, W35) → b_1
  289. Segment(W35, W36) → c_1
  290. Segment(B1, F1) → d_1
  291. Segment(B2, F2) → e_1
  292. Segment(B3, F3) → f_2
  293. Segment(B4, F4) → g_2
  294. Segment(B5, F5) → h_2
  295. Segment(B6, F6) → i_2
  296. Segment(B7, F7) → j_2
  297. Segment(B8, F8) → k_2
  298. Segment(B9, F9) → l_2
  299. Segment(B10, F10) → m_2
  300. Segment(B11, F11) → n_2
  301. Segment(B12, F12) → p_2
  302. Segment(B13, F13) → q_2
  303. Segment(B14, F14) → r_2
  304. Segment(B15, F15) → s_2
  305. Segment(B16, F16) → t_2
  306. Segment(B17, F17) → a_2
  307. Segment(B18, F18) → b_2
  308. Segment(B19, F19) → c_2
  309. Segment(B20, F20) → d_2
  310. Segment(B21, F21) → e_2
  311. Segment(B22, F22) → f_3
  312. Segment(B23, F23) → g_3
  313. Segment(B24, F24) → h_3
  314. Segment(B25, F25) → i_3
  315. Segment(B26, F26) → j_3
  316. Segment(B27, F27) → k_3
  317. Segment(B28, F28) → l_3
  318. Segment(B29, F29) → m_3
  319. Segment(B30, F30) → n_3
  320. Segment(B31, F31) → p_3
  321. Segment(B32, F32) → q_3
  322. Segment(B33, F33) → r_3
  323. Segment(B34, F34) → s_3
  324. Segment(B35, F35) → t_3
  325. Intersect(c35, rp0, 1) → B36
  326. Segment(P0, W2) → s2
  327. Segment(P0, W3) → s3
  328. Segment(P0, W4) → s4
  329. Segment(P0, W5) → s5
  330. Segment(P0, W6) → s6
  331. Segment(P0, W7) → s7
  332. Segment(P0, W8) → s8
  333. Segment(P0, W9) → s9
  334. Segment(P0, W10) → s10
  335. Segment(P0, W11) → s11
  336. Segment(P0, W12) → s12
  337. Segment(P0, W13) → s13
  338. Segment(P0, W14) → s14
  339. Segment(P0, W15) → s15
  340. Segment(P0, W16) → s16
  341. Segment(P0, W17) → s17
  342. Segment(P0, W18) → s18
  343. Segment(P0, W19) → s19
  344. Segment(P0, W20) → s20
  345. Segment(P0, W21) → s21
  346. Segment(P0, W22) → s22
  347. Segment(P0, W23) → s23
  348. Segment(P0, W24) → s24
  349. Segment(P0, W25) → s25
  350. Segment(P0, W26) → s26
  351. Segment(P0, W27) → s27
  352. Segment(P0, W28) → s28
  353. Segment(P0, W29) → s29
  354. Segment(P0, W30) → s30
  355. Segment(P0, W31) → s31
  356. Segment(P0, W32) → s32
  357. Segment(P0, W33) → s33
  358. Segment(P0, W34) → s34
  359. Segment(P0, W35) → s35
  360. Segment(P0, W36) → s36
  361. If(x(B36) > x(P1), (-x(B36)), (-x(P1))) → y1
  362. If(x(B36) > x(P1), x(B36), x(P1)) → y2
  363. Segment(P0, F35) → s1
  364. Segment(P0, B36) → s0
  365. Segment(P0, P1) → s0'
  366. Segment(P0, W1) → s1'
  367. Textfield(r0) → CampoInserimento1
  368. Textfield(α) → CampoInserimento2

AI Drawing Prompt Reference

Draw a logarithmic/golden polygonal spiral. Start by rotating point A around point P0 to generate a series of polygons. Then, draw rays from point P0 through the rotated points A, intersecting with the previous ray to create new points. These points form the spiral, with each polygon having an increasing number of sides, following a logarithmic growth pattern.
The prompt above was compiled from the original drawing commands and is for reference only.