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12- and 9-pointed Stars

12- and 9-pointed Stars
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2019-12-31 20:23
PolygonTransformationGeometric Construction
This example demonstrates Islamic geometric patterns of 12- and 9-pointed stars. It involves concepts of symmetry, circles, and polygons. Using GeoGebra commands such as Intersect, Point, Circle, Polygon, and Line, complex star patterns can be constructed. Suitable for middle school and above students to learn geometric construction and symmetry.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. ConstructionStep() → stap
  3. Point(xAxis) → B
  4. Circle(A, B) → c
  5. Circle(B, A) → d
  6. Intersect(c, d, 2) → C
  7. Circle(C, A) → e
  8. Polygon(A, B, C) → driehoek1, c_1, a, b
  9. Intersect(d, e, 2) → D
  10. Segment(B, D) → f
  11. Intersect(c, e, 2) → E
  12. Segment(E, D) → g
  13. Line(A, B) → h
  14. OrthogonalLine(A, h) → i
  15. Intersect(i, g) → F
  16. Segment(A, F) → j
  17. OrthogonalLine(D, h) → k
  18. Intersect(h, k) → G
  19. Segment(D, G) → l
  20. Polygon(A, F, D, G) → vierhoek1, a_1, f_1, d_1, g_1
  21. Segment(A, C) → a_5
  22. Segment(A, D) → m
  23. AngularBisector(F, A, C) → n
  24. AngularBisector(C, A, D) → p
  25. AngularBisector(F, D, A) → q
  26. AngularBisector(A, D, B) → r
  27. AngularBisector(B, D, G) → s
  28. Intersect(n, f_1) → K
  29. Intersect(p, f_1) → L
  30. Segment(A, K) → t
  31. Segment(A, L) → h_1
  32. AngularBisector(D, A, G) → i_1
  33. Intersect(i_1, l) → N
  34. Segment(A, N) → j_1
  35. Intersect(r, c_1) → I
  36. Intersect(s, g_1) → J
  37. Intersect(q, j) → M
  38. Segment(D, J) → k_1
  39. Segment(D, I) → l_1
  40. Segment(D, M) → m_1
  41. Circle(A, M) → e_1
  42. Circle(D, N) → p_1
  43. Intersect(e_1, j_1, 1) → O
  44. Intersect(e_1, h_1, 1) → P
  45. Intersect(e_1, t, 1) → Q
  46. Polygon(Q, P, 12) → veelhoek1, n_1, i_2, j_2, k_2, R, S, T, U, V, W, Z, q_1, A_1, B_1, C_1, r_1, s_1, t_1, b_1, f_2, g_2, h_2
  47. Segment(V, B_1) → l_2
  48. Segment(W, C_1) → m_2
  49. Segment(Z, Q) → n_2
  50. Segment(A_1, P) → p_2
  51. Segment(B_1, O) → q_2
  52. Segment(C_1, S) → r_2
  53. Segment(Q, T) → s_2
  54. Segment(P, U) → t_2
  55. Segment(O, V) → a_2
  56. Segment(S, W) → b_2
  57. Segment(T, Z) → c_2
  58. Segment(U, A_1) → d_2
  59. Intersect(b_2, d_2) → D_1
  60. Intersect(l_2, c_2) → E_1
  61. Intersect(m_2, d_2) → F_1
  62. Intersect(l_2, n_2) → G_1
  63. Intersect(m_2, p_2) → H_1
  64. Intersect(n_2, q_2) → I_1
  65. Intersect(t, p_2) → J_1
  66. Intersect(h_1, q_2) → K_1
  67. Intersect(j_1, r_2) → L_1
  68. Intersect(s_2, a_2) → M_1
  69. Intersect(t_2, b_2) → N_1
  70. Intersect(a_2, c_2) → O_1
  71. PolyLine(V, E_1, T, O_1, V, Z, G_1, B_1, I_1, Q, K_1, O, M_1) → e_2
  72. PolyLine(U, D_1, S, N_1, U, W, F_1, A_1, H_1, C_1, J_1, P, L_1) → f_3
  73. Translate(V, Vector[A, D]) → V'
  74. Translate(E_1, Vector[A, D]) → Q_1
  75. Translate(Z, Vector[A, D]) → Z'
  76. Translate(G_1, Vector[A, D]) → R_1
  77. Translate(B_1, Vector[A, D]) → S_1
  78. Translate(I_1, Vector[A, D]) → T_1
  79. Translate(Q, Vector[A, D]) → Q'
  80. Translate(K_1, Vector[A, D]) → U_1
  81. Translate(O, Vector[A, D]) → O'
  82. Translate(M_1, Vector[A, D]) → V_1
  83. Translate(T, Vector[A, D]) → T'
  84. Translate(O_1, Vector[A, D]) → W_1
  85. Translate(V, Vector[A, D]) → V'_1
  86. PolyLine(V', Q_1, T', W_1, V'_1, Z', R_1, S_1, T_1, Q', U_1, O', V_1) → g_3
  87. Translate(U, Vector[A, D]) → U'
  88. Translate(D_1, Vector[A, D]) → Z_1
  89. Translate(W, Vector[A, D]) → W'
  90. Translate(F_1, Vector[A, D]) → A_2
  91. Translate(A_1, Vector[A, D]) → B_2
  92. Translate(H_1, Vector[A, D]) → C_2
  93. Translate(C_1, Vector[A, D]) → D_2
  94. Translate(J_1, Vector[A, D]) → E_2
  95. Translate(P, Vector[A, D]) → P'
  96. Translate(L_1, Vector[A, D]) → F_2
  97. Translate(S, Vector[A, D]) → S'
  98. Translate(N_1, Vector[A, D]) → G_2
  99. Translate(U, Vector[A, D]) → U'_1
  100. PolyLine(U', Z_1, S', G_2, U'_1, W', A_2, B_2, C_2, D_2, E_2, P', F_2) → h_3
  101. Segment(M, C) → i_3
  102. Segment(B, N) → n_3
  103. AngularBisector(M, C, F) → p_3
  104. AngularBisector(B, C, P) → q_3
  105. AngularBisector(B, C, W') → r_3
  106. AngularBisector(N, B, G) → s_3
  107. Intersect(e_1, m, 1) → I_2
  108. Intersect(p_1, m, 1) → J_2
  109. Segment(C, I_2) → j_3
  110. Segment(I_2, B) → k_3
  111. Segment(B, J_2) → l_3
  112. Segment(J_2, C) → m_3
  113. AngularBisector(I_2, C, B) → t_3
  114. Intersect(p_3, j) → K_2
  115. Intersect(s_3, l) → H_2
  116. AngularBisector(B, C, J_2) → a_3
  117. AngularBisector(I_2, B, C) → b_3
  118. AngularBisector(C, B, J_2) → c_3
  119. Intersect(t_3, b_3) → L_2
  120. Intersect(a_3, c_3) → M_2
  121. Segment(B, H_2) → d_3
  122. Segment(B, M_2) → e_3
  123. Segment(B, L_2) → f_4
  124. Segment(C, L_2) → g_4
  125. Segment(C, M_2) → h_4
  126. Segment(C, K_2) → i_4
  127. Intersect(a, l_1) → O_2
  128. Circle(B, O_2) → c_4
  129. Intersect(c_4, k_3, 1) → P_2
  130. Intersect(c_4, l_3) → N_2,
  131. Intersect(c_4, n_3, 1) → Q_2
  132. Line(L_2, B) → j_4
  133. Intersect(c_4, j_4, 2) → R_2
  134. Intersect(m, e_3) → S_2
  135. Line(M_2, B) → k_4
  136. Intersect(c_4, k_4, 2) → T_2
  137. Line(D, B) → l_4
  138. Intersect(c_4, l_4, 2) → U_2
  139. Line(H_2, B) → m_4
  140. Intersect(c_4, m_4, 2) → V_2
  141. Segment(O_2, J) → n_4
  142. Segment(N_2, R_2) → p_4
  143. Segment(Q_2, T_2) → q_4
  144. Segment(J, U_2) → r_4
  145. Segment(R_2, V_2) → s_4
  146. Segment(T_2, P_2) → t_4
  147. Segment(U_2, O_2) → a_4
  148. Segment(V_2, N_2) → b_4
  149. Segment(P_2, Q_2) → d_4
  150. Intersect(r_4, s_4) → W_2
  151. Intersect(t_4, a_4) → Z_2
  152. Intersect(k_3, a_4) → A_3
  153. Intersect(a, b_4) → B_3
  154. Intersect(l_3, n_4) → C_3
  155. Intersect(n_3, n_4) → D_3
  156. Intersect(p_4, q_4) → E_3
  157. Intersect(q_4, r_4) → F_3
  158. Intersect(s_4, t_4) → G_3
  159. PolyLine(T_2, G_3, U_2, Z_2, P_2, B_3, N_2, D_3, J, F_3, T_2, V_2, A_3, O_2, C_3, Q_2, E_3, R_2, W_2) → e_4
  160. Circle(C, K) → c_5
  161. Intersect(c_5, i_3, 1) → H_3
  162. Intersect(c_5, j_3, 1) → I_3
  163. Intersect(h_1, m_1) → J_3
  164. Intersect(c_5, m_3, 1) → K_3
  165. Line(K_2, C) → f_5
  166. Intersect(c_5, f_5, 2) → L_3
  167. Line(A, C) → g_5
  168. Intersect(c_5, g_5, 2) → M_3
  169. Line(L_2, C) → h_5
  170. Intersect(c_5, h_5, 2) → N_3
  171. Line(M_2, C) → i_5
  172. Intersect(c_5, i_5, 2) → O_3
  173. Segment(K, M_3) → j_5
  174. Segment(O_3, L_3) → k_5
  175. Segment(N_3, K_3) → l_5
  176. Segment(M_3, J_3) → m_5
  177. Segment(L_3, I_3) → n_5
  178. Segment(K_3, H_3) → p_5
  179. Segment(J_3, K) → q_5
  180. Segment(I_3, O_3) → r_5
  181. Segment(H_3, N_3) → s_5
  182. Intersect(r_5, s_5) → P_3
  183. Intersect(j_5, s_5) → Q_3
  184. Intersect(j_5, k_5) → R_3
  185. Intersect(k_5, l_5) → S_3
  186. Intersect(l_5, m_5) → T_3
  187. Intersect(m_3, m_5) → U_3
  188. Intersect(a, n_5) → V_3
  189. Intersect(j_3, p_5) → W_3
  190. Intersect(i_3, q_5) → Z_3
  191. PolyLine(I_3, Z_3, H_3, P_3, O_3, R_3, M_3, T_3, K_3, V_3, I_3, K, Q_3, N_3, S_3, L_3, U_3, J_3, W_3) → t_5
  192. Ray(M_1, O) → b_5
  193. Intersect(kk_3, b_5) → B_4
  194. Circle(A, B_4) → d_5
  195. Ray(K_1, O) → e_5
  196. Intersect(d_5, e_5, 1) → C_4
  197. Ray(J_1, P) → f_6
  198. Intersect(jj_3, f_6) → D_4
  199. Intersect(d_5, ii_3, 1) → E_4
  200. Intersect(d_5, mm_1, 1) → F_4
  201. Segment(E_4, Q) → g_6
  202. Segment(Q, F_4) → h_6
  203. Segment(D_4, P) → i_6
  204. Ray(L_1, P) → j_6
  205. Intersect(d_5, j_6, 1) → G_4
  206. Segment(G_4, P) → k_6
  207. Segment(B_4, O) → l_6
  208. Segment(O, C_4) → m_6
  209. Distance(M, M) → MM
  210. Midpoint(A, D) → MI
  211. Mirror(d_5, MI) → c_6
  212. Mirror(E_4, MI) → H_4
  213. Mirror(F_4, MI) → I_4
  214. Mirror(G_4, MI) → J_4
  215. Mirror(D_4, MI) → K_4
  216. Mirror(B_4, MI) → L_4
  217. Mirror(C_4, MI) → M_4
  218. Ray(Z_3, I_3) → n_6
  219. Intersect(hh_1, n_6) → N_4
  220. Circle(C, N_4) → d_6
  221. Intersect(d_6, tt, 1) → O_4
  222. Ray(P_3, H_3) → p_6
  223. Intersect(d_6, p_6, 1) → P_4
  224. Ray(V_3, I_3) → q_6
  225. Intersect(d_6, q_6, 1) → Q_4
  226. Ray(Q_3, K) → r_6
  227. Intersect(d_6, r_6, 1) → R_4
  228. Ray(U_3, J_3) → s_6
  229. Ray(W_3, J_3) → t_6
  230. Intersect(d_6, s_6, 1) → S_4
  231. Intersect(d_6, t_6, 1) → T_4
  232. Ray(T_3, K_3) → a_6
  233. Intersect(mm_1, a_6) → U_4
  234. Ray(V_3, K_3) → b_6
  235. Intersect(d_6, b_6, 1) → V_4
  236. Segment(K, R_4) → e_6
  237. Segment(O_4, H_3) → f_7
  238. Segment(H_3, P_4) → g_7
  239. Segment(Q_4, I_3) → h_7
  240. Segment(I_3, N_4) → i_7
  241. Segment(S_4, J_3) → j_7
  242. Segment(J_3, T_4) → k_7
  243. Segment(K_3, U_4) → l_7
  244. Segment(K_3, V_4) → m_7
  245. Mirror(E_4, MI) → W_4
  246. Mirror(Q, MI) → Q'_1
  247. Segment(W_4, Q'_1) → n_7
  248. Mirror(Q, MI) → Q'_2
  249. Mirror(F_4, MI) → Z_4
  250. Segment(Q'_2, Z_4) → p_7
  251. Mirror(G_4, MI) → A_5
  252. Mirror(P, MI) → P'_1
  253. Segment(A_5, P'_1) → q_7
  254. Mirror(D_4, MI) → B_5
  255. Mirror(P, MI) → P'_2
  256. Segment(B_5, P'_2) → r_7
  257. Mirror(B_4, MI) → C_5
  258. Mirror(O, MI) → O'_1
  259. Segment(C_5, O'_1) → s_7
  260. Mirror(O, MI) → O'_2
  261. Mirror(C_4, MI) → D_5
  262. Segment(O'_2, D_5) → t_7
  263. Mirror(K, MI) → K'
  264. Mirror(R_4, MI) → E_5
  265. Segment(K', E_5) → a_7
  266. Mirror(O_4, MI) → F_5
  267. Mirror(H_3, MI) → G_5
  268. Segment(F_5, G_5) → b_7
  269. Mirror(H_3, MI) → H_5
  270. Mirror(P_4, MI) → I_5
  271. Segment(H_5, I_5) → c_7
  272. Mirror(Q_4, MI) → J_5
  273. Mirror(I_3, MI) → K_5
  274. Segment(J_5, K_5) → d_7
  275. Mirror(I_3, MI) → L_5
  276. Mirror(N_4, MI) → M_5
  277. Segment(L_5, M_5) → e_7
  278. Mirror(S_4, MI) → N_5
  279. Mirror(J_3, MI) → O_5
  280. Segment(N_5, O_5) → f_8
  281. Mirror(J_3, MI) → P_5
  282. Mirror(T_4, MI) → Q_5
  283. Segment(P_5, Q_5) → g_8
  284. Mirror(K_3, MI) → R_5
  285. Mirror(U_4, MI) → S_5
  286. Segment(R_5, S_5) → h_8
  287. Mirror(K_3, MI) → T_5
  288. Mirror(V_4, MI) → U_5
  289. Segment(T_5, U_5) → i_8
  290. Mirror(d_6, MI) → c_8
  291. Segment(A_5, P'_1) → m_{11}
  292. Segment(F_4, Q_4) → j_8
  293. Segment(G_4, P_4) → k_8
  294. Segment(D_4, S_5) → l_8
  295. Segment(B_4, N_4) → m_8
  296. Segment(N_4, N_5) → n_8
  297. Segment(S_5, T_4) → p_8
  298. Segment(Q_5, U_4) → q_8
  299. Segment(S_4, M_5) → r_8
  300. Segment(U_4, K_4) → s_8
  301. Segment(M_5, L_4) → t_8
  302. Segment(J_5, I_4) → a_8
  303. Segment(I_5, J_4) → b_8
  304. Intersect(a_5, j_8) → W_5
  305. Intersect(f, a_8) → Z_5
  306. Circle(A, W_5) → d_8
  307. Circle(D, Z_5) → e_8
  308. Circle(C, W_5) → c_9
  309. Circle(B, Z_5) → d_9
  310. Intersect(c_1, g_1) → A_6
  311. Intersect(c_1, g_1) → B_6
  312. Intersect(c_1, g_1) → C_6
  313. Intersect(d_8, c_1) → D_6,
  314. Segment(CC4, D_6) → f_9
  315. Segment(UU_5, D_6) → g_9
  316. Intersect(j, a_1) → E_6
  317. Intersect(d_8, j) → F_6,
  318. Segment(F_6, EE_4) → h_9
  319. Intersect(e_8, c_9, 1) → G_6
  320. Segment(G_6, D_5) → i_9
  321. Segment(VV_4, G_6) → j_9
  322. Intersect(e_8, l) → H_6,
  323. Segment(W_4, H_6) → k_9
  324. Intersect(d_9, dd_3, 1) → I_6
  325. Intersect(c_6, f) → J_6,
  326. Segment(EE_5, I_6) → l_9
  327. Segment(FF_5, I_6) → m_9
  328. Intersect(ff_4, p_8) → K_6
  329. Intersect(gg_8, q_8) → L_6
  330. Segment(K_6, L_6) → n_9
  331. Intersect(c_8, ff_8, 1) → M_6
  332. Intersect(d_9, ee_3, 1) → N_6
  333. Segment(M_6, N_6) → p_9
  334. Intersect(c_9, ii_4, 1) → O_6
  335. Intersect(gg_4, n_8) → P_6
  336. Intersect(hh_4, q_8) → Q_6
  337. Point(d_6) → R_6
  338. Segment(O_6, R_6) → q_9
  339. Intersect(d_6, tt, 1) → S_6
  340. Segment(O_6, S_6) → r_9
  341. Intersect(d_6, jj_7, 1) → T_6
  342. Segment(P_6, T_6) → s_9
  343. Intersect(d_6, kk_7, 1) → U_6
  344. Segment(Q_6, U_6) → t_9
  345. Ray(S_6, O_6) → a_9
  346. Intersect(j, a_1) → V_6
  347. Intersect(j, a_9) → W_6
  348. Segment(O_6, W_6) → b_9
  349. Ray(R_6, O_6) → e_9
  350. Intersect(d_6, tt, 1) → Z_6
  351. Segment(F_6, Z_6) → f_{10}
  352. Intersect(e_9, f_{10}) → A_7
  353. Segment(O_6, A_7) → g_{10}
  354. Ray(L_6, K_6) → h_{10}
  355. Intersect(l_8, h_{10}) → B_7
  356. Segment(B_7, K_6) → i_{10}
  357. Ray(S_4, P_6) → j_{10}
  358. Intersect(m_8, j_{10}) → C_7
  359. Segment(C_7, P_6) → k_{10}
  360. Ray(M_6, N_6) → l_{10}
  361. Intersect(t_8, l_{10}) → D_7
  362. Segment(N_6, D_7) → m_{10}
  363. Segment(A_5, P'_1) → n_{10}
  364. Ray(U_6, Q_6) → p_{10}
  365. Intersect(s_8, p_{10}) → E_7
  366. Segment(Q_6, E_7) → q_{10}
  367. Ray(EE_5, I_6) → r_{10}
  368. Segment(FF_5, H_6) → s_{10}
  369. Intersect(r_{10}, s_{10}) → F_7
  370. Segment(I_6, F_7) → t_{10}
  371. Ray(FF_5, I_6) → a_{10}
  372. Intersect(l, d_1) → G_7
  373. Intersect(l, d_1) → H_7
  374. Intersect(l, a_{10}) → I_7
  375. Segment(I_6, I_7) → b_{10}
  376. Segment(D_4, B_7) → c_{10}
  377. Segment(F_6, A_7) → d_{10}
  378. Intersect(e_8, mm, 1) → J_7
  379. Segment(F_7, H_6) → f_{11}
  380. Segment(B_5, E_7) → g_{11}
  381. Segment(D_7, C_5) → e_{10}
  382. Intersect(d_5, kk_3, 1) → K_7
  383. Segment(C_7, K_7) → h_{11}
  384. Intersect(d_6, hh_1, 1) → L_7
  385. Intersect(a, q_8) → M_7
  386. Segment(L_7, M_7) → i_{11}
  387. Intersect(d_6, mm_1, 1) → N_7
  388. Segment(M_7, N_7) → j_{11}
  389. Intersect(c_8, jj_1, 1) → O_7
  390. Intersect(a, r_8) → P_7
  391. Segment(O_7, P_7) → k_{11}
  392. Intersect(c_8, ll_1, 1) → Q_7
  393. Segment(Q_7, P_7) → l_{11}
  394. Polygon(A, F, D, G) → vierhoek2, a_{11}, f_{12}, d_{11}, g_{12}

AI Drawing Prompt Reference

Draw an Islamic geometric pattern featuring 12-pointed and 9-pointed stars. Start by placing two points on the x-axis, use each as center to draw a circle through the other, and mark their upper intersection to form an equilateral triangle. From this, construct perpendiculars, angle bisectors, and intersections to build two quadrilaterals and auxiliary lines. Finally, use the quadrilateral vertices as centers to draw circles that intersect the bisectors, yielding 12 equally spaced points; connect these to form a dodecagon and add two diagonals, completing the star pattern with symmetry and polygon construction.
The prompt above was compiled from the original drawing commands and is for reference only.