MathPlotAI Logo MathPlotAI

10- and 11-pointed stars

10- and 11-pointed stars
Loading the math board and drawing, please wait…
2019-12-31 20:29
PolygonTrigonometryTransformationGeometric Construction
This example demonstrates how to construct 10- and 11-pointed star patterns using rotations and reflections. Key math concepts include symmetry, polygon construction, rotational transformations, and geometric constructions. Construction steps involve commands like Point, Rotate, Polygon, Segment, Sequence, Circle, Mirror, Ray, Intersect, Midpoint, and Distance. Suitable for high school mathematics education or for enthusiasts of geometric constructions.

GeoGebra Commands & Parameters

  1. ConstructionStep() → stap
  2. Point(xAxis) → A
  3. Rotate(A, 18°) → A'
  4. Polygon(A, A', 20) → veelhoek1, f, p, q, r, s, t, a, b, c, d, e, g, f_1, B, C, D, E, F, G, H, I, J, h, K, L, M, N, P, Q, R, S, T, i, j, k, l, m, n
  5. Segment(O, A) → g_1
  6. Sequence(Segment[Rotate[O, (nn * 36°)], Rotate[A, (nn * 36°)]], nn, 1, 10) → l1
  7. Circle(O, 4) → c_{15}
  8. Sequence(Segment[Rotate[O, (nn * 36°)], Rotate[A', (nn * 36°)]], nn, 1, 10) → l2
  9. Mirror(l1, y = 4) → l1'
  10. Circle((0, 8), 4) → d_{15}
  11. Mirror(l2, y = 4) → l2'
  12. Mirror(veelhoek1, y = 4) → veelhoek1'
  13. Ray(O, B) → h_1
  14. Mirror(O, y = 4) → O'
  15. Mirror(B, y = 4) → B'
  16. Ray(O', B') → i_1
  17. Ray(O, D) → j_1
  18. Mirror(O, y = 4) → O'_1
  19. Mirror(D, y = 4) → D'
  20. Ray(O'_1, D') → k_1
  21. Intersect(j_1, k_1) → V
  22. Rotate(W, ((-(360°))) / 11, U) → W'
  23. Polygon(W', W, 11) → veelhoek2, l_1, b_1, c_1, Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, m_1, H_1, n_1, p_1, q_1, r_1, s_1, t_1, a_1
  24. Ray(U, G_1) → d_1
  25. Ray(O', B') → e_1
  26. Ray(O, B) → f_2
  27. Ray(U, F_1) → g_2
  28. Ray(U, H_1) → h_2
  29. Circle(U, 4) → c_2
  30. Segment(U, B') → i_2
  31. Rotate(U, (360°) / 11, U) → U'
  32. Rotate(B', (360°) / 11, U) → B''
  33. Segment(U', B'') → j_2
  34. Rotate(U, (2 * (360°) / 11), U) → U'_1
  35. Rotate(B', (2 * (360°) / 11), U) → B''_1
  36. Segment(U'_1, B''_1) → k_2
  37. Sequence(Segment[Rotate[U, (nn * (360°) / 11), U'_1], Rotate[B', (nn * (360°) / 11), U'_1]], nn, 1, 11) → l3
  38. Sequence(Segment[Rotate[U, (360°) / 22 + (nn * (360°) / 11), U'_1], Rotate[B', (360°) / 22 + (nn * (360°) / 11), U'_1]], nn, 1, 11) → l4
  39. Circle(U, B') → e_{15}
  40. Mirror(l3, y = 8) → l3'
  41. Mirror(l4, y = 8) → l4'
  42. Point(yAxis) → I_1
  43. Point(yAxis) → J_1
  44. Rotate(J_1, 18°, I_1) → K_1
  45. Polygon(J_1, K_1, 20) → veelhoek3, l_2, b_2, d_2, e_2, f_3, g_3, h_3, i_3, j_3, k_3, l_3, m_2, m_3, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, S_1, T_1, n_2, U_1, V_1, W_1, Z_1, A_2, B_2, C_2, D_2, E_2, p_2, q_2, r_2, s_2, t_2, a_2
  46. Polygon(O, RO, RB, I_1) → vierhoek1, n_3, p_3, q_3, r_3
  47. Mirror(U, y = 8) → UU
  48. Midpoint(q_3) → F_2
  49. Segment(UU, F_2) → s_3
  50. Rotate(F_2, (360°) / 11, UU) → G_2
  51. Ray(UU, G_2) → t_3
  52. Intersect(q_3, t_3) → H_2
  53. Rotate(F_2, (360°) / 22, UU) → I_2
  54. Ray(UU, I_2) → a_3
  55. Intersect(a_3, q_3) → J_2
  56. Segment(UU, H_2) → g_{15}
  57. Segment(UU, J_2) → h_{15}
  58. Circle(J_2, H_2) → c_3
  59. Intersect(c_3, a_3, 1) → K_2
  60. Circle(UU, K_2) → d_3
  61. Rotate(K_2, ((-(360°))) / 11, UU) → L_2
  62. Polygon(L_2, K_2, 11) → veelhoek4, b_3, m_4, n_4, M_2, N_2, O_2, P_2, Q_2, R_2, S_2, T_2, e_3, U_2, f_4, g_4, h_4, i_4, j_4, k_4, l_4
  63. Ray(UU, P_2) → p_4
  64. Ray(O', V_2) → q_4
  65. Intersect(q_4, p_4) → W_2
  66. Rotate(Q_2, ((-(360°))) / 22, UU) → Z_2
  67. Ray(UU, Z_2) → r_4
  68. Intersect(s_3, g_{15}) → D_{10}
  69. Segment(D_{10}, W_2) → i_{15}
  70. Ray(O', A_3) → s_4
  71. Intersect(r_4, s_4) → B_3
  72. Segment(O', B_3) → t_4
  73. Segment(UU, B_3) → a_4
  74. Polygon(B_3, W_2, 5) → veelhoek5, b_4, c_4, d_4, e_4, f_5, C_3, D_3, E_3
  75. Polygon(D_3, C_3, 10) → veelhoek6, g_5, q_5, F_3, G_3, H_3, I_3, J_3, K_3, L_3, M_3, h_5, i_5, j_5, k_5, l_5, m_5, n_5, p_5
  76. Distance(O', G_3) → straal
  77. Circle(I_1, straal) → c_5
  78. Segment(I_1, L_1) → r_5
  79. Intersect(c_5, r_5, 1) → N_3
  80. Intersect(c_5, r_3, 1) → O_3
  81. Polygon(O_3, N_3, 10) → veelhoek7, s_5, i_6, P_3, Q_3, R_3, S_3, T_3, U_3, V_3, W_3, t_5, a_5, b_5, d_5, e_5, f_6, g_6, h_6
  82. Circle(O', W_2) → c_6
  83. Ray(O', F_3) → j_6
  84. Intersect(c_6, j_6) → Z_3,
  85. Polygon(C_3, W_2, P_2, O_2, Z_3, F_3) → veelhoek8, k_6, w_2, l_6, o_2, z_3, m_6
  86. Ray(UU, N_2) → n_6
  87. Intersect(r_5, n_6) → A_4
  88. Rotate(O_3, (360°) / 20, I_1) → B_4
  89. Ray(I_1, B_4) → p_6
  90. Polygon(N_3, O_3, 5) → veelhoek9, q_6, r_6, s_6, t_6, a_6, C_4, D_4, E_4
  91. Polygon(G_3, F_3, 5) → veelhoek10, b_6, d_6, e_6, f_7, g_7, F_4, G_4, H_4
  92. Midpoint(G_4, D_4) → I_4
  93. Polygon(F_3, F_4, I_4, H_4, G_3) → veelhoek11, h_7, i_7, j_7, k_7, l_7
  94. Polygon(N_3, O_3, C_4, I_4, E_4) → veelhoek12, m_7, o_3, n_7, p_7, q_7
  95. Polygon(O_2, N_2, A_4, I_4, Z_3) → veelhoek13, r_7, s_7, t_7, a_7, b_7
  96. Ray(I_1, P_3) → c_7
  97. Ray(UU, M_2) → d_7
  98. Ray(I_1, P_3) → e_7
  99. Circle(H_2, J_2) → c_8
  100. Point(e_7) → J_4
  101. Polygon(M_2, K_2, J_2, H_2, J_4) → veelhoek14, f_8, g_8, h_8, i_8, j_8
  102. Polygon(A_4, N_2, M_2, J_4, P_3, N_3) → veelhoek15, k_8, l_8, m_8, n_8, p_8, q_8
  103. Ray(UU, L_2) → r_8
  104. Intersect(q_3, r_8) → K_4
  105. Mirror(L_2, q_3) → L_4
  106. Mirror(K_2, q_3) → M_4
  107. Polygon(K_2, L_2, K_4, L_4, M_4, J_2) → veelhoek16, s_8, t_8, a_8, b_8, d_8, e_8
  108. Mirror(J_4, q_3) → N_4
  109. Polygon(P_3, J_4, H_2, N_4, Q_3) → veelhoek17, f_9, g_9, h_9, i_9, j_9
  110. Circle(UU, W_2) → c_9
  111. Ray(UU, Q_2) → k_9
  112. Intersect(c_9, k_9, 1) → O_4
  113. Polygon(Q_2, P_2, W_2, B_3, O_4) → veelhoek18, l_9, m_9, n_9, p_9, o_4
  114. Mirror(M_2, vert) → P_4
  115. Mirror(K_2, vert) → Q_4
  116. Mirror(J_2, vert) → R_4
  117. Mirror(H_2, vert) → S_4
  118. Mirror(J_4, vert) → T_4
  119. Polygon(P_4, Q_4, R_4, S_4, T_4) → veelhoek14', q_9, r_9, s_9, t_9, a_9
  120. Mirror(P_3, vert) → U_4
  121. Mirror(J_4, vert) → V_4
  122. Mirror(H_2, vert) → W_4
  123. Mirror(N_4, vert) → Z_4
  124. Mirror(Q_3, vert) → A_5
  125. Polygon(U_4, V_4, W_4, Z_4, A_5) → veelhoek17', u_4, v_4, w_4, z_4, b_9
  126. Mirror(A_4, vert) → B_5
  127. Mirror(N_2, vert) → C_5
  128. Mirror(M_2, vert) → D_5
  129. Mirror(J_4, vert) → E_5
  130. Mirror(P_3, vert) → F_5
  131. Mirror(N_3, vert) → G_5
  132. Polygon(B_5, C_5, D_5, E_5, F_5, G_5) → veelhoek15', d_9, e_9, f_{10}, g_{10}, h_{10}, i_{10}
  133. Mirror(O_2, vert) → H_5
  134. Mirror(N_2, vert) → I_5
  135. Mirror(A_4, vert) → J_5
  136. Mirror(I_4, vert) → K_5
  137. Mirror(Z_3, vert) → L_5
  138. Polygon(H_5, I_5, J_5, K_5, L_5) → veelhoek13', j_{10}, k_{10}, l_{10}, m_{10}, n_{10}
  139. Mirror(N_3, vert) → M_5
  140. Mirror(O_3, vert) → N_5
  141. Mirror(C_4, vert) → O_5
  142. Mirror(I_4, vert) → P_5
  143. Mirror(E_4, vert) → Q_5
  144. Polygon(M_5, N_5, O_5, P_5, Q_5) → veelhoek12', p_{10}, q_{10}, o_5, r_{10}, s_{10}
  145. Mirror(C_3, vert) → R_5
  146. Mirror(W_2, vert) → S_5
  147. Mirror(P_2, vert) → T_5
  148. Mirror(O_2, vert) → U_5
  149. Mirror(Z_3, vert) → V_5
  150. Mirror(F_3, vert) → W_5
  151. Polygon(R_5, S_5, T_5, U_5, V_5, W_5) → veelhoek8', t_{10}, a_{10}, b_{10}, u_5, v_5, w_5
  152. Mirror(F_3, vert) → Z_5
  153. Mirror(F_4, vert) → A_6
  154. Mirror(I_4, vert) → B_6
  155. Mirror(H_4, vert) → C_6
  156. Mirror(G_3, vert) → D_6
  157. Polygon(Z_5, A_6, B_6, C_6, D_6) → veelhoek11', z_5, c_{10}, d_{10}, e_{10}, f_{11}
  158. Mirror(veelhoek6, vert) → veelhoek6'
  159. Mirror(veelhoek7, vert) → veelhoek7'
  160. Mirror(Q_2, vert) → E_6
  161. Mirror(P_2, vert) → F_6
  162. Mirror(W_2, vert) → G_6
  163. Mirror(B_3, vert) → H_6
  164. Mirror(O_4, vert) → I_6
  165. Polygon(E_6, F_6, G_6, H_6, I_6) → veelhoek18', g_{11}, h_{11}, i_{11}, j_{11}, k_{11}
  166. Mirror(veelhoek5, vert) → veelhoek5'
  167. Mirror(Q_2, hor) → J_6
  168. Mirror(P_2, hor) → K_6
  169. Mirror(W_2, hor) → L_6
  170. Mirror(B_3, hor) → M_6
  171. Mirror(O_4, hor) → N_6
  172. Polygon(J_6, K_6, L_6, M_6, N_6) → veelhoek18'_1, l_{11}, m_{11}, n_{11}, p_{11}, q_{11}
  173. Mirror(E_6, hor) → O_6
  174. Mirror(F_6, hor) → P_6
  175. Mirror(G_6, hor) → Q_6
  176. Mirror(H_6, hor) → R_6
  177. Mirror(I_6, hor) → S_6
  178. Polygon(O_6, P_6, Q_6, R_6, S_6) → veelhoek18'', o_6, r_{11}, s_{11}, t_{11}, a_{11}
  179. Mirror(R_5, hor) → T_6
  180. Mirror(S_5, hor) → U_6
  181. Mirror(T_5, hor) → V_6
  182. Mirror(U_5, hor) → W_6
  183. Mirror(V_5, hor) → Z_6
  184. Mirror(W_5, hor) → A_7
  185. Polygon(T_6, U_6, V_6, W_6, Z_6, A_7) → veelhoek8'', b_{11}, u_6, v_6, w_6, z_6, c_{11}
  186. Mirror(C_3, hor) → B_7
  187. Mirror(W_2, hor) → C_7
  188. Mirror(P_2, hor) → D_7
  189. Mirror(O_2, hor) → E_7
  190. Mirror(Z_3, hor) → F_7
  191. Mirror(F_3, hor) → G_7
  192. Polygon(B_7, C_7, D_7, E_7, F_7, G_7) → veelhoek8'_1, d_{11}, e_{11}, f_{12}, g_{12}, h_{12}, i_{12}
  193. Mirror(H_5, hor) → H_7
  194. Mirror(I_5, hor) → I_7
  195. Mirror(J_5, hor) → J_7
  196. Mirror(K_5, hor) → K_7
  197. Mirror(L_5, hor) → L_7
  198. Polygon(H_7, I_7, J_7, K_7, L_7) → veelhoek13'', j_{12}, k_{12}, l_{12}, m_{12}, n_{12}
  199. Mirror(B_5, hor) → M_7
  200. Mirror(C_5, hor) → N_7
  201. Mirror(D_5, hor) → O_7
  202. Mirror(E_5, hor) → P_7
  203. Mirror(F_5, hor) → Q_7
  204. Mirror(G_5, hor) → R_7
  205. Polygon(M_7, N_7, O_7, P_7, Q_7, R_7) → veelhoek15'', p_{12}, q_{12}, o_7, r_{12}, s_{12}, t_{12}
  206. Mirror(P_4, hor) → S_7
  207. Mirror(Q_4, hor) → T_7
  208. Mirror(R_4, hor) → U_7
  209. Mirror(S_4, hor) → V_7
  210. Mirror(T_4, hor) → W_7
  211. Polygon(S_7, T_7, U_7, V_7, W_7) → veelhoek14'', a_{12}, b_{12}, u_7, v_7, w_7
  212. Mirror(K_2, hor) → Z_7
  213. Mirror(L_2, hor) → A_8
  214. Mirror(K_4, hor) → B_8
  215. Mirror(L_4, hor) → C_8
  216. Mirror(M_4, hor) → D_8
  217. Mirror(J_2, hor) → E_8
  218. Polygon(Z_7, A_8, B_8, C_8, D_8, E_8) → veelhoek16', z_7, c_{12}, d_{12}, e_{12}, f_{13}, g_{13}
  219. Mirror(M_2, hor) → F_8
  220. Mirror(K_2, hor) → G_8
  221. Mirror(J_2, hor) → H_8
  222. Mirror(H_2, hor) → I_8
  223. Mirror(J_4, hor) → J_8
  224. Polygon(F_8, G_8, H_8, I_8, J_8) → veelhoek14'_1, h_{13}, i_{13}, j_{13}, k_{13}, l_{13}
  225. Mirror(A_4, hor) → K_8
  226. Mirror(N_2, hor) → L_8
  227. Mirror(M_2, hor) → M_8
  228. Mirror(J_4, hor) → N_8
  229. Mirror(P_3, hor) → O_8
  230. Mirror(N_3, hor) → P_8
  231. Polygon(K_8, L_8, M_8, N_8, O_8, P_8) → veelhoek15'_1, m_{13}, n_{13}, p_{13}, q_{13}, o_8, r_{13}
  232. Mirror(N_3, hor) → Q_8
  233. Mirror(O_3, hor) → R_8
  234. Mirror(C_4, hor) → S_8
  235. Mirror(I_4, hor) → T_8
  236. Mirror(E_4, hor) → U_8
  237. Polygon(Q_8, R_8, S_8, T_8, U_8) → veelhoek12'_1, s_{13}, t_{13}, a_{13}, b_{13}, u_8
  238. Mirror(O_2, hor) → V_8
  239. Mirror(N_2, hor) → W_8
  240. Mirror(A_4, hor) → Z_8
  241. Mirror(I_4, hor) → A_9
  242. Mirror(Z_3, hor) → B_9
  243. Polygon(V_8, W_8, Z_8, A_9, B_9) → veelhoek13'_1, v_8, w_8, z_8, c_{13}, d_{13}
  244. Mirror(F_3, hor) → C_9
  245. Mirror(F_4, hor) → D_9
  246. Mirror(I_4, hor) → E_9
  247. Mirror(H_4, hor) → F_9
  248. Mirror(G_3, hor) → G_9
  249. Polygon(C_9, D_9, E_9, F_9, G_9) → veelhoek11'_1, e_{13}, f_{14}, g_{14}, h_{14}, i_{14}
  250. Mirror(M_5, hor) → H_9
  251. Mirror(N_5, hor) → I_9
  252. Mirror(O_5, hor) → J_9
  253. Mirror(P_5, hor) → K_9
  254. Mirror(Q_5, hor) → L_9
  255. Polygon(H_9, I_9, J_9, K_9, L_9) → veelhoek12'', j_{14}, k_{14}, l_{14}, m_{14}, n_{14}
  256. Mirror(Z_5, hor) → M_9
  257. Mirror(A_6, hor) → N_9
  258. Mirror(B_6, hor) → O_9
  259. Mirror(C_6, hor) → P_9
  260. Mirror(D_6, hor) → Q_9
  261. Polygon(M_9, N_9, O_9, P_9, Q_9) → veelhoek11'', p_{14}, q_{14}, o_9, r_{14}, s_{14}
  262. Mirror(P_3, hor) → R_9
  263. Mirror(J_4, hor) → S_9
  264. Mirror(H_2, hor) → T_9
  265. Mirror(N_4, hor) → U_9
  266. Mirror(Q_3, hor) → V_9
  267. Polygon(R_9, S_9, T_9, U_9, V_9) → veelhoek17'_1, t_{14}, a_{14}, b_{14}, u_9, v_9
  268. Mirror(U_4, hor) → W_9
  269. Mirror(V_4, hor) → Z_9
  270. Mirror(W_4, hor) → A_{10}
  271. Mirror(Z_4, hor) → B_{10}
  272. Mirror(A_5, hor) → C_{10}
  273. Polygon(W_9, Z_9, A_{10}, B_{10}, C_{10}) → veelhoek17'', w_9, z_9, c_{14}, d_{14}, e_{14}
  274. Mirror(veelhoek7, hor) → veelhoek7'_1
  275. Mirror(veelhoek7'_1, vert) → veelhoek19
  276. Mirror(veelhoek4, hor) → veelhoek4'

AI Drawing Prompt Reference

Draw 10- and 11-pointed star patterns by first creating a point and rotating it by 18 degrees, then constructing a 20-sided polygon with these two points. Next, use rotation and reflection commands to create multiple segments and circles, and reflect the polygon across the y-axis. Finally, construct an 11-pointed star using rotations and reflections, and reflect it across the y-axis to complete the pattern.
The prompt above was compiled from the original drawing commands and is for reference only.