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Cellular Geometric Construction and Plane Transformations

Cellular Geometric Construction and Plane Transformations
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2020-08-17 14:58
PolygonSequenceTransformationGeometric Construction
In collections Fatin Uluengin - muqarnas
This example demonstrates how cellular arrangements and geometric transformations can be used to construct complex planar patterns. Core mathematical concepts include polygon construction, plane transformations (mirror, rotate, translate), sequence generation, and relationships between intersections and diagonals. Construction highlights utilize GeoGebra commands such as Polygon, Mirror, Rotate, Translate, Sequence, Intersect, and OrthogonalLine, generating regular patterns through cell repetition and symmetry. Suitable for high school geometry, plane transformation units, or introductory mathematical modeling, it helps students explore symmetry, regular arrangements, and the construction of parametric or parameterized shapes.

GeoGebra Commands & Parameters

  1. ConstructionStep() → stap
  2. Point(yAxis) → B
  3. Point(yAxis) → A
  4. Polygon(D, C, 4) → bas1, f, g, h, i, E, F
  5. Line(G, E) → di1
  6. Mirror(B, di1) → B'
  7. Mirror(B', yAxis) → B''
  8. Midpoint(B', B'') → H_6
  9. Mirror(A, di1) → A'
  10. Mirror(K_1, spieg) → Q_1
  11. Mirror(L, di1) → N
  12. Mirror(I_6, yAxis) → N_8
  13. Mirror(O_2, spieg) → R_2
  14. Rotate(I_6, 45°, H_6) → A_7
  15. Mirror(A_7, yAxis) → L_8
  16. Mirror(L_2, spieg) → P_2
  17. Rotate(I_6, 22.5°, I_6) → Z_6
  18. Mirror(L, Line[F, C]) → L'_1
  19. Mirror(W_1, spieg) → K_2
  20. Rotate(L'_1, (-(22.5°)), A_7) → B_7
  21. Segment(B_7, A_7) → f_{11}
  22. Rotate(L'_1, 22.5°, I_6) → W_6
  23. Segment(W_6, Z_6) → e_{10}
  24. Intersect(e_{10}, f_{11}) → C_7
  25. Rotate(C_7, (-(45°)), H_6) → Q_9
  26. Mirror(Q_9, yAxis) → R_9
  27. Mirror(S_2, spieg) → T_2
  28. Rotate(C_7, 45°, H_6) → O_8
  29. Mirror(C_7, yAxis) → M_8
  30. Mirror(N_2, spieg) → Q_2
  31. Rotate(L, 22.5°, D) → L'
  32. Segment(D, L') → n
  33. Line(L, xAxis) → p
  34. Intersect(p, n) → Q
  35. Mirror(B_2, spieg) → H_2
  36. Mirror(Q, x = ((-v2)) / 2) → U
  37. Mirror(D_2, spieg) → F_2
  38. Mirror(U, di1) → V
  39. Mirror(L_1, spieg) → R_1
  40. Mirror(Q, di1) → S
  41. Mirror(N_1, spieg) → T_1
  42. Mirror(Q, Line[F, C]) → Q'_1
  43. Mirror(Z_1, spieg) → J_2
  44. Mirror(Q'_1, di1) → J_3
  45. Mirror(O_1, spieg) → V_1
  46. Mirror(C, di1) → C'
  47. Polygon(E, L'_1, J_3, C') → bas11, e_5, d_4, f_5, g_5
  48. Polygon(A, C, D, B) → basvierhoek1, a, c, d, b
  49. Mirror(bas1, di1) → bas7
  50. Mirror(D, di1) → D'
  51. Polygon(A', C', D', B') → bas16, a', c', d', b'
  52. Mirror(bas1, yAxis) → bas9
  53. Mirror(A, yAxis) → A'_1
  54. Mirror(C, yAxis) → C'_1
  55. Mirror(D, yAxis) → D'_1
  56. Mirror(B, yAxis) → B'_1
  57. Polygon(A'_1, C'_1, D'_1, B'_1) → bas18, a'_1, c'_1, d'_1, b'_1
  58. Mirror(A', yAxis) → A''
  59. Mirror(C', yAxis) → C''
  60. Mirror(D', yAxis) → D''
  61. Polygon(A'', C'', D'', B'') → bas17, a'', c'', d'', b''
  62. Mirror(bas7, yAxis) → bas8
  63. Mirror(L, yAxis) → M
  64. Mirror(N, yAxis) → P
  65. Segment(D, L) → j
  66. Segment(L, B) → k
  67. Segment(B, M) → l
  68. Segment(M, D'_1) → m
  69. Mirror(Q, yAxis) → R
  70. Mirror(S, yAxis) → T
  71. Segment(B', N) → q
  72. Segment(N, S) → r
  73. Segment(S, C') → s
  74. Segment(N, D') → t
  75. Segment(C, Q) → e
  76. Segment(Q, L) → f_1
  77. Mirror(V, yAxis) → W
  78. Mirror(U, yAxis) → Z
  79. Segment(L, U) → g_1
  80. Segment(U, A) → h_1
  81. Segment(A, Z) → i_1
  82. Segment(Z, M) → j_1
  83. Segment(M, R) → k_1
  84. Segment(R, C'_1) → l_1
  85. Segment(C'', T) → m_1
  86. Segment(T, P) → n_1
  87. Segment(P, W) → p_1
  88. Segment(W, A'') → q_1
  89. Segment(P, B'') → r_1
  90. Segment(P, D'') → s_1
  91. Segment(N, V) → j_2
  92. Segment(V, A') → k_2
  93. Plane(L, L + (0, 0, 1), L + (0, 1, 0)) → vlakL
  94. Mirror(D, vlakL) → D'_2
  95. Polygon(A, B, D'_1, C'_1, R, M, Z) → bas2, a_2, b_2, r_2, s_2, r_3, m_3, z_1
  96. Polygon(D, B, A, U, Q, C) → bas21, d_2, b_3, a_3, u, q_3, c_2
  97. Mirror(A, di1) → A'_2
  98. Mirror(C, di1) → C'_2
  99. Polygon(C'', D'', B'', A'', W, T) → bas22, t_2, e_2, f_3, g_3, w, t_3
  100. Polygon(E, C, Q'_1, L'_1) → bas10, e_4, c_5, q'_1, l'_1
  101. Mirror(E, yAxis) → E'
  102. Mirror(C, yAxis) → C'_3
  103. Mirror(Q'_1, yAxis) → K_3
  104. Mirror(L'_1, yAxis) → L_3
  105. Polygon(E', C'_3, K_3, L_3) → bas19, e', c'_3, h_5, i_5
  106. Mirror(E, yAxis) → E'_1
  107. Mirror(L'_1, yAxis) → M_3
  108. Mirror(J_3, yAxis) → N_3
  109. Mirror(C', yAxis) → C''_1
  110. Polygon(E'_1, M_3, N_3, C''_1) → bas20, e'_1, j_5, k_5, c''_1
  111. Intersect(l'_1, e_5) → A_4
  112. Intersect(l'_1, e_5) → B_4
  113. Midpoint(d'') → T_5
  114. PolyLine(B', H, E, F, K, J, I, B'') → h_{10}
  115. Line(H_6, H_6 + (0, 0, 1)) → rot
  116. Polygon(I_6, A', V, N) → bas12, m_{10}, n_{10}, v_{10}, n_{11}
  117. Sequence(Polygon[Rotate[I_6, (nn * 45°), rot], Rotate[A', (nn * 45°), rot], Rotate[V, (nn * 45°), rot], Rotate[N, (nn * 45°), rot]], nn, 1, 3) → l1
  118. Polygon(I_6, N, S, C') → bas13, p_{10}, n_{12}, s_{10}, q_{10}
  119. Sequence(Polygon[Rotate[I_6, (nn * 45°), rot], Rotate[N, (nn * 45°), rot], Rotate[S, (nn * 45°), rot], Rotate[C', (nn * 45°), rot]], nn, 1, 3) → l2
  120. Polygon(I_6, C', J_3, L'_1) → bas14, r_{10}, t_{10}, a_{10}, b_{10}
  121. Sequence(Polygon[Rotate[I_6, (nn * 45°), rot], Rotate[C', (nn * 45°), rot], Rotate[J_3, (nn * 45°), rot], Rotate[L'_1, (nn * 45°), rot]], nn, 1, 3) → l3
  122. Polygon(I_6, L'_1, A_7, C_7) → bas15, g_{11}, h_{11}, i_{11}, j_{11}
  123. Sequence(Polygon[Rotate[I_6, (nn * 45°), rot], Rotate[L'_1, (nn * 45°), rot], Rotate[A_7, (nn * 45°), rot], Rotate[C_7, (nn * 45°), rot]], nn, 1, 2) → l4
  124. Polygon(A', B', W, T, N_3, L_3, K_3, R, Z, A, U, Q, H, Q'_1, L'_1, J_3, S, V, E, F, K, J, I, B'', A'') → basveelhoek5, p_{11}, s_{11}, w_9, t_{11}, a_{11}, b_{11}, c_{11}, r_{12}, z_2, a_{12}, u_{10}, q_{11}, q_{12}, d_{11}, g_{12}, l_{12}, s_{12}, v_{11}, h_{12}, e_{11}, f_{12}, k_{12}, j_{12}, i_{12}, r_{11}
  125. Translate(E, Vector[(0, 0, 2)]) → E'_2
  126. Translate(F, Vector[(0, 0, 2)]) → F'
  127. Polygon(A', V, R, K_3, L_3, N_3, T, W, A'', N_8, M_8, O_8, S, A_7, C_7, I_6, J_3, L'_1, Q'_1, Q, U, A, Z) → bas3, e_{12}, z_3, r_{13}, i_{13}, j_{13}, k_{13}, t_{13}, w_{10}, l_{13}, m_{13}, n_{13}, v_{12}, o_8, p_{13}, b_{13}, c_{13}, s_{13}, f_{13}, g_{13}, h_{13}, q_{13}, u_{11}, a_{13}
  128. Translate(E, Vector[(0, 0, 2)]) → E'_3
  129. Translate(F, Vector[(0, 0, 2)]) → F'_1
  130. Rotate(P_9, (-(45°)), H_6) → S_9
  131. Ray(C, A_7) → i_{16}
  132. Intersect(i_{16}, yAxis) → T_9
  133. Segment(I_6, S_9) → j_{16}
  134. Segment(I_6, P_9) → k_{16}
  135. OrthogonalLine(P_9, j_{16}) → l_{16}
  136. OrthogonalLine(S_9, k_{16}) → m_{16}
  137. Intersect(l_{16}, m_{16}) → U_9
  138. Segment(S_9, U_9) → n_{16}
  139. Segment(U_9, P_9) → p_{16}
  140. Segment(Q_9, S_9) → q_{16}
  141. Segment(I_6, Q_9) → r_{16}
  142. Segment(A', Q_9) → s_{16}
  143. Mirror(A_7, yAxis) → V_9
  144. Mirror(C_7, yAxis) → W_9
  145. Mirror(I_6, yAxis) → Z_9
  146. Segment(A'', R_9) → t_{16}
  147. Segment(R_9, Z_9) → a_{16}
  148. Mirror(P_9, yAxis) → A_{10}
  149. Mirror(U_9, yAxis) → B_{10}
  150. Mirror(S_9, yAxis) → C_{10}
  151. Segment(C_{10}, R_9) → b_{16}
  152. Segment(C_{10}, B_{10}) → c_{16}
  153. Segment(B_{10}, A_{10}) → d_{16}
  154. Segment(A_{10}, Z_9) → e_{16}
  155. Segment(C_{10}, Z_9) → f_{17}
  156. Segment(P_9, A_7) → g_{17}
  157. Segment(A_{10}, V_9) → h_{17}
  158. Segment(T_9, V_9) → j_{17}
  159. Segment(A_7, T_9) → i_{17}
  160. Rotate(U_9, 45°, H_6) → D_{10}
  161. Rotate(U_9, 90°, H_6) → E_{10}
  162. Segment(P_9, D_{10}) → k_{17}
  163. Segment(D_{10}, T_9) → l_{17}
  164. Segment(T_9, E_{10}) → m_{17}
  165. Segment(E_{10}, A_{10}) → n_{17}
  166. Segment(S_9, C_{10}) → i_{21}
  167. Segment(P_9, H_6) → j_{21}
  168. Segment(T_9, H_6) → k_{21}
  169. Polygon(A', I_6, Q_9) → basdriehoek1, r_{17}, s_{17}, t_{17}
  170. Mirror(A', yAxis) → A''_5
  171. Segment(H_6, A_{10}) → l_{21}
  172. Mirror(I_6, yAxis) → A_{11}
  173. Mirror(Q_9, yAxis) → B_{11}
  174. Polygon(A''_5, A_{11}, B_{11}) → basdriehoek1', c_{17}, a''_5, d_{17}
  175. PolyLine(S_9, U_9, D_{10}, E_{10}, B_{10}, C_{10}) → e_{17}
  176. PolyLine(Q_9, C_7, W_{11}, W_9, R_9) → g_{18}
  177. Polygon(Q_9, C_7, S_9, W_{11}, W_9, R_9, C_{10}, B_{10}, E_{10}, D_{10}, U_9) → bas4, j_{18}, t_{18}, a_{18}, k_{18}, l_{18}, m_{18}, n_{18}, p_{18}, q_{18}, r_{18}, s_{18}
  178. Polygon(A', V, R, K_3, L_3, N_3, T, W, A'', R_9, W_9, W_{11}, S, C_7, Q_9, J_3, L'_1, Q'_1, Q, U, A, Z) → bas5, b_{18}, z_4, r_{19}, f_{19}, g_{19}, h_{19}, t_{19}, w_{16}, i_{19}, j_{19}, k_{19}, v_{16}, l_{19}, m_{19}, n_{19}, s_{19}, c_{18}, d_{18}, e_{18}, q_{19}, u_{16}, a_{19}
  179. Polygon(S_9, U_9, P_9, D_{10}, E_{10}, B_{10}, C_{10}) → bas6, g_{20}, h_{20}, i_{20}, j_{20}, k_{20}, l_{20}, m_{20}
  180. Segment(S_9, C_{10}) → n_{20}
  181. Mirror(C'', yAxis) → C'''
  182. Mirror(D'', yAxis) → D'''
  183. Mirror(B'', yAxis) → B'''
  184. Mirror(A'', yAxis) → A'''
  185. Mirror(W, yAxis) → W'
  186. Mirror(T, yAxis) → T'
  187. Polygon(C''', D''', B''', A''', W', T') → bas23, c''', d''', b''', a''', w', t'
  188. Translate(baspa, Vector[(-ver)]) → baspa'
  189. Translate(bas5, Vector[(-ver)]) → bas5'
  190. Translate(bas4, Vector[(-ver)]) → bas4'
  191. Mirror(N, yAxis) → N'
  192. Mirror(I_6, yAxis) → G_3
  193. Segment(N', G_3) → t_{26}
  194. Mirror(C', yAxis) → C''_2
  195. Mirror(I_6, yAxis) → H_3
  196. Segment(C''_2, H_3) → a_{26}
  197. Mirror(L'_1, yAxis) → I_3
  198. Mirror(I_6, yAxis) → O_3
  199. Segment(I_3, O_3) → b_{26}
  200. Mirror(L'_1, yAxis) → P_3
  201. Mirror(A_7, yAxis) → Q_3
  202. Segment(P_3, Q_3) → c_{26}
  203. Intersect(n_{13}, o_8) → R_3
  204. Intersect(h_1, i_1) → S_3
  205. Segment(R_3, S_3) → d_{26}
  206. Segment(A_7, C) → e_{26}
  207. Segment(A_7, L) → f_{27}
  208. Mirror(A_7, yAxis) → T_3
  209. Mirror(C, yAxis) → C'_4
  210. Segment(T_3, C'_4) → g_{27}
  211. Mirror(A_7, yAxis) → U_3
  212. Mirror(L, yAxis) → L'_2
  213. Segment(U_3, L'_2) → h_{27}
  214. Translate(S_9, Vector[(-ver)]) → V_3
  215. Translate(U_9, Vector[(-ver)]) → W_3
  216. Translate(P_9, Vector[(-ver)]) → Z_3
  217. Translate(D_{10}, Vector[(-ver)]) → C_4
  218. Translate(E_{10}, Vector[(-ver)]) → D_4
  219. Translate(B_{10}, Vector[(-ver)]) → E_4
  220. Translate(C_{10}, Vector[(-ver)]) → F_4
  221. Polygon(V_3, W_3, Z_3, C_4, D_4, E_4, F_4) → bas6', v_3, i_{27}, j_{27}, k_{27}, l_{27}, m_{27}, n_{27}
  222. Polygon(H, D_1, C_1, B') → pa44, h_4, f_2, g_2, i_2
  223. Polygon(H, E, E + ver, D_1) → pa42, h_2, e_1, t_1, d_1
  224. Polygon(E, E_1, F_1, F) → pa43, e_3, a_1, b_1, f_4
  225. Polygon(F, K, G_1, F_1) → pa45, f_6, k_3, c_1, l_2
  226. Polygon(K, J, H_1, G_1) → pa46, k_4, j_3, m_2, n_2
  227. Polygon(J, I, I_1, H_1) → pa47, j_4, i_3, p_2, q_2
  228. Polygon(I, B'', J_1, I_1) → pa48, i_4, h_3, l_3, n_3
  229. Polygon(B', K_1, L_1) → pa1, p_3, s_3, c_3
  230. Polygon(B', L_1, M_1) → pa2, d_3, g_4, l_4
  231. Polygon(D', B', M_1) → pa3, m_4, n_4, p_4
  232. Polygon(D', M_1, N_1) → pa4, q_4, r_4, s_4
  233. Polygon(D', N_1, P_1) → pa5, t_4, a_4, b_4
  234. Polygon(D', P_1, H) → pa6, h_6, c_4, l_5
  235. Polygon(E, H, P_1) → pa7, m_5, e_6, h_7
  236. Mirror(M_1, spieg) → S_1
  237. Mirror(P_1, spieg) → U_1
  238. Polygon(E, P_1, O_1) → pa8, o_1, e_7, d_5
  239. Polygon(E, O_1, W_1) → pa9, w_1, e_8, g_6
  240. Polygon(E, W_1, Z_1) → pa10, i_6, e_9, j_6
  241. Polygon(E, Z_1, A_2) → pa11, k_6, e_{13}, l_6
  242. Polygon(F, E, A_2) → pa12, m_6, f_7, e_{14}
  243. Polygon(F, A_2, D) → pa13, d_6, f_8, n_6
  244. Polygon(F, A_2, D) → pa14, d_7, f_9, p_6
  245. Polygon(D, A_2, B_2) → pa15, n_5, d_8, p_5
  246. Polygon(D, B_2, C_2) → pa16, q_5, d_9, r_5
  247. Polygon(D, C_2, B) → pa17, b_5, d_{10}, s_5
  248. Mirror(pa1, spieg) → pa31
  249. Mirror(pa2, spieg) → pa34
  250. Mirror(pa3, spieg) → pa35
  251. Mirror(pa4, spieg) → pa36
  252. Mirror(pa5, spieg) → pa37
  253. Mirror(pa6, spieg) → pa38
  254. Mirror(pa7, spieg) → pa39
  255. Mirror(pa8, spieg) → pa40
  256. Mirror(pa9, spieg) → pa21
  257. Mirror(pa10, spieg) → pa22
  258. Mirror(pa11, spieg) → pa23
  259. Mirror(pa12, spieg) → pa24
  260. Mirror(pa13, spieg) → pa25
  261. Mirror(pa14, spieg) → pa26
  262. Mirror(pa15, spieg) → pa27
  263. Mirror(pa16, spieg) → pa28
  264. Mirror(pa17, spieg) → pa29
  265. Polygon(B, D_2, C_2) → pa18, t_5, b_6, a_5
  266. Polygon(B, E_2, D_2) → pa19, q_6, b_7, r_6
  267. Mirror(pa18, spieg) → pa30
  268. Mirror(pa19, spieg) → pa32
  269. Polygon(B', C_1, K_1) → pa20, s_6, t_6, a_6
  270. Mirror(pa20, spieg) → pa33
  271. Mirror(C_2, spieg) → G_2
  272. Mirror(A_2, spieg) → I_2
  273. Polygon(K_1, L_1, E_2, F_2, H_2, J_2, K_2, V_1, T_1, R_1, Q_1, J_1, M_1, I_1, H_1, G_1, F_1, E_1, D_1, C_1, N_1, P_1, O_1, W_1, Z_1, B_2, D_2) → pa41, c_6, q_7, r_7, s_7, t_7, a_7, c_7, v_1, g_8, h_8, i_8, g_7, j_8, k_8, l_8, m_8, n_8, p_8, q_8, r_8, i_7, j_7, k_7, l_7, m_7, n_7, p_7
  274. Polygon(R_9, N_8 + ver, A'') → driehoek22, s_8, r_9, t_8
  275. Polygon(A, Z, L_8 + ver) → ro1, a_8, b_8, c_8
  276. Polygon(M, Z, L_8 + ver) → ro2, g_9, m_9, z_5
  277. Polygon(M, R, L_8 + ver) → ro3, h_9, m_{11}, r_{14}
  278. Polygon(R, C'_3, L_8 + ver) → ro4, i_9, r_{15}, j_9
  279. Mirror(ro1, spieg) → driehoek23'
  280. Mirror(ro2, spieg) → driehoek24'
  281. Mirror(ro3, spieg) → driehoek25'
  282. Mirror(ro4, spieg) → driehoek26'
  283. Polygon(C'_3, K_3, L_8 + ver) → ro5, k_9, l_9, n_9
  284. Polygon(L_3, K_3, L_8 + ver) → ro6, p_9, q_9, s_9
  285. Mirror(ro5, spieg) → driehoek27'
  286. Mirror(ro6, spieg) → driehoek28'
  287. Polygon(A, L_2, M_2) → ro7, t_9, a_9, b_9
  288. Mirror(ro7, spieg) → driehoek29'
  289. Polygon(L_3, N_3, O_2) → ro8, o_2, c_9, f_{10}
  290. Polygon(L_2, L_3, N_2) → ro9, g_{10}, i_{10}, j_{10}
  291. Polygon(L_3, O_2, N_2) → ro10, k_{10}, l_{10}, c_{10}
  292. Polygon(N_3, C'', O_2) → ro11, k_{11}, l_{11}, m_{12}
  293. Polygon(C'', T, O_2) → ro12, p_{12}, t_{12}, t_{14}
  294. Polygon(T, P, O_2) → ro13, b_{12}, t_{15}, p_{14}
  295. Polygon(P, W, O_2) → ro14, c_{12}, p_{15}, w_2
  296. Polygon(W, A'', O_2) → ro15, d_{12}, w_3, d_{13}
  297. Polygon(A'', R_9, O_2) → driehoek38, f_{14}, g_{14}, h_{14}
  298. Mirror(ro8, spieg) → driehoek30'
  299. Mirror(ro9, spieg) → driehoek31'
  300. Mirror(ro10, spieg) → driehoek32'
  301. Mirror(ro11, spieg) → driehoek33'
  302. Mirror(ro12, spieg) → driehoek34'
  303. Mirror(ro13, spieg) → driehoek35'
  304. Mirror(ro14, spieg) → driehoek36'
  305. Mirror(ro15, spieg) → driehoek37'
  306. Mirror(driehoek38, spieg) → driehoek38'
  307. Polygon(A'', Q_1, R_1, W) → ro20, i_{14}, j_{14}, k_{14}, w_4
  308. Polygon(O_2, S_2, A'') → ro16, s_{14}, a_{14}, b_{14}
  309. Mirror(ro16, spieg) → driehoek39'
  310. Polygon(A'', S_2, Q_1) → ro17, c_{14}, d_{14}, f_{15}
  311. Mirror(ro17, spieg) → driehoek40'
  312. Polygon(W, T, T_1, R_1) → ro22, w_5, t_{20}, g_{15}, h_{15}
  313. Mirror(ro22, spieg) → ro30
  314. Mirror(ro20, spieg) → ro28
  315. Polygon(T, N_3, V_1, T_1) → ro21, t_{21}, l_{14}, m_{14}, n_{14}
  316. Mirror(ro21, spieg) → ro29
  317. Polygon(N_3, L_3, K_2, V_1) → ro23, q_{14}, i_{15}, j_{15}, k_{15}
  318. Mirror(ro23, spieg) → ro31
  319. Polygon(L_3, K_3, J_2, K_2) → ro24, l_{15}, m_{15}, n_{15}, q_{15}
  320. Mirror(ro24, spieg) → ro32
  321. Polygon(K_3, R, H_2, J_2) → ro25, s_{15}, r_{20}, a_{15}, b_{15}
  322. Mirror(ro25, spieg) → ro33
  323. Polygon(R, Z, F_2, H_2) → ro26, r_{21}, z_6, c_{15}, d_{15}
  324. Mirror(ro26, spieg) → ro34
  325. Polygon(Z, A, E_2, F_2) → ro27, z_7, a_{17}, e_{15}, f_{16}
  326. Polygon(S_2, O_2, L_1, N_1, O_1, W_1, Z_1, B_2, D_2, E_2, F_2, H_2, N_2, J_2, K_2, V_1, T_1, R_1, Q_1, L_2, M_2, P_2, Q_2, R_2, T_2, K_1) → ro19, g_{16}, b_{19}, c_{19}, d_{19}, e_{19}, f_{20}, p_{20}, q_{20}, s_{20}, a_{20}, b_{20}, h_{16}, c_{20}, d_{20}, e_{20}, f_{21}, g_{21}, h_{21}, q_{21}, p_{17}, q_{17}, b_{17}, f_{18}, h_{18}, i_{18}, p_{19}
  327. Mirror(ro27, spieg) → ro35
  328. Mirror(ro2, spieg) → ro2_2
  329. Mirror(ro3, spieg) → ro3_2
  330. Mirror(ro1, spieg) → ro1_2
  331. Mirror(ro4, spieg) → ro4_2
  332. Mirror(ro5, spieg) → ro5_2
  333. Mirror(ro6, spieg) → ro6_2
  334. Mirror(ro7, spieg) → ro7_2
  335. Mirror(ro8, spieg) → ro8_2
  336. Mirror(ro9, spieg) → ro9_2
  337. Mirror(ro10, spieg) → ro10_2
  338. Mirror(ro11, spieg) → ro11_2
  339. Mirror(ro12, spieg) → ro12_2
  340. Mirror(ro13, spieg) → ro13_2
  341. Mirror(ro14, spieg) → ro14_2
  342. Mirror(ro15, spieg) → ro15_2
  343. Mirror(ro16, spieg) → ro16_2
  344. Polygon(A', K_1, T_2) → ro18, m_{21}, n_{21}, p_{21}
  345. Polygon(R_9, N_8, U_2) → geel1, u_2, s_{21}, a_{21}
  346. Polygon(N_8, V_2, U_2) → geel2, b_{21}, c_{21}, v_2
  347. Polygon(N_8, W_2, V_2) → geel3, d_{21}, e_{21}, f_{22}
  348. Polygon(N_8, M_8, W_2) → geel4, g_{22}, h_{22}, i_{22}
  349. Polygon(M_8, L_8, W_2) → geel5, j_{22}, k_{22}, l_{22}
  350. Polygon(L_8, Z_2, W_2) → geel6, m_{22}, n_{22}, p_{22}
  351. Polygon(L_8, B_3, Z_2) → geel7, q_{22}, r_{22}, s_{22}
  352. Polygon(L_8, O_8, B_3) → geel8, t_{22}, a_{22}, b_{22}
  353. Mirror(geel1, spieg) → geel10
  354. Mirror(geel2, spieg) → geel11
  355. Mirror(geel3, spieg) → geel12
  356. Mirror(geel4, spieg) → geel13
  357. Mirror(geel5, spieg) → geel14
  358. Mirror(geel6, spieg) → geel15
  359. Mirror(geel7, spieg) → geel16
  360. Mirror(geel8, spieg) → geel17
  361. Polygon(R_9, U_2, S_2) → geel9, c_{22}, d_{22}, e_{22}
  362. Mirror(geel9, spieg) → geel18
  363. Polygon(R_9, N_8, O_2, S_2) → geel19, f_{23}, g_{23}, h_{23}, i_{23}
  364. Mirror(geel19, spieg) → geel22
  365. Polygon(N_8, M_8, N_2, O_2) → geel20, j_{23}, k_{23}, l_{23}, m_{23}
  366. Mirror(geel20, spieg) → geel23
  367. Polygon(M_8, O_8, M_2, N_2) → geel21, n_{23}, p_{23}, q_{23}, r_{23}
  368. Mirror(geel21, spieg) → geel24
  369. Polygon(U_2, V_2, S_2, Z_2, A_3, D_3, E_3, T_2, Q_2, M_2, N_2) → geel25, s_{23}, h_{24}, i_{24}, t_{23}, a_{23}, b_{23}, c_{23}, d_{23}, e_{23}, f_{24}, g_{24}
  370. Polygon(C_{10}, F_3, B_{10}) → groen1, j_{24}, k_{24}, l_{24}
  371. Polygon(B_{10}, A_{10}, F_3) → groen2, m_{24}, n_{24}, p_{24}
  372. Polygon(A_{10}, E_{10}, F_3) → groen3, q_{24}, r_{24}, s_{24}
  373. Polygon(E_{10}, T_9, F_3) → groen4, t_{24}, a_{24}, b_{24}
  374. Polygon(T_9, D_{10}, F_3) → groen5, c_{24}, d_{24}, e_{24}
  375. Polygon(D_{10}, F_3, P_9) → groen6, f_{25}, g_{25}, h_{25}
  376. Polygon(P_9, F_3, U_9) → groen7, u_9, i_{25}, j_{25}
  377. Polygon(S_9, U_9, F_3) → groen8, k_{25}, l_{25}, m_{25}
  378. Polygon(C_{10}, F_3, U_2) → groen9, n_{25}, p_{25}, q_{25}
  379. Polygon(S_9, E_3, F_3) → groen10, r_{25}, s_{25}, t_{25}
  380. Polygon(C_{10}, B_{10}, V_2, U_2) → groen11, a_{25}, b_{25}, c_{25}, d_{25}
  381. Mirror(groen11, spieg) → groen14
  382. Polygon(B_{10}, E_{10}, Z_2, V_2) → groen12, e_{25}, f_{26}, g_{26}, h_{26}
  383. Mirror(groen12, spieg) → groen15
  384. Polygon(E_{10}, D_{10}, A_3, Z_2) → groen13, i_{26}, j_{26}, k_{26}, l_{26}
  385. Polygon(U_2, E_3, D_3, A_3, Z_2, V_2) → groenboven, m_{26}, n_{26}, p_{26}, q_{26}, r_{26}, s_{26}
  386. Translate(groewand, Vector[(-ver)]) → groewand'
  387. Translate(gewand, Vector[(-ver)]) → gewand'
  388. Translate(rowand, Vector[(-ver)]) → rowand'
  389. Translate(pawand, Vector[(-ver)]) → pawand'
  390. If(ss < 1, Translate[rolijst, Vector[((2 * ss) * ver)]], Translate[rolijst, Vector[(2 * ver)]]) → rolijstdyn
  391. If(ss < 1, Translate[rowand', Vector[((2 * ss) * ver)]], Translate[rowand', Vector[(2 * ver)]]) → rowanddyn
  392. If(ss ≤ 1, geellijst, If[1 < ss < 2, Translate[geellijst, Vector[(((ss - 1) * 4) * ver)]], Translate[geellijst, Vector[(4 * ver)]]]) → geellijstdyn
  393. If(ss ≤ 1, gewand', If[1 < ss < 2, Translate[gewand', Vector[(((ss - 1) * 4) * ver)]], Translate[gewand', Vector[(4 * ver)]]]) → gewanddyn
  394. If(ss ≤ 2, groelijst, If[2 < ss, Translate[groelijst, Vector[(((ss - 2) * 6) * ver)]]]) → groelijstdyn
  395. If(ss ≤ 2, groewand', If[2 < ss, Translate[groewand', Vector[(((ss - 2) * 6) * ver)]]]) → groewanddyn

AI Drawing Prompt Reference

Construct complex planar patterns using polygon construction and geometric transformations such as mirroring, rotating, and translating. Focus on cellular repetition and symmetry, generating regular patterns through intersections and diagonals. The construction sequence involves initial point placement, polygon creation, mirroring and rotating operations, intersection calculations, and final polygon generation. Suitable for exploring symmetry, regular arrangements, and parameterized shapes in mathematical modeling. This approach helps in understanding the relationships between geometric elements and their transformations.
The prompt above was compiled from the original drawing commands and is for reference only.