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Irregular Polygon Tessellation: Exploring Geometric Transformations and Construc

Irregular Polygon Tessellation: Exploring Geometric Transformations and Construc
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2022-11-25 10:39
PolygonTransformationGeometric Construction
This example demonstrates how irregular polygons can be tessellated, highlighting key mathematical concepts such as geometric transformations and construction. The construction uses GeoGebra commands like Polygon, Segment, and Translate, allowing users to adjust polygon shapes and positions to achieve tessellation. Suitable for middle school and above students learning geometry transformations and construction, useful for math teaching and geometric drawing exercises.

GeoGebra Commands & Parameters

  1. Polygon(G, H, C, B) → q1, g, h, c, b
  2. Segment(A, F) → f
  3. Segment(E, F) → i
  4. Segment(E, D) → j
  5. Vector(I, J) → u
  6. Translate(A, u) → A'
  7. Translate(F, u) → F'
  8. Segment(A', F') → f'
  9. Translate(E, u) → E'
  10. Translate(F, u) → F'_{1}
  11. Segment(E', F'_{1}) → i'
  12. Translate(E, u) → E'_{1}
  13. Translate(D, u) → D'
  14. Segment(E'_{1}, D') → j'
  15. Segment(D, K) → k
  16. Segment(K, C) → l
  17. Vector(I, L) → v
  18. Translate(D, v) → D'_{1}
  19. Translate(K, v) → K'
  20. Segment(D'_{1}, K') → k'
  21. Translate(K, v) → K'_{1}
  22. Translate(C, v) → C'
  23. Segment(K'_{1}, C') → l'
  24. Polygon(K', A, F, E, D, K, C, E', F', B) → poly1, m, b_1, a, f_1, e, d, k_1, c_1, e', n
  25. Translate(K', u) → K''
  26. Translate(A, u) → A'_{1}
  27. Translate(F, u) → F'_{2}
  28. Translate(E, u) → E'_{2}
  29. Translate(D, u) → D'_{2}
  30. Translate(K, u) → K'_{2}
  31. Translate(C, u) → C'_{1}
  32. Translate(E', u) → E''
  33. Translate(F', u) → F''
  34. Translate(B, u) → B'
  35. Translate(K', u) → K''_{1}
  36. Translate(A, u) → A'_{2}
  37. Translate(F, u) → F'_{3}
  38. Translate(E, u) → E'_{3}
  39. Translate(D, u) → D'_{3}
  40. Translate(K, u) → K'_{3}
  41. Translate(C, u) → C'_{2}
  42. Translate(E', u) → E''_{1}
  43. Translate(F', u) → F''_{1}
  44. Translate(B, u) → B'_{1}
  45. Polygon(K''_{1}, A'_{2}, F'_{3}, E'_{3}, D'_{3}, K'_{3}, C'_{2}, E''_{1}, F''_{1}, B'_{1}) → poly1', k''_{1}, b'_{1}, a'_{2}, f'_{3}, e'_{3}, d'_{3}, k'_{3}, c'_{2}, e''_{1}, f''_{1}
  46. Translate(K''_{1}, u) → M
  47. Translate(A'_{2}, u) → N
  48. Translate(F'_{3}, u) → O
  49. Translate(E'_{3}, u) → P
  50. Translate(D'_{3}, u) → Q
  51. Translate(K'_{3}, u) → R
  52. Translate(C'_{2}, u) → S
  53. Translate(E''_{1}, u) → T
  54. Translate(F''_{1}, u) → U
  55. Translate(B'_{1}, u) → V
  56. Polygon(M, N, O, P, Q, R, S, T, U, V) → poly1'', m_1, v_1, n_1, o, p, q, r, s, t, u_1
  57. Translate(M, u) → M'
  58. Translate(N, u) → N'
  59. Translate(O, u) → O'
  60. Translate(P, u) → P'
  61. Translate(Q, u) → Q'
  62. Translate(R, u) → R'
  63. Translate(S, u) → S'
  64. Translate(T, u) → T'
  65. Translate(U, u) → U'
  66. Translate(V, u) → V'
  67. Polygon(M', N', O', P', Q', R', S', T', U', V') → poly1''', m', v', n', o', p', q', r', s', t', u'
  68. Translate(M', u) → M''
  69. Translate(N', u) → N''
  70. Translate(O', u) → O''
  71. Translate(P', u) → P''
  72. Translate(Q', u) → Q''
  73. Translate(R', u) → R''
  74. Translate(S', u) → S''
  75. Translate(T', u) → T''
  76. Translate(U', u) → U''
  77. Translate(V', u) → V''
  78. Polygon(M'', N'', O'', P'', Q'', R'', S'', T'', U'', V'') → poly2, m'', v'', n'', o'', p'', q'', r'', s'', t'', u''
  79. Vector(I, W) → w
  80. Translate(K', w) → K''_{2}
  81. Translate(A, w) → A'_{3}
  82. Translate(F, w) → F'_{4}
  83. Translate(E, w) → E'_{4}
  84. Translate(D, w) → D'_{4}
  85. Translate(K, w) → K'_{4}
  86. Translate(C, w) → C'_{3}
  87. Translate(E', w) → E''_{2}
  88. Translate(F', w) → F''_{2}
  89. Translate(B, w) → B'_{2}
  90. Polygon(K''_{2}, A'_{3}, F'_{4}, E'_{4}, D'_{4}, K'_{4}, C'_{3}, E''_{2}, F''_{2}, B'_{2}) → poly1'_{1}, k''_{2}, b'_{2}, a'_{3}, f'_{4}, e'_{4}, d'_{4}, k'_{4}, c'_{3}, e''_{2}, f''_{2}
  91. Translate(K''_{1}, w) → Z
  92. Translate(A'_{2}, w) → A_1
  93. Translate(F'_{3}, w) → B_1
  94. Translate(E'_{3}, w) → C_1
  95. Translate(D'_{3}, w) → D_1
  96. Translate(K'_{3}, w) → E_1
  97. Translate(C'_{2}, w) → F_1
  98. Translate(E''_{1}, w) → G_1
  99. Translate(F''_{1}, w) → H_1
  100. Translate(B'_{1}, w) → I_1
  101. Polygon(Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1) → poly1''_{1}, z_1, p_1, a_1, g_1, h_1, d_1, e_1, i_1, j_1, l_1
  102. Translate(M, w) → M'_{1}
  103. Translate(N, w) → N'_{1}
  104. Translate(O, w) → O'_{1}
  105. Translate(P, w) → P'_{1}
  106. Translate(Q, w) → Q'_{1}
  107. Translate(R, w) → R'_{1}
  108. Translate(S, w) → S'_{1}
  109. Translate(T, w) → T'_{1}
  110. Translate(U, w) → U'_{1}
  111. Translate(V, w) → V'_{1}
  112. Polygon(M'_{1}, N'_{1}, O'_{1}, P'_{1}, Q'_{1}, R'_{1}, S'_{1}, T'_{1}, U'_{1}, V'_{1}) → poly1'''_{1}, m'_{1}, v'_{1}, n'_{1}, o'_{1}, p'_{1}, q'_{1}, r'_{1}, s'_{1}, t'_{1}, u'_{1}
  113. Translate(M', w) → M''_{1}
  114. Translate(N', w) → N''_{1}
  115. Translate(O', w) → O''_{1}
  116. Translate(P', w) → P''_{1}
  117. Translate(Q', w) → Q''_{1}
  118. Translate(R', w) → R''_{1}
  119. Translate(S', w) → S''_{1}
  120. Translate(T', w) → T''_{1}
  121. Translate(U', w) → U''_{1}
  122. Translate(V', w) → V''_{1}
  123. Polygon(M''_{1}, N''_{1}, O''_{1}, P''_{1}, Q''_{1}, R''_{1}, S''_{1}, T''_{1}, U''_{1}, V''_{1}) → poly3, m''_{1}, v''_{1}, n''_{1}, o''_{1}, p''_{1}, q''_{1}, r''_{1}, s''_{1}, t''_{1}, u''_{1}
  124. Translate(M'', w) → M'''
  125. Translate(N'', w) → N'''
  126. Translate(O'', w) → O'''
  127. Translate(P'', w) → P'''
  128. Translate(Q'', w) → Q'''
  129. Translate(R'', w) → R'''
  130. Translate(S'', w) → S'''
  131. Translate(T'', w) → T'''
  132. Translate(U'', w) → U'''
  133. Translate(V'', w) → V'''
  134. Polygon(M''', N''', O''', P''', Q''', R''', S''', T''', U''', V''') → poly2', m''', v''', n''', o''', p''', q''', r''', s''', t''', u'''
  135. Translate(K''_{2}, w) → J_1
  136. Translate(A'_{3}, w) → K_1
  137. Translate(F'_{4}, w) → L_1
  138. Translate(E'_{4}, w) → M_1
  139. Translate(D'_{4}, w) → N_1
  140. Translate(K'_{4}, w) → O_1
  141. Translate(C'_{3}, w) → P_1
  142. Translate(E''_{2}, w) → Q_1
  143. Translate(F''_{2}, w) → R_1
  144. Translate(B'_{2}, w) → S_1
  145. Polygon(J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, S_1) → poly4, q_1, j_2, r_1, s_1, t_1, f_2, o_1, g_2, h_2, i_2
  146. Translate(Z, w) → Z'
  147. Translate(A_1, w) → T_1
  148. Translate(B_1, w) → U_1
  149. Translate(C_1, w) → V_1
  150. Translate(D_1, w) → W_1
  151. Translate(E_1, w) → Z_1
  152. Translate(F_1, w) → A_2
  153. Translate(G_1, w) → B_2
  154. Translate(H_1, w) → C_2
  155. Translate(I_1, w) → D_2
  156. Polygon(Z', T_1, U_1, V_1, W_1, Z_1, A_2, B_2, C_2, D_2) → poly5, z', d_2, k_2, l_2, m_2, w_1, n_2, a_2, b_2, c_2
  157. Translate(M'_{1}, w) → E_2
  158. Translate(N'_{1}, w) → F_2
  159. Translate(O'_{1}, w) → G_2
  160. Translate(P'_{1}, w) → H_2
  161. Translate(Q'_{1}, w) → I_2
  162. Translate(R'_{1}, w) → J_2
  163. Translate(S'_{1}, w) → K_2
  164. Translate(T'_{1}, w) → L_2
  165. Translate(U'_{1}, w) → M_2
  166. Translate(V'_{1}, w) → N_2
  167. Polygon(E_2, F_2, G_2, H_2, I_2, J_2, K_2, L_2, M_2, N_2) → poly6, e_2, i_3, p_2, q_2, r_2, s_2, t_2, f_3, g_3, h_3
  168. Translate(M''_{1}, w) → O_2
  169. Translate(N''_{1}, w) → P_2
  170. Translate(O''_{1}, w) → Q_2
  171. Translate(P''_{1}, w) → R_2
  172. Translate(Q''_{1}, w) → S_2
  173. Translate(R''_{1}, w) → T_2
  174. Translate(S''_{1}, w) → U_2
  175. Translate(T''_{1}, w) → V_2
  176. Translate(U''_{1}, w) → W_2
  177. Translate(V''_{1}, w) → Z_2
  178. Polygon(O_2, P_2, Q_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2) → poly3', o_2, z_2, j_3, k_3, l_3, m_3, n_3, u_2, v_2, w_2
  179. Translate(M''', w) → A_3
  180. Translate(N''', w) → B_3
  181. Translate(O''', w) → C_3
  182. Translate(P''', w) → D_3
  183. Translate(Q''', w) → E_3
  184. Translate(R''', w) → F_3
  185. Translate(S''', w) → G_3
  186. Translate(T''', w) → H_3
  187. Translate(U''', w) → I_3
  188. Translate(V''', w) → J_3
  189. Polygon(A_3, B_3, C_3, D_3, E_3, F_3, G_3, H_3, I_3, J_3) → poly2'', a_3, t_3, b_3, c_3, d_3, e_3, p_3, q_3, r_3, s_3
  190. Translate(M'', u) → M'''_{1}
  191. Translate(N'', u) → N'''_{1}
  192. Translate(O'', u) → O'''_{1}
  193. Translate(P'', u) → P'''_{1}
  194. Translate(Q'', u) → Q'''_{1}
  195. Translate(R'', u) → R'''_{1}
  196. Translate(S'', u) → S'''_{1}
  197. Translate(T'', u) → T'''_{1}
  198. Translate(U'', u) → U'''_{1}
  199. Translate(V'', u) → V'''_{1}
  200. Polygon(M'''_{1}, N'''_{1}, O'''_{1}, P'''_{1}, Q'''_{1}, R'''_{1}, S'''_{1}, T'''_{1}, U'''_{1}, V'''_{1}) → poly2'_{1}, m'''_{1}, v'''_{1}, n'''_{1}, o'''_{1}, p'''_{1}, q'''_{1}, r'''_{1}, s'''_{1}, t'''_{1}, u'''_{1}
  201. Translate(M''', u) → K_3
  202. Translate(N''', u) → L_3
  203. Translate(O''', u) → M_3
  204. Translate(P''', u) → N_3
  205. Translate(Q''', u) → O_3
  206. Translate(R''', u) → P_3
  207. Translate(S''', u) → Q_3
  208. Translate(T''', u) → R_3
  209. Translate(U''', u) → S_3
  210. Translate(V''', u) → T_3
  211. Polygon(K_3, L_3, M_3, N_3, O_3, P_3, Q_3, R_3, S_3, T_3) → poly2''_{1}, f_4, n_4, g_4, h_4, i_4, o_3, j_4, k_4, l_4, m_4
  212. Translate(A_3, u) → U_3
  213. Translate(B_3, u) → V_3
  214. Translate(C_3, u) → W_3
  215. Translate(D_3, u) → Z_3
  216. Translate(E_3, u) → A_4
  217. Translate(F_3, u) → B_4
  218. Translate(G_3, u) → C_4
  219. Translate(H_3, u) → D_4
  220. Translate(I_3, u) → E_4
  221. Translate(J_3, u) → F_4
  222. Polygon(U_3, V_3, W_3, Z_3, A_4, B_4, C_4, D_4, E_4, F_4) → poly2''', u_3, p_4, v_3, w_3, z_3, a_4, b_4, c_4, d_4, e_4
  223. Translate(J_1, w) → G_4
  224. Translate(K_1, w) → H_4
  225. Translate(L_1, w) → I_4
  226. Translate(M_1, w) → J_4
  227. Translate(N_1, w) → K_4
  228. Translate(O_1, w) → L_4
  229. Translate(P_1, w) → M_4
  230. Translate(Q_1, w) → N_4
  231. Translate(R_1, w) → O_4
  232. Translate(S_1, w) → P_4
  233. Polygon(G_4, H_4, I_4, J_4, K_4, L_4, M_4, N_4, O_4, P_4) → poly4', q_4, j_5, r_4, s_4, t_4, f_5, g_5, h_5, i_5, o_4
  234. Translate(Z', w) → Z''
  235. Translate(T_1, w) → Q_4
  236. Translate(U_1, w) → R_4
  237. Translate(V_1, w) → S_4
  238. Translate(W_1, w) → T_4
  239. Translate(Z_1, w) → U_4
  240. Translate(A_2, w) → V_4
  241. Translate(B_2, w) → W_4
  242. Translate(C_2, w) → Z_4
  243. Translate(D_2, w) → A_5
  244. Polygon(Z'', Q_4, R_4, S_4, T_4, U_4, V_4, W_4, Z_4, A_5) → poly5', z'', a_5, k_5, l_5, m_5, n_5, u_4, v_4, w_4, z_4
  245. Translate(E_2, w) → B_5
  246. Translate(F_2, w) → C_5
  247. Translate(G_2, w) → D_5
  248. Translate(H_2, w) → E_5
  249. Translate(I_2, w) → F_5
  250. Translate(J_2, w) → G_5
  251. Translate(K_2, w) → H_5
  252. Translate(L_2, w) → I_5
  253. Translate(M_2, w) → J_5
  254. Translate(N_2, w) → K_5
  255. Polygon(B_5, C_5, D_5, E_5, F_5, G_5, H_5, I_5, J_5, K_5) → poly6', b_5, f_6, c_5, d_5, e_5, p_5, q_5, r_5, s_5, t_5
  256. Translate(O_2, w) → L_5
  257. Translate(P_2, w) → M_5
  258. Translate(Q_2, w) → N_5
  259. Translate(R_2, w) → O_5
  260. Translate(S_2, w) → P_5
  261. Translate(T_2, w) → Q_5
  262. Translate(U_2, w) → R_5
  263. Translate(V_2, w) → S_5
  264. Translate(W_2, w) → T_5
  265. Translate(Z_2, w) → U_5
  266. Polygon(L_5, M_5, N_5, O_5, P_5, Q_5, R_5, S_5, T_5, U_5) → poly3'', g_6, u_5, h_6, i_6, o_5, j_6, k_6, l_6, m_6, n_6
  267. Translate(A_3, w) → V_5
  268. Translate(B_3, w) → W_5
  269. Translate(C_3, w) → Z_5
  270. Translate(D_3, w) → A_6
  271. Translate(E_3, w) → B_6
  272. Translate(F_3, w) → C_6
  273. Translate(G_3, w) → D_6
  274. Translate(H_3, w) → E_6
  275. Translate(I_3, w) → F_6
  276. Translate(J_3, w) → G_6
  277. Polygon(V_5, W_5, Z_5, A_6, B_6, C_6, D_6, E_6, F_6, G_6) → poly2'''_{1}, v_5, q_6, w_5, z_5, a_6, b_6, c_6, d_6, e_6, p_6
  278. Translate(U_3, w) → H_6
  279. Translate(V_3, w) → I_6
  280. Translate(W_3, w) → J_6
  281. Translate(Z_3, w) → K_6
  282. Translate(A_4, w) → L_6
  283. Translate(B_4, w) → M_6
  284. Translate(C_4, w) → N_6
  285. Translate(D_4, w) → O_6
  286. Translate(E_4, w) → P_6
  287. Translate(F_4, w) → Q_6
  288. Polygon(H_6, I_6, J_6, K_6, L_6, M_6, N_6, O_6, P_6, Q_6) → poly7, r_6, k_7, s_6, t_6, f_7, g_7, h_7, i_7, o_6, j_7
  289. Translate(M'''_{1}, u) → R_6
  290. Translate(N'''_{1}, u) → S_6
  291. Translate(O'''_{1}, u) → T_6
  292. Translate(P'''_{1}, u) → U_6
  293. Translate(Q'''_{1}, u) → V_6
  294. Translate(R'''_{1}, u) → W_6
  295. Translate(S'''_{1}, u) → Z_6
  296. Translate(T'''_{1}, u) → A_7
  297. Translate(U'''_{1}, u) → B_7
  298. Translate(V'''_{1}, u) → C_7
  299. Polygon(R_6, S_6, T_6, U_6, V_6, W_6, Z_6, A_7, B_7, C_7) → poly8, l_7, c_7, m_7, n_7, u_6, v_6, w_6, z_6, a_7, b_7
  300. Translate(K_3, u) → D_7
  301. Translate(L_3, u) → E_7
  302. Translate(M_3, u) → F_7
  303. Translate(N_3, u) → G_7
  304. Translate(O_3, u) → H_7
  305. Translate(P_3, u) → I_7
  306. Translate(Q_3, u) → J_7
  307. Translate(R_3, u) → K_7
  308. Translate(S_3, u) → L_7
  309. Translate(T_3, u) → M_7
  310. Polygon(D_7, E_7, F_7, G_7, H_7, I_7, J_7, K_7, L_7, M_7) → poly9, d_7, h_8, e_7, p_7, q_7, r_7, s_7, t_7, f_8, g_8
  311. Translate(U_3, u) → N_7
  312. Translate(V_3, u) → O_7
  313. Translate(W_3, u) → P_7
  314. Translate(Z_3, u) → Q_7
  315. Translate(A_4, u) → R_7
  316. Translate(B_4, u) → S_7
  317. Translate(C_4, u) → T_7
  318. Translate(D_4, u) → U_7
  319. Translate(E_4, u) → V_7
  320. Translate(F_4, u) → W_7
  321. Polygon(N_7, O_7, P_7, Q_7, R_7, S_7, T_7, U_7, V_7, W_7) → poly10, i_8, w_7, o_7, j_8, k_8, l_8, m_8, n_8, u_7, v_7
  322. Translate(H_6, u) → Z_7
  323. Translate(I_6, u) → A_8
  324. Translate(J_6, u) → B_8
  325. Translate(K_6, u) → C_8
  326. Translate(L_6, u) → D_8
  327. Translate(M_6, u) → E_8
  328. Translate(N_6, u) → F_8
  329. Translate(O_6, u) → G_8
  330. Translate(P_6, u) → H_8
  331. Translate(Q_6, u) → I_8
  332. Polygon(Z_7, A_8, B_8, C_8, D_8, E_8, F_8, G_8, H_8, I_8) → poly7', z_7, s_8, a_8, b_8, c_8, d_8, e_8, p_8, q_8, r_8

AI Drawing Prompt Reference

Draw an irregular polygon using a set of given vertices, and construct the key sides as segments. Determine a translation direction from a pair of points to obtain a displacement vector. Translate the relevant vertices by this vector, then connect the translated vertices to form an adjacent polygon that shares matching edge structure with the original. To build an irregular tessellation, repeatedly translate the same key points along the same vector to create additional polygon copies in new positions. Keep the resulting combined arrangement so the tiling region appears continuous and the overall outline is clear.
The prompt above was compiled from the original drawing commands and is for reference only.