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Translation Tessellation Step 11

Translation Tessellation Step 11
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2014-03-28 10:14
PolygonVectorTransformationGeometric Construction
In collections Translation Tesselation
This example demonstrates constructing a tessellation pattern through translation operations, specifically showing how vector transformations generate repeated geometric arrangements. Core math concepts include vectors, translation transformations, polygon construction, and coordinate geometry. Key GeoGebra commands and objects used, as derived from the construction summary, are Polygon, Segment, Vector, Translate, Intersect, Point, and Circle. Suitable for high school students or geometry enthusiasts studying transformation geometry and tessellation design.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Segment(A, E) → e
  3. Segment(E, F) → f
  4. Segment(F, G) → g
  5. Segment(G, H) → h
  6. Segment(J, B) → k
  7. Segment(H, J) → i
  8. Vector(A, D) → u
  9. Translate(A, u) → A'
  10. Translate(E, u) → E'
  11. Segment(A', E') → e'
  12. Translate(E, u) → E'_1
  13. Translate(F, u) → F'
  14. Segment(E'_1, F') → f'
  15. Translate(F, u) → F'_1
  16. Translate(G, u) → G'
  17. Segment(F'_1, G') → g'
  18. Translate(G, u) → G'_1
  19. Translate(H, u) → H'
  20. Segment(G'_1, H') → h'
  21. Translate(J, u) → J'
  22. Translate(B, u) → B'
  23. Segment(J', B') → k'
  24. Translate(H, u) → H'_1
  25. Translate(J, u) → J'_1
  26. Segment(H'_1, J'_1) → i'
  27. Segment(D, I) → j
  28. Segment(I, K) → l
  29. Segment(K, L) → m
  30. Segment(L, M) → n
  31. Segment(M, N) → p
  32. Segment(N, A) → q
  33. Vector(A, B) → v
  34. Translate(A, v) → A'_1
  35. Translate(E, v) → E'_2
  36. Segment(A'_1, E'_2) → e'_1
  37. Translate(A', v) → A''
  38. Translate(E', v) → E''
  39. Segment(A'', E'') → e''
  40. Translate(D, v) → D'
  41. Translate(I, v) → I'
  42. Segment(D', I') → j'
  43. Translate(I, v) → I'_1
  44. Translate(K, v) → K'
  45. Segment(I'_1, K') → l'
  46. Translate(K, v) → K'_1
  47. Translate(L, v) → L'
  48. Segment(K'_1, L') → m'
  49. Translate(L, v) → L'_1
  50. Translate(M, v) → M'
  51. Segment(L'_1, M') → n'
  52. Translate(M, v) → M'_1
  53. Translate(N, v) → N'
  54. Segment(M'_1, N') → p'
  55. Translate(N, v) → N'_1
  56. Translate(A, v) → A'_2
  57. Segment(N'_1, A'_2) → q'
  58. Intersect(f, g) → O
  59. Intersect(g, h) → P
  60. Point(i) → Q
  61. Intersect(k, i) → R
  62. Intersect(p', q') → S
  63. Intersect(n', p') → T
  64. Intersect(m', n') → U
  65. Intersect(l', m') → V
  66. Intersect(j', l') → W
  67. Intersect(k', i') → Z
  68. Intersect(h', i') → A_1
  69. Intersect(g', h') → B_1
  70. Intersect(f', g') → C_1
  71. Point(l) → D_1
  72. Intersect(l, m) → E_1
  73. Intersect(m, n) → F_1
  74. Intersect(n, p) → G_1
  75. Intersect(p, q) → H_1
  76. Polygon(A, O, W, C, Z, A_1, B_1, C_1, D, D_1, E_1, F_1, P, G_1, H_1, Q, R, B, S, T, U, V) → poly2, a_1, v_1, w, c_1, z, a_2, b_2, c_2, d_1, d_2, e_1, o, f_1, g_1, h_1, p_1, q_1, r, b_1, s, t, u_1
  77. Circle(I_1, J_1) → k_1
  78. Intersect(n', p') → K_1
  79. Intersect(m', n') → L_1
  80. Point(m') → M_1
  81. Point(n') → N_1
  82. Point(n') → P_1
  83. Point(n') → R_1
  84. Polygon(K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, S_1) → poly3, k_2, l_1, m_1, n_1, o_1, p_2, q_2, r_1, s_1
  85. Intersect(n, p) → T_1
  86. Intersect(m, n) → U_1
  87. Polygon(T_1, U_1, V_1, W_1, Z_1, A_2, B_2) → poly4, t_1, u_2, v_2, w_1, z_1, a_3, b_3
  88. Translate(A, v) → A'_3
  89. Translate(O, v) → O'
  90. Translate(P, v) → P'
  91. Translate(Q, v) → Q'
  92. Translate(R, v) → R'
  93. Translate(B, v) → B'_1
  94. Translate(S, v) → S'
  95. Translate(T, v) → T'
  96. Translate(U, v) → U'
  97. Translate(V, v) → V'
  98. Translate(W, v) → W'
  99. Translate(C, v) → C'
  100. Translate(Z, v) → Z'
  101. Translate(A_1, v) → C_2
  102. Translate(B_1, v) → D_2
  103. Translate(C_1, v) → E_2
  104. Translate(D, v) → D'_1
  105. Translate(D_1, v) → F_2
  106. Translate(E_1, v) → G_2
  107. Translate(F_1, v) → H_2
  108. Translate(G_1, v) → I_2
  109. Translate(H_1, v) → J_2
  110. Polygon(A'_3, O', W', C', Z', C_2, D_2, E_2, D'_1, F_2, G_2, H_2, P', I_2, J_2, Q', R', B'_1, S', T', U', V') → poly2', a'_3, v', w', c', z', c_3, d_3, e_2, d'_1, f_2, g_2, o', h_2, i_2, j_2, p'_1, q'_1, r', b'_1, s', t', u'
  111. Translate(K_1, v) → K_2
  112. Translate(L_1, v) → L_2
  113. Translate(M_1, v) → M_2
  114. Translate(N_1, v) → N_2
  115. Translate(O_1, v) → O_2
  116. Translate(P_1, v) → P_2
  117. Translate(Q_1, v) → Q_2
  118. Translate(R_1, v) → R_2
  119. Translate(S_1, v) → S_2
  120. Polygon(K_2, L_2, M_2, N_2, O_2, P_2, Q_2, R_2, S_2) → poly3', k_3, l_2, m_2, n_2, o_2, p_3, q_3, r_2, s_2
  121. Translate(T_1, v) → T_2
  122. Translate(U_1, v) → U_2
  123. Translate(V_1, v) → V_2
  124. Translate(W_1, v) → W_2
  125. Translate(Z_1, v) → Z_2
  126. Translate(A_2, v) → A_3
  127. Translate(B_2, v) → B_3
  128. Polygon(T_2, U_2, V_2, W_2, Z_2, A_3, B_3) → poly4', t_2, u_3, v_3, w_2, z_2, a_4, b_4
  129. Translate(k_1, v) → e_3
  130. Translate(A, v) → A'_4
  131. Translate(E, v) → E'_3
  132. Segment(A'_4, E'_3) → e'_2
  133. Translate(E, v) → E'_4
  134. Translate(F, v) → F'_2
  135. Segment(E'_4, F'_2) → f'_1
  136. Translate(F, v) → F'_3
  137. Translate(G, v) → G'_2
  138. Segment(F'_3, G'_2) → g'_1
  139. Translate(G, v) → G'_3
  140. Translate(H, v) → H'_2
  141. Segment(G'_3, H'_2) → h'_1
  142. Translate(J, v) → J'_2
  143. Translate(B, v) → B'_2
  144. Segment(J'_2, B'_2) → k'_1
  145. Translate(H, v) → H'_3
  146. Translate(J, v) → J'_3
  147. Segment(H'_3, J'_3) → i'_1
  148. Translate(A', v) → A''_1
  149. Translate(E', v) → E''_1
  150. Segment(A''_1, E''_1) → e''_1
  151. Translate(E'_1, v) → C_3
  152. Translate(F', v) → F''
  153. Segment(C_3, F'') → f''
  154. Translate(F'_1, v) → D_3
  155. Translate(G', v) → G''
  156. Segment(D_3, G'') → g''
  157. Translate(G'_1, v) → E_3
  158. Translate(H', v) → H''
  159. Segment(E_3, H'') → h''
  160. Translate(J', v) → J''
  161. Translate(B', v) → B''
  162. Segment(J'', B'') → k''
  163. Translate(H'_1, v) → F_3
  164. Translate(J'_1, v) → G_3
  165. Segment(F_3, G_3) → i''
  166. Translate(D, v) → D'_2
  167. Translate(I, v) → I'_2
  168. Segment(D'_2, I'_2) → j'_1
  169. Translate(I, v) → I'_3
  170. Translate(K, v) → K'_2
  171. Segment(I'_3, K'_2) → l'_1
  172. Translate(K, v) → K'_3
  173. Translate(L, v) → L'_2
  174. Segment(K'_3, L'_2) → m'_1
  175. Translate(L, v) → L'_3
  176. Translate(M, v) → M'_2
  177. Segment(L'_3, M'_2) → n'_1
  178. Translate(M, v) → M'_3
  179. Translate(N, v) → N'_2
  180. Segment(M'_3, N'_2) → p'_2
  181. Translate(N, v) → N'_3
  182. Translate(A, v) → A'_5
  183. Segment(N'_3, A'_5) → q'_2
  184. Translate(D', v) → D''
  185. Translate(I', v) → I''
  186. Segment(D'', I'') → j''
  187. Translate(I'_1, v) → H_3
  188. Translate(K', v) → K''
  189. Segment(H_3, K'') → l''
  190. Translate(K'_1, v) → I_3
  191. Translate(L', v) → L''
  192. Segment(I_3, L'') → m''
  193. Translate(L'_1, v) → J_3
  194. Translate(M', v) → M''
  195. Segment(J_3, M'') → n''
  196. Translate(M'_1, v) → K_3
  197. Translate(N', v) → N''
  198. Segment(K_3, N'') → p''
  199. Translate(N'_1, v) → L_3
  200. Translate(A'_2, v) → M_3
  201. Segment(L_3, M_3) → q''
  202. Translate(I_1, v) → N_3
  203. Translate(A, u) → A'_6
  204. Translate(O, u) → O'_1
  205. Translate(P, u) → P'_1
  206. Translate(Q, u) → Q'_1
  207. Translate(R, u) → R'_1
  208. Translate(B, u) → B'_3
  209. Translate(S, u) → S'_1
  210. Translate(T, u) → T'_1
  211. Translate(U, u) → U'_1
  212. Translate(V, u) → V'_1
  213. Translate(W, u) → W'_1
  214. Translate(C, u) → C'_1
  215. Translate(Z, u) → Z'_1
  216. Translate(A_1, u) → O_3
  217. Translate(B_1, u) → P_3
  218. Translate(C_1, u) → Q_3
  219. Translate(D, u) → D'_3
  220. Translate(D_1, u) → R_3
  221. Translate(E_1, u) → S_3
  222. Translate(F_1, u) → T_3
  223. Translate(G_1, u) → U_3
  224. Translate(H_1, u) → V_3
  225. Polygon(A'_6, O'_1, W'_1, C'_1, Z'_1, O_3, P_3, Q_3, D'_3, R_3, S_3, T_3, P'_1, U_3, V_3, Q'_1, R'_1, B'_3, S'_1, T'_1, U'_1, V'_1) → poly2'_1, a'_6, v'_1, w'_1, c'_1, z'_1, o_3, p_4, q_4, d'_3, r_3, s_3, o'_1, t_3, u_4, v_4, p'_3, q'_3, r'_1, b'_3, s'_1, t'_1, u'_1
  226. Translate(K_1, u) → W_3
  227. Translate(L_1, u) → Z_3
  228. Translate(M_1, u) → A_4
  229. Translate(N_1, u) → B_4
  230. Translate(O_1, u) → C_4
  231. Translate(P_1, u) → D_4
  232. Translate(Q_1, u) → E_4
  233. Translate(R_1, u) → F_4
  234. Translate(S_1, u) → G_4
  235. Polygon(W_3, Z_3, A_4, B_4, C_4, D_4, E_4, F_4, G_4) → poly3'_1, w_3, z_3, a_5, b_5, c_4, d_4, e_4, f_4, g_4
  236. Translate(T_1, u) → H_4
  237. Translate(U_1, u) → I_4
  238. Translate(V_1, u) → J_4
  239. Translate(W_1, u) → K_4
  240. Translate(Z_1, u) → L_4
  241. Translate(A_2, u) → M_4
  242. Translate(B_2, u) → N_4
  243. Polygon(H_4, I_4, J_4, K_4, L_4, M_4, N_4) → poly4'_1, h_4, i_4, j_4, k_4, l_4, m_4, n_4
  244. Translate(A'_3, u) → O_4
  245. Translate(O', u) → O''
  246. Translate(P', u) → P''
  247. Translate(Q', u) → Q''
  248. Translate(R', u) → R''
  249. Translate(B'_1, u) → P_4
  250. Translate(S', u) → S''
  251. Translate(T', u) → T''
  252. Translate(U', u) → U''
  253. Translate(V', u) → V''
  254. Translate(W', u) → W''
  255. Translate(C', u) → C''
  256. Translate(Z', u) → Z''
  257. Translate(C_2, u) → Q_4
  258. Translate(D_2, u) → R_4
  259. Translate(E_2, u) → S_4
  260. Translate(D'_1, u) → T_4
  261. Translate(F_2, u) → U_4
  262. Translate(G_2, u) → V_4
  263. Translate(H_2, u) → W_4
  264. Translate(I_2, u) → Z_4
  265. Translate(J_2, u) → A_5
  266. Polygon(O_4, O'', W'', C'', Z'', Q_4, R_4, S_4, T_4, U_4, V_4, W_4, P'', Z_4, A_5, Q'', R'', P_4, S'', T'', U'', V'') → poly2'', o_4, v'', w'', c'', z'', q_5, r_4, s_4, t_4, u_5, v_5, o'', w_4, z_4, a_6, p''_1, q''_1, r'', p_5, s'', t'', u''
  267. Translate(K_2, u) → B_5
  268. Translate(L_2, u) → C_5
  269. Translate(M_2, u) → D_5
  270. Translate(N_2, u) → E_5
  271. Translate(O_2, u) → F_5
  272. Translate(P_2, u) → G_5
  273. Translate(Q_2, u) → H_5
  274. Translate(R_2, u) → I_5
  275. Translate(S_2, u) → J_5
  276. Polygon(B_5, C_5, D_5, E_5, F_5, G_5, H_5, I_5, J_5) → poly3'', b_6, c_5, d_5, e_5, f_5, g_5, h_5, i_5, j_5
  277. Translate(T_2, u) → K_5
  278. Translate(U_2, u) → L_5
  279. Translate(V_2, u) → M_5
  280. Translate(W_2, u) → N_5
  281. Translate(Z_2, u) → O_5
  282. Translate(A_3, u) → P_5
  283. Translate(B_3, u) → Q_5
  284. Polygon(K_5, L_5, M_5, N_5, O_5, P_5, Q_5) → poly4'', k_5, l_5, m_5, n_5, o_5, p_6, q_6
  285. Translate(k_1, u) → f_3
  286. Translate(e_3, u) → g_3
  287. Translate(A, u) → A'_7
  288. Translate(E, u) → E'_5
  289. Segment(A'_7, E'_5) → e'_3
  290. Translate(E, u) → E'_6
  291. Translate(F, u) → F'_4
  292. Segment(E'_6, F'_4) → f'_2
  293. Translate(F, u) → F'_5
  294. Translate(G, u) → G'_4
  295. Segment(F'_5, G'_4) → g'_2
  296. Translate(G, u) → G'_5
  297. Translate(H, u) → H'_4
  298. Segment(G'_5, H'_4) → h'_2
  299. Translate(J, u) → J'_4
  300. Translate(B, u) → B'_4
  301. Segment(J'_4, B'_4) → k'_2
  302. Translate(H, u) → H'_5
  303. Translate(J, u) → J'_5
  304. Segment(H'_5, J'_5) → i'_2
  305. Translate(A', u) → A''_2
  306. Translate(E', u) → E''_2
  307. Segment(A''_2, E''_2) → e''_2
  308. Translate(E'_1, u) → R_5
  309. Translate(F', u) → F''_1
  310. Segment(R_5, F''_1) → f''_1
  311. Translate(F'_1, u) → S_5
  312. Translate(G', u) → G''_1
  313. Segment(S_5, G''_1) → g''_1
  314. Translate(G'_1, u) → T_5
  315. Translate(H', u) → H''_1
  316. Segment(T_5, H''_1) → h''_1
  317. Translate(J', u) → J''_1
  318. Translate(B', u) → B''_1
  319. Segment(J''_1, B''_1) → k''_1
  320. Translate(H'_1, u) → U_5
  321. Translate(J'_1, u) → V_5
  322. Segment(U_5, V_5) → i''_1
  323. Translate(D, u) → D'_4
  324. Translate(I, u) → I'_4
  325. Segment(D'_4, I'_4) → j'_2
  326. Translate(I, u) → I'_5
  327. Translate(K, u) → K'_4
  328. Segment(I'_5, K'_4) → l'_2
  329. Translate(K, u) → K'_5
  330. Translate(L, u) → L'_4
  331. Segment(K'_5, L'_4) → m'_2
  332. Translate(L, u) → L'_5
  333. Translate(M, u) → M'_4
  334. Segment(L'_5, M'_4) → n'_2
  335. Translate(M, u) → M'_5
  336. Translate(N, u) → N'_4
  337. Segment(M'_5, N'_4) → p'_4
  338. Translate(N, u) → N'_5
  339. Translate(A, u) → A'_8
  340. Segment(N'_5, A'_8) → q'_4
  341. Translate(D', u) → D''_1
  342. Translate(I', u) → I''_1
  343. Segment(D''_1, I''_1) → j''_1
  344. Translate(I'_1, u) → W_5
  345. Translate(K', u) → K''_1
  346. Segment(W_5, K''_1) → l''_1
  347. Translate(K'_1, u) → Z_5
  348. Translate(L', u) → L''_1
  349. Segment(Z_5, L''_1) → m''_1
  350. Translate(L'_1, u) → A_6
  351. Translate(M', u) → M''_1
  352. Segment(A_6, M''_1) → n''_1
  353. Translate(M'_1, u) → B_6
  354. Translate(N', u) → N''_1
  355. Segment(B_6, N''_1) → p''_2
  356. Translate(N'_1, u) → C_6
  357. Translate(A'_2, u) → D_6
  358. Segment(C_6, D_6) → q''_2
  359. Translate(A'_4, u) → E_6
  360. Translate(E'_3, u) → F_6
  361. Segment(E_6, F_6) → i_1
  362. Translate(E'_4, u) → G_6
  363. Translate(F'_2, u) → H_6
  364. Segment(G_6, H_6) → j_1
  365. Translate(F'_3, u) → I_6
  366. Translate(G'_2, u) → J_6
  367. Segment(I_6, J_6) → h_3
  368. Translate(G'_3, u) → K_6
  369. Translate(H'_2, u) → L_6
  370. Segment(K_6, L_6) → i_3
  371. Translate(J'_2, u) → M_6
  372. Translate(B'_2, u) → N_6
  373. Segment(M_6, N_6) → j_3
  374. Translate(H'_3, u) → O_6
  375. Translate(J'_3, u) → P_6
  376. Segment(O_6, P_6) → l_3
  377. Translate(A''_1, u) → Q_6
  378. Translate(E''_1, u) → R_6
  379. Segment(Q_6, R_6) → m_3
  380. Translate(C_3, u) → S_6
  381. Translate(F'', u) → F'''
  382. Segment(S_6, F''') → f'''
  383. Translate(D_3, u) → T_6
  384. Translate(G'', u) → G'''
  385. Segment(T_6, G''') → g'''
  386. Translate(E_3, u) → U_6
  387. Translate(H'', u) → H'''
  388. Segment(U_6, H''') → h'''
  389. Translate(J'', u) → J'''
  390. Translate(B'', u) → B'''
  391. Segment(J''', B''') → k'''
  392. Translate(F_3, u) → V_6
  393. Translate(G_3, u) → W_6
  394. Segment(V_6, W_6) → i'''
  395. Translate(D'_2, u) → Z_6
  396. Translate(I'_2, u) → A_7
  397. Segment(Z_6, A_7) → n_3
  398. Translate(I'_3, u) → B_7
  399. Translate(K'_2, u) → C_7
  400. Segment(B_7, C_7) → r_5
  401. Translate(K'_3, u) → D_7
  402. Translate(L'_2, u) → E_7
  403. Segment(D_7, E_7) → s_5
  404. Translate(L'_3, u) → F_7
  405. Translate(M'_2, u) → G_7
  406. Segment(F_7, G_7) → t_5
  407. Translate(M'_3, u) → H_7
  408. Translate(N'_2, u) → I_7
  409. Segment(H_7, I_7) → c_6
  410. Translate(N'_3, u) → J_7
  411. Translate(A'_5, u) → K_7
  412. Segment(J_7, K_7) → d_6
  413. Translate(D'', u) → D'''
  414. Translate(I'', u) → I'''
  415. Segment(D''', I''') → j'''
  416. Translate(H_3, u) → L_7
  417. Translate(K'', u) → K'''
  418. Segment(L_7, K''') → l'''
  419. Translate(I_3, u) → M_7
  420. Translate(L'', u) → L'''
  421. Segment(M_7, L''') → m'''
  422. Translate(J_3, u) → N_7
  423. Translate(M'', u) → M'''
  424. Segment(N_7, M''') → n'''
  425. Translate(K_3, u) → O_7
  426. Translate(N'', u) → N'''
  427. Segment(O_7, N''') → p'''
  428. Translate(L_3, u) → P_7
  429. Translate(M_3, u) → Q_7
  430. Segment(P_7, Q_7) → q'''

AI Drawing Prompt Reference

Start with a square as the base shape. Inside it, construct a polyline path connecting the square's vertices to interior points. Then, using one side of the square as a translation vector, copy the entire path to create a first translated set. Next, using another side of the square as a second vector, translate again to form a second set. Finally, extend a separate polyline outside the square and translate it along both vectors, completing a periodic tessellation pattern. All translated segments remain parallel and equal in length to the originals, creating a repeating geometric arrangement.
The prompt above was compiled from the original drawing commands and is for reference only.