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Pythagorean Tiling

Pythagorean Tiling
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2015-01-11 04:20
PolygonTransformationGeometric Construction
This example demonstrates the Pythagorean tiling, a pattern composed of squares and isosceles right triangles. It involves key mathematical concepts such as squares, isosceles right triangles, and rotation. Using GeoGebra commands like Line, OrthogonalLine, Point, Segment, and Rotate, this beautiful tiling pattern can be constructed. Suitable for middle school and above students to learn geometric construction and transformations.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. OrthogonalLine(B, a) → b
  3. OrthogonalLine(A, a) → c
  4. Point(c) → C
  5. OrthogonalLine(C, c) → d
  6. Point(c) → D
  7. OrthogonalLine(D, c) → e
  8. Point(b) → E
  9. OrthogonalLine(E, b) → f
  10. Point(b) → F
  11. OrthogonalLine(F, b) → g
  12. Point(b) → G
  13. OrthogonalLine(G, b) → h
  14. Point(b) → H
  15. OrthogonalLine(H, b) → i
  16. Point(i) → I
  17. OrthogonalLine(I, i) → j
  18. Point(h) → J
  19. OrthogonalLine(J, h) → k
  20. Point(g) → K
  21. OrthogonalLine(K, g) → l
  22. Point(i) → L
  23. OrthogonalLine(L, i) → m
  24. Point(h) → M
  25. OrthogonalLine(M, h) → n
  26. Point(g) → N
  27. OrthogonalLine(N, g) → p
  28. Point(c) → O
  29. Segment(O, P) → q
  30. Rotate(O, -45*°, P) → O'
  31. Angle(O', P, O) → α
  32. Segment(P, O') → r
  33. Point(b) → Q
  34. Segment(Q, R) → s
  35. Segment(R, O') → t
  36. Segment(O, S) → a_1
  37. Point(d) → T
  38. Segment(S, T) → b_1
  39. Segment(T, U) → c_1
  40. Segment(U, Q) → d_1
  41. Segment(V, T) → e_1
  42. Point(b) → W
  43. Segment(V, W) → f_1
  44. Point(c) → A_1
  45. Segment(Z, A_1) → g_1
  46. Point(e) → B_1
  47. Segment(Z, B_1) → h_1
  48. Segment(C_1, T) → i_1
  49. Segment(C_1, A_1) → j_1
  50. Segment(D_1, W) → k_1
  51. Segment(B_1, D_1) → l_1
  52. Point(j) → F_1
  53. Segment(E_1, F_1) → m_1
  54. Point(d) → G_1
  55. Segment(G_1, E_1) → n_1
  56. Segment(H_1, O) → p_1
  57. Point(a) → I_1
  58. Segment(H_1, I_1) → q_1
  59. Segment(J_1, O) → r_1
  60. Segment(J_1, G_1) → s_1
  61. Segment(K_1, I_1) → t_1
  62. Segment(K_1, F_1) → a_2
  63. Point(j) → M_1
  64. Segment(L_1, M_1) → b_2
  65. Point(e) → N_1
  66. Segment(N_1, L_1) → c_2
  67. Segment(O_1, G_1) → d_2
  68. Segment(O_1, M_1) → e_2
  69. Segment(P_1, A_1) → f_2
  70. Segment(P_1, G_1) → g_2
  71. Segment(Q_1, A_1) → h_2
  72. Segment(Q_1, N_1) → i_2
  73. Point(c) → S_1
  74. Segment(R_1, S_1) → j_2
  75. Point(f) → T_1
  76. Segment(T_1, R_1) → k_2
  77. Segment(U_1, B_1) → l_2
  78. Segment(U_1, S_1) → m_2
  79. Segment(V_1, B_1) → n_2
  80. Point(b) → W_1
  81. Segment(V_1, W_1) → p_2
  82. Segment(Z_1, W_1) → q_2
  83. Segment(T_1, Z_1) → r_2
  84. Point(c) → B_2
  85. Segment(A_2, B_2) → s_2
  86. Point(g) → C_2
  87. Segment(C_2, A_2) → t_2
  88. Segment(D_2, T_1) → a_3
  89. Segment(B_2, D_2) → b_3
  90. Segment(E_2, T_1) → c_3
  91. Point(b) → F_2
  92. Segment(E_2, F_2) → d_3
  93. Segment(G_2, F_2) → e_3
  94. Segment(C_2, G_2) → f_3
  95. Point(c) → I_2
  96. Segment(H_2, I_2) → g_3
  97. Segment(I_2, J_2) → h_3
  98. Segment(J_2, C_2) → i_3
  99. Point(h) → K_2
  100. Segment(H_2, K_2) → j_3
  101. Segment(K_2, L_2) → k_3
  102. Point(b) → M_2
  103. Segment(L_2, M_2) → l_3
  104. Segment(M_2, N_2) → m_3
  105. Segment(N_2, C_2) → n_3
  106. Point(c) → P_2
  107. Segment(O_2, P_2) → p_3
  108. Point(i) → Q_2
  109. Segment(Q_2, O_2) → q_3
  110. Segment(R_2, K_2) → r_3
  111. Segment(R_2, P_2) → s_3
  112. Point(b) → T_2
  113. Segment(S_2, T_2) → t_3
  114. Segment(S_2, K_2) → a_4
  115. Segment(U_2, T_2) → b_4
  116. Segment(Q_2, U_2) → c_4
  117. Point(j) → W_2
  118. Segment(V_2, W_2) → d_4
  119. Segment(W_2, Z_2) → e_4
  120. Segment(Z_2, N_1) → f_4
  121. Point(f) → A_3
  122. Segment(A_3, V_2) → g_4
  123. Segment(A_3, B_3) → h_4
  124. Segment(B_3, S_1) → i_4
  125. Segment(S_1, C_3) → j_4
  126. Segment(C_3, N_1) → k_4
  127. Point(g) → E_3
  128. Segment(D_3, E_3) → l_4
  129. Segment(E_3, F_3) → m_4
  130. Segment(F_3, B_2) → n_4
  131. Segment(B_2, G_3) → p_4
  132. Segment(G_3, A_3) → q_4
  133. Point(j) → H_3
  134. Segment(D_3, H_3) → r_4
  135. Segment(H_3, I_3) → s_4
  136. Segment(I_3, A_3) → t_4
  137. Point(j) → K_3
  138. Segment(J_3, K_3) → a_5
  139. Segment(K_3, L_3) → b_5
  140. Segment(L_3, E_3) → c_5
  141. Segment(E_3, M_3) → d_5
  142. Segment(M_3, I_2) → e_5
  143. Point(h) → N_3
  144. Segment(N_3, J_3) → f_5
  145. Segment(N_3, O_3) → g_5
  146. Segment(O_3, I_2) → h_5
  147. Point(j) → Q_3
  148. Segment(P_3, Q_3) → i_5
  149. Segment(Q_3, R_3) → j_5
  150. Segment(R_3, N_3) → k_5
  151. Segment(N_3, S_3) → l_5
  152. Segment(S_3, P_2) → m_5
  153. Segment(P_2, T_3) → n_5
  154. Point(i) → U_3
  155. Segment(T_3, U_3) → p_5
  156. Segment(U_3, P_3) → q_5
  157. Point(k) → W_3
  158. Segment(V_3, W_3) → r_5
  159. Segment(W_3, Z_3) → s_5
  160. Point(a) → A_4
  161. Segment(Z_3, A_4) → t_5
  162. Segment(A_4, B_4) → a_6
  163. Segment(B_4, F_1) → b_6
  164. Segment(F_1, C_4) → c_6
  165. Point(d) → D_4
  166. Segment(C_4, D_4) → d_6
  167. Segment(D_4, V_3) → e_6
  168. Point(k) → F_4
  169. Segment(E_4, F_4) → f_6
  170. Point(e) → G_4
  171. Segment(G_4, E_4) → g_6
  172. Segment(G_4, H_4) → h_6
  173. Segment(H_4, M_1) → i_6
  174. Segment(M_1, I_4) → j_6
  175. Segment(I_4, D_4) → k_6
  176. Segment(D_4, J_4) → l_6
  177. Segment(J_4, F_4) → m_6
  178. Point(k) → L_4
  179. Segment(K_4, L_4) → n_6
  180. Segment(L_4, M_4) → p_6
  181. Segment(M_4, G_4) → q_6
  182. Segment(G_4, N_4) → r_6
  183. Segment(N_4, W_2) → s_6
  184. Segment(W_2, O_4) → t_6
  185. Point(f) → P_4
  186. Segment(O_4, P_4) → a_7
  187. Segment(P_4, K_4) → b_7
  188. Point(k) → R_4
  189. Segment(Q_4, R_4) → c_7
  190. Segment(R_4, S_4) → d_7
  191. Segment(S_4, P_4) → e_7
  192. Segment(P_4, T_4) → f_7
  193. Segment(T_4, H_3) → g_7
  194. Segment(H_3, U_4) → h_7
  195. Point(g) → V_4
  196. Segment(U_4, V_4) → i_7
  197. Segment(V_4, Q_4) → j_7
  198. Point(k) → Z_4
  199. Segment(W_4, Z_4) → k_7
  200. Segment(Z_4, A_5) → l_7
  201. Segment(A_5, V_4) → m_7
  202. Segment(V_4, B_5) → n_7
  203. Segment(B_5, K_3) → p_7
  204. Segment(K_3, C_5) → q_7
  205. Point(h) → D_5
  206. Segment(C_5, D_5) → r_7
  207. Segment(D_5, W_4) → s_7
  208. Point(k) → F_5
  209. Segment(E_5, F_5) → t_7
  210. Segment(F_5, G_5) → a_8
  211. Segment(G_5, D_5) → b_8
  212. Segment(D_5, H_5) → c_8
  213. Segment(H_5, Q_3) → d_8
  214. Segment(Q_3, I_5) → e_8
  215. Point(i) → J_5
  216. Segment(I_5, J_5) → f_8
  217. Segment(J_5, E_5) → g_8
  218. Point(l) → L_5
  219. Segment(K_5, L_5) → h_8
  220. Segment(L_5, M_5) → i_8
  221. Point(a) → N_5
  222. Segment(M_5, N_5) → j_8
  223. Segment(N_5, O_5) → k_8
  224. Segment(O_5, W_3) → l_8
  225. Segment(W_3, P_5) → m_8
  226. Point(d) → Q_5
  227. Segment(P_5, Q_5) → n_8
  228. Segment(Q_5, K_5) → p_8
  229. Point(l) → S_5
  230. Segment(R_5, S_5) → q_8
  231. Segment(S_5, T_5) → r_8
  232. Segment(T_5, Q_5) → s_8
  233. Segment(Q_5, U_5) → t_8
  234. Segment(U_5, F_4) → a_9
  235. Segment(F_4, V_5) → b_9
  236. Point(e) → W_5
  237. Segment(V_5, W_5) → c_9
  238. Segment(W_5, R_5) → d_9
  239. Point(l) → A_6
  240. Segment(Z_5, A_6) → e_9
  241. Segment(A_6, B_6) → f_9
  242. Segment(B_6, W_5) → g_9
  243. Point(f) → C_6
  244. Segment(Z_5, C_6) → h_9
  245. Segment(C_6, D_6) → i_9
  246. Segment(D_6, L_4) → j_9
  247. Segment(L_4, E_6) → k_9
  248. Segment(E_6, W_5) → l_9
  249. Point(l) → G_6
  250. Segment(F_6, G_6) → m_9
  251. Segment(G_6, H_6) → n_9
  252. Segment(H_6, C_6) → p_9
  253. Segment(C_6, I_6) → q_9
  254. Segment(I_6, R_4) → r_9
  255. Segment(R_4, J_6) → s_9
  256. Point(g) → K_6
  257. Segment(J_6, K_6) → t_9
  258. Segment(K_6, F_6) → a_{10}
  259. Point(l) → M_6
  260. Segment(L_6, M_6) → b_{10}
  261. Segment(M_6, N_6) → c_{10}
  262. Segment(N_6, K_6) → d_{10}
  263. Point(h) → O_6
  264. Segment(L_6, O_6) → e_{10}
  265. Segment(O_6, P_6) → f_{10}
  266. Segment(P_6, Z_4) → g_{10}
  267. Segment(Z_4, Q_6) → h_{10}
  268. Segment(Q_6, K_6) → i_{10}
  269. Point(i) → S_6
  270. Segment(R_6, S_6) → j_{10}
  271. Segment(S_6, T_6) → k_{10}
  272. Segment(T_6, F_5) → l_{10}
  273. Segment(F_5, U_6) → m_{10}
  274. Segment(U_6, O_6) → n_{10}
  275. Segment(O_6, V_6) → p_{10}
  276. Point(l) → W_6
  277. Segment(V_6, W_6) → q_{10}
  278. Segment(W_6, R_6) → r_{10}
  279. Point(m) → A_7
  280. Segment(Z_6, A_7) → s_{10}
  281. Segment(A_7, B_7) → t_{10}
  282. Point(a) → C_7
  283. Segment(B_7, C_7) → a_{11}
  284. Segment(C_7, D_7) → b_{11}
  285. Segment(D_7, L_5) → c_{11}
  286. Segment(L_5, E_7) → d_{11}
  287. Point(d) → F_7
  288. Segment(E_7, F_7) → e_{11}
  289. Segment(F_7, Z_6) → f_{11}
  290. Point(m) → H_7
  291. Segment(G_7, H_7) → g_{11}
  292. Segment(H_7, I_7) → h_{11}
  293. Segment(I_7, F_7) → i_{11}
  294. Segment(F_7, J_7) → j_{11}
  295. Segment(J_7, S_5) → k_{11}
  296. Segment(S_5, K_7) → l_{11}
  297. Point(e) → L_7
  298. Segment(K_7, L_7) → m_{11}
  299. Segment(L_7, G_7) → n_{11}
  300. Point(m) → O_7
  301. Segment(N_7, O_7) → p_{11}
  302. Segment(O_7, P_7) → q_{11}
  303. Segment(P_7, L_7) → r_{11}
  304. Segment(L_7, Q_7) → s_{11}
  305. Segment(Q_7, A_6) → t_{11}
  306. Segment(A_6, R_7) → a_{12}
  307. Point(f) → S_7
  308. Segment(R_7, S_7) → b_{12}
  309. Segment(S_7, N_7) → c_{12}
  310. Point(m) → U_7
  311. Segment(T_7, U_7) → d_{12}
  312. Segment(U_7, V_7) → e_{12}
  313. Segment(V_7, S_7) → f_{12}
  314. Segment(W_7, T_7) → g_{12}
  315. Segment(W_7, Z_7) → h_{12}
  316. Segment(Z_7, G_6) → i_{12}
  317. Segment(G_6, A_8) → j_{12}
  318. Segment(A_8, S_7) → k_{12}
  319. Point(b_9) → B_8
  320. Point(m) → D_8
  321. Segment(C_8, D_8) → l_{12}
  322. Segment(D_8, E_8) → m_{12}
  323. Segment(E_8, W_7) → n_{12}
  324. Segment(W_7, F_8) → p_{12}
  325. Segment(F_8, M_6) → q_{12}
  326. Segment(M_6, G_8) → r_{12}
  327. Point(h) → H_8
  328. Segment(G_8, H_8) → s_{12}
  329. Segment(H_8, C_8) → t_{12}
  330. Point(m) → J_8
  331. Segment(I_8, J_8) → a_{13}
  332. Segment(J_8, K_8) → b_{13}
  333. Segment(K_8, H_8) → c_{13}
  334. Segment(H_8, L_8) → d_{13}
  335. Segment(L_8, W_6) → e_{13}
  336. Segment(W_6, M_8) → f_{13}
  337. Point(i) → N_8
  338. Segment(M_8, N_8) → g_{13}
  339. Segment(N_8, I_8) → h_{13}
  340. Point(n) → P_8
  341. Segment(P_8, O_8) → i_{13}
  342. Point(d) → Q_8
  343. Segment(O_8, Q_8) → j_{13}
  344. Segment(Q_8, R_8) → k_{13}
  345. Segment(R_8, A_7) → l_{13}
  346. Segment(A_7, S_8) → m_{13}
  347. Point(a) → T_8
  348. Segment(S_8, T_8) → n_{13}
  349. Segment(T_8, U_8) → p_{13}
  350. Segment(U_8, P_8) → q_{13}
  351. Point(n) → W_8
  352. Segment(V_8, W_8) → r_{13}
  353. Segment(W_8, Z_8) → s_{13}
  354. Segment(Z_8, Q_8) → t_{13}
  355. Point(e) → A_9
  356. Segment(V_8, A_9) → a_{14}
  357. Segment(A_9, B_9) → b_{14}
  358. Segment(B_9, H_7) → c_{14}
  359. Segment(H_7, C_9) → d_{14}
  360. Segment(C_9, Q_8) → e_{14}
  361. Point(n) → E_9
  362. Segment(D_9, E_9) → f_{14}
  363. Segment(E_9, F_9) → g_{14}
  364. Segment(F_9, A_9) → h_{14}
  365. Segment(A_9, G_9) → i_{14}
  366. Segment(G_9, O_7) → j_{14}
  367. Segment(O_7, H_9) → k_{14}
  368. Point(f) → I_9
  369. Segment(H_9, I_9) → l_{14}
  370. Segment(I_9, D_9) → m_{14}
  371. Point(n) → K_9
  372. Segment(J_9, K_9) → n_{14}
  373. Segment(K_9, L_9) → p_{14}
  374. Segment(L_9, I_9) → q_{14}
  375. Segment(I_9, M_9) → r_{14}
  376. Segment(M_9, U_7) → s_{14}
  377. Segment(U_7, N_9) → t_{14}
  378. Point(g) → O_9
  379. Segment(N_9, O_9) → a_{15}
  380. Segment(O_9, J_9) → b_{15}
  381. Point(n) → Q_9
  382. Segment(P_9, Q_9) → c_{15}
  383. Segment(Q_9, R_9) → d_{15}
  384. Segment(R_9, O_9) → e_{15}
  385. Segment(O_9, S_9) → f_{15}
  386. Segment(S_9, D_8) → g_{15}
  387. Segment(D_8, T_9) → h_{15}
  388. Point(h) → U_9
  389. Segment(T_9, U_9) → i_{15}
  390. Segment(U_9, P_9) → j_{15}
  391. Segment(V_9, W_9) → k_{15}
  392. Segment(W_9, Z_9) → l_{15}
  393. Segment(Z_9, U_9) → m_{15}
  394. Segment(U_9, A_{10}) → n_{15}
  395. Segment(A_{10}, J_8) → p_{15}
  396. Segment(J_8, B_{10}) → q_{15}
  397. Point(i) → C_{10}
  398. Segment(B_{10}, C_{10}) → r_{15}
  399. Segment(V_9, C_{10}) → s_{15}
  400. Intersect(i, n) → D_{10}
  401. Intersect(a, n) → E_{10}
  402. Segment(D_{10}, E_{10}) → t_{15}
  403. Segment(D_{10}, H) → a_{16}
  404. Segment(G, M) → b_{16}
  405. Point(g) → F_{10}
  406. Segment(F_{10}, F) → c_{16}
  407. Point(f) → G_{10}
  408. Segment(E, G_{10}) → d_{16}
  409. Point(e) → H_{10}
  410. Point(e) → I_{10}
  411. Segment(H_{10}, I_{10}) → e_{16}
  412. Point(d) → J_{10}
  413. Point(d) → K_{10}
  414. Segment(J_{10}, K_{10}) → f_{16}
  415. Segment(E_{10}, B) → g_{16}
  416. Segment(B, H) → h_{16}
  417. Point(c) → L_{10}
  418. Segment(L_{10}, A) → i_{16}
  419. Point(j) → M_{10}
  420. Segment(M_{10}, I) → j_{16}
  421. Point(k) → N_{10}
  422. Point(k) → O_{10}
  423. Segment(N_{10}, O_{10}) → k_{16}
  424. Point(l) → P_{10}
  425. Point(l) → Q_{10}
  426. Segment(P_{10}, Q_{10}) → l_{16}
  427. Segment(K_{10}, J_{10}) → m_{16}
  428. Segment(I_{10}, H_{10}) → n_{16}
  429. Point(m) → R_{10}
  430. Segment(L, R_{10}) → p_{16}
  431. Intersect(n_6, b_7) → S_{10}
  432. Intersect(q_8, d_9) → T_{10}
  433. Point(l_{16}) → U_{10}
  434. Intersect(d_9, e_{16}) → V_{10}
  435. Angle(U_{10}, T_{10}, V_{10}) → β
  436. Rotate(U_{10}, 45*°, T_{10}) → W_{10}
  437. Angle(U_{10}, T_{10}, W_{10}) → γ