MathPlotAI Logo MathPlotAI

Pythagorean Tessellation #133 Tiling

Pythagorean Tessellation #133 Tiling
Loading the math board and drawing, please wait…
2015-01-11 04:50
TriangleCircleGeometric Construction
This example demonstrates the Pythagorean Theorem through tessellation, visually illustrating the relationship between the sides of a right triangle. Core mathematical concepts include the Pythagorean Theorem, circumcircle, geometric mean, tree diagrams, root calculations, and standard deviation. The construction utilizes GeoGebra commands and objects such as Line, Point, OrthogonalLine, Segment, Rotate, Angle, Intersect, and Polygon. Suitable for high school mathematics education to enhance understanding of the Pythagorean Theorem and geometric constructions.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(C, D) → b
  3. Point(b) → E
  4. Point(b) → F
  5. OrthogonalLine(E, b) → c
  6. OrthogonalLine(D, b) → d
  7. Point(c) → G
  8. Point(c) → H
  9. OrthogonalLine(H, c) → e
  10. Point(c) → I
  11. Point(c) → J
  12. Point(c) → K
  13. Point(c) → L
  14. Point(d) → M
  15. Line(J, M) → f
  16. Point(d) → N
  17. Line(K, N) → g
  18. Point(d) → O
  19. Line(L, O) → h
  20. Point(f) → P
  21. Line(C, P) → i
  22. Point(d) → Q
  23. Point(d) → R
  24. Point(d) → S
  25. Point(i) → T
  26. Line(Q, T) → j
  27. Point(i) → U
  28. Line(R, U) → k
  29. Point(i) → V
  30. Line(S, V) → l
  31. Point(j) → W
  32. Point(h) → Z
  33. Line(W, Z) → m
  34. Point(j) → A_1
  35. Point(h) → B_1
  36. Line(A_1, B_1) → n
  37. Point(e) → C_1
  38. Point(g) → D_1
  39. Point(c) → E_1
  40. Point(d) → F_1
  41. Segment(H, G_1) → p
  42. Rotate(H, 270*°, G_1) → H'
  43. Angle(H', G_1, H) → α
  44. Segment(G_1, H') → q
  45. Segment(H_1, D_1) → r
  46. Segment(O, H_1) → s
  47. Segment(I_1, N) → t
  48. Segment(I_1, C_1) → a_1
  49. Segment(J_1, C_1) → b_1
  50. Segment(J_1, E) → c_1
  51. Segment(K_1, H) → d_1
  52. Segment(K_1, L_1) → e_1
  53. Segment(L_1, C) → f_1
  54. Segment(M_1, D) → g_1
  55. Point(c) → N_1
  56. Segment(M_1, N_1) → h_1
  57. Intersect(d, e) → O_1
  58. Segment(O_1, P_1) → i_1
  59. Point(b) → Q_1
  60. Segment(P_1, Q_1) → j_1
  61. Segment(R_1, N_1) → k_1
  62. Segment(R_1, E) → l_1
  63. Segment(D, S_1) → m_1
  64. Segment(S_1, Q_1) → n_1
  65. Point(e_1) → T_1
  66. Segment(T_1, O_1) → p_1
  67. Segment(U_1, O_1) → q_1
  68. Segment(U_1, V_1) → r_1
  69. Segment(V_1, H) → s_1
  70. Segment(W_1, K) → t_1
  71. Segment(Z_1, W_1) → a_2
  72. Intersect(e, i) → A_2
  73. Segment(A_2, Z_1) → b_2
  74. Segment(B_2, O_1) → c_2
  75. Segment(C_2, B_2) → d_2
  76. Intersect(h, i) → D_2
  77. Segment(D_2, C_2) → e_2
  78. Segment(E_2, K) → f_2
  79. Segment(E_2, F_2) → g_2
  80. Segment(F_2, Q) → h_2
  81. Segment(O, G_2) → i_2
  82. Point(c) → H_2
  83. Segment(G_2, H_2) → j_2
  84. Segment(L, I_2) → k_2
  85. Point(j) → J_2
  86. Segment(I_2, J_2) → l_2
  87. Segment(L, K_2) → m_2
  88. Segment(K_2, L_2) → n_2
  89. Intersect(g, i) → M_2
  90. Segment(L_2, M_2) → p_2
  91. Intersect(b_1, h_1) → N_2
  92. Polygon(R_1, E, J_1, N_2) → poly1, r_2, e_3, j_3, n_3
  93. Intersect(h_1, j_1) → O_2
  94. Polygon(M_1, O_2, S_1, D) → poly2, m_3, o_2, s_2, d_3
  95. Intersect(e_1, j_1) → P_2
  96. Polygon(P_1, P_2, T_1, O_1) → poly3, p_3, p_4, t_2, o_1
  97. Intersect(b_1, e_1) → Q_2
  98. Intersect(p, a_1) → R_2
  99. Polygon(H, K_1, Q_2, R_2) → poly4, h_3, k_3, q_2, r_3
  100. Intersect(q, a_2) → S_2
  101. Intersect(r, f_2) → T_2
  102. Polygon(K, W_1, S_2, T_2) → poly5, k_4, w_1, s_3, t_3
  103. Point(a_2) → U_2
  104. Intersect(a_1, a_2) → V_2
  105. Polygon(N, U_2, V_2, I_1) → poly6, n_4, u_2, v_2, i_3
  106. Intersect(b_1, i_1) → W_2
  107. Intersect(a_1, r_1) → Z_2
  108. Polygon(W_2, O_1, U_1, Z_2) → poly7, w_2, o_3, u_1, z_2
  109. Intersect(q, r_1) → A_3
  110. Polygon(V_1, A_3, S_2, G_1, H) → poly8, v_1, a_3, s_4, g_3, h_4
  111. Point(n_2) → B_3
  112. Intersect(r, n_2) → C_3
  113. Polygon(O, B_3, C_3, H_1) → poly9, o, b_3, c_3, h_5
  114. Intersect(g_2, n_2) → D_3
  115. Intersect(g_2, k_2) → E_3
  116. Polygon(D_3, E_3, L, K_2) → poly10, d_4, e_4, l_3, k_5
  117. Segment(N, F_3) → f_3
  118. Point(c) → G_3
  119. Segment(G_3, F_3) → q_3
  120. Segment(H_3, G_3) → a_4
  121. Segment(K, H_3) → b_4
  122. Point(g_2) → I_3
  123. Polygon(H_3, I_3, E_2, K) → poly11, h_6, i_4, e_5, k_6
  124. Point(r) → J_3
  125. Intersect(q, t) → K_3
  126. Polygon(J_3, K_3, N, F_3) → poly12, j_4, k_7, n_5, f_4
  127. Intersect(c, j) → L_3
  128. Segment(L_3, M_3) → c_4
  129. Segment(M_3, N_3) → g_4
  130. Segment(N_3, Q) → l_4
  131. Segment(J_2, O_3) → m_4
  132. Segment(O_3, L_3) → q_4
  133. Segment(H_2, P_3) → r_4
  134. Segment(P_3, L) → t_4
  135. Intersect(l_2, g_4) → Q_3
  136. Polygon(L_3, M_3, Q_3, O_3) → poly13, l_5, m_5, q_5, o_4
  137. Intersect(j_2, l_2) → R_3
  138. Polygon(P_3, R_3, I_2, L) → poly14, p_5, r_5, i_5, l_6
  139. Intersect(g_2, j_2) → S_3
  140. Intersect(s, g_2) → T_3
  141. Polygon(S_3, T_3, O, G_2) → poly15, s_5, t_5, o_5, g_5
  142. Intersect(g_2, g_4) → U_3
  143. Polygon(N_3, Q, F_2, U_3) → poly16, n_6, q_6, f_5, u_3
  144. Segment(D, V_3) → a_5
  145. Segment(V_3, W_3) → b_5
  146. Segment(W_3, M_2) → c_5
  147. Segment(O_1, Z_3) → d_5
  148. Point(i) → A_4
  149. Segment(Z_3, A_4) → j_5
  150. Segment(A_2, B_4) → a_6
  151. Point(b) → C_4
  152. Segment(B_4, C_4) → b_6
  153. Segment(C_4, D_4) → c_6
  154. Segment(D_4, C) → d_6
  155. Segment(A_4, E_4) → e_6
  156. Segment(E_4, A_2) → f_6
  157. Segment(N, F_4) → g_6
  158. Segment(F_4, G_4) → i_6
  159. Segment(G_4, M_2) → j_6
  160. Segment(O, H_4) → m_6
  161. Point(i) → I_4
  162. Segment(H_4, I_4) → p_6
  163. Segment(N, J_4) → r_6
  164. Segment(J_4, K_4) → s_6
  165. Segment(K_4, T) → t_6
  166. Intersect(e_1, g_1) → L_4
  167. Intersect(e_1, b_5) → M_4
  168. Polygon(L_4, D, V_3, M_4) → poly17, l_7, d_7, v_3, m_7
  169. Intersect(e_1, b_6) → N_4
  170. Polygon(D_4, C, L_1, N_4) → poly18, d_8, c_7, l_8, n_7
  171. Intersect(j_5, b_6) → O_4
  172. Polygon(E_4, A_2, B_4, O_4) → poly19, e_7, a_7, b_7, o_6
  173. Intersect(b_5, j_5) → P_4
  174. Intersect(c_2, b_5) → Q_4
  175. Polygon(O_1, Z_3, P_4, Q_4) → poly20, o_7, z_3, p_7, q_7
  176. Intersect(b_5, a_6) → R_4
  177. Intersect(a_2, b_5) → S_4
  178. Polygon(R_4, A_2, Z_1, S_4) → poly21, r_7, a_8, z_1, s_7
  179. Intersect(b_5, i_6) → T_4
  180. Polygon(G_4, M_2, W_3, T_4) → poly22, g_7, m_8, w_3, t_7
  181. Intersect(d_2, i_6) → U_4
  182. Intersect(d_2, r_6) → V_4
  183. Polygon(N, F_4, U_4, V_4) → poly23, n_8, f_7, u_4, v_4
  184. Intersect(q_1, a_2) → W_4
  185. Intersect(a_2, d_2) → Z_4
  186. Polygon(W_4, O_1, B_2, Z_4) → poly24, w_4, o_8, b_8, z_4
  187. Intersect(d_2, c_5) → A_5
  188. Intersect(d_2, n_2) → B_5
  189. Polygon(A_5, B_5, L_2, M_2) → poly25, a_9, b_9, l_9, m_9
  190. Intersect(d_2, p_6) → D_5
  191. Polygon(C_5, D_5, C_2, D_2) → poly26, c_8, d_9, c_9, d_{10}
  192. Point(s_6) → F_5
  193. Segment(F_5, O) → h_7
  194. Intersect(p_6, s_6) → G_5
  195. Polygon(O, H_4, G_5, F_5) → poly27, o_9, h_8, g_8, f_8
  196. Point(n_2) → H_5
  197. Intersect(n_2, s_6) → I_5
  198. Polygon(H_5, I_5, J_4, N) → poly28, h_9, i_7, j_7, n_9
  199. Polygon(H_5, I_5, J_4, N) → poly29, h_{10}, i_8, j_8, n_{10}
  200. Polygon(O, H_4, G_5, F_5) → poly30, o_{10}, h_{11}, g_9, f_9
  201. Polygon(A_5, M_2, L_2, B_5) → poly31, a_{10}, m_{10}, l_{10}, b_{10}
  202. Segment(N_3, J_5) → e_8
  203. Point(i) → K_5
  204. Segment(J_5, K_5) → k_8
  205. Segment(D_2, L_5) → p_8
  206. Intersect(d, g_4) → M_5
  207. Segment(L_5, M_5) → q_8
  208. Segment(K_5, N_5) → r_8
  209. Segment(N_5, T) → s_8
  210. Point(k_8) → P_5
  211. Polygon(Q, O_5, P_5, J_5) → poly32, q_9, o_{11}, p_9, j_9
  212. Intersect(i_2, g_4) → Q_5
  213. Intersect(q_8, o_{11}) → S_5
  214. Polygon(Q_5, O, R_5, S_5) → poly33, q_{10}, o_{12}, r_9, s_9
  215. Intersect(e_2, s_6) → T_5
  216. Intersect(s_6, q_8) → U_5
  217. Polygon(T_5, D_2, L_5, U_5) → poly34, t_8, d_{11}, l_{11}, u_5
  218. Intersect(s_6, k_8) → V_5
  219. Polygon(N_5, T, K_4, V_5) → poly35, n_{11}, t_9, k_9, v_5
  220. Segment(C, W_5) → e_9
  221. Segment(W_5, Z_5) → i_9
  222. Intersect(g, m) → A_6
  223. Segment(Z_5, A_6) → c_{10}
  224. Intersect(e, m) → C_6
  225. Segment(B_6, C_6) → e_{10}
  226. Intersect(b_2, i_6) → D_6
  227. Segment(B_6, D_6) → f_{10}
  228. Segment(A_2, E_6) → g_{10}
  229. Segment(E_6, F_6) → i_{10}
  230. Segment(F_6, Z) → j_{10}
  231. Segment(D_2, G_6) → k_{10}
  232. Point(m) → H_6
  233. Segment(G_6, H_6) → p_{10}
  234. Segment(A_6, I_6) → r_{10}
  235. Intersect(p_2, p_6) → J_6
  236. Segment(I_6, J_6) → s_{10}
  237. Segment(M_2, K_6) → t_{10}
  238. Segment(K_6, L_6) → a_{11}
  239. Segment(L_6, W) → b_{11}
  240. Segment(T, M_6) → c_{11}
  241. Point(m) → N_6
  242. Segment(M_6, N_6) → e_{11}
  243. Segment(Z, O_6) → f_{11}
  244. Intersect(k_8, p_8) → P_6
  245. Segment(O_6, P_6) → g_{11}
  246. Segment(D_2, Q_6) → i_{11}
  247. Point(j) → R_6
  248. Segment(Q_6, R_6) → j_{11}
  249. Segment(R_6, S_6) → k_{11}
  250. Segment(S_6, T) → m_{11}
  251. Segment(T_6, N_6) → p_{11}
  252. Segment(T_6, W) → q_{11}
  253. Segment(U_6, H_6) → r_{11}
  254. Segment(U_6, Z) → s_{11}
  255. Segment(M_2, V_6) → t_{11}
  256. Segment(V_6, W_6) → a_{12}
  257. Segment(W_6, A_6) → b_{12}
  258. Segment(C_6, Z_6) → c_{12}
  259. Point(b) → A_7
  260. Segment(Z_6, A_7) → d_{12}
  261. Intersect(b, m) → B_7
  262. Segment(B_7, C_7) → e_{12}
  263. Segment(C_7, A_4) → f_{12}
  264. Segment(A_7, D_7) → g_{12}
  265. Segment(D_7, B_7) → h_{12}
  266. Segment(E_4, E_7) → i_{12}
  267. Segment(E_7, F_7) → j_{12}
  268. Segment(F_7, C_6) → k_{12}
  269. OrthogonalLine(R, d) → l_{12}
  270. Intersect(f_1, j_5) → G_7
  271. Intersect(i_9, f_{12}) → H_7
  272. Polygon(G_7, C, W_5, H_7) → poly36, g_{13}, c_{13}, w_5, h_{13}
  273. Intersect(d_{12}, f_{12}) → I_7
  274. Polygon(I_7, D_7, B_7, C_7) → poly37, i_{13}, d_{13}, b_{13}, c_{14}
  275. Intersect(d_{12}, j_{12}) → J_7
  276. Polygon(Z_6, J_7, F_7, C_6) → poly38, z_6, j_{13}, f_{13}, c_{15}
  277. Intersect(i_9, j_{12}) → K_7
  278. Intersect(i_9, g_{10}) → L_7
  279. Polygon(A_2, E_7, K_7, L_7) → poly39, a_{13}, e_{13}, k_{13}, l_{13}
  280. Intersect(i_9, f_{10}) → M_7
  281. Intersect(i_9, c_{12}) → N_7
  282. Polygon(M_7, N_7, C_6, B_6) → poly40, m_{12}, n_{12}, c_{16}, b_{14}
  283. Intersect(i_9, a_{12}) → O_7
  284. Polygon(O_7, W_6, A_6, Z_5) → poly41, o_{13}, w_6, a_{14}, z_5
  285. Intersect(i_{10}, a_{12}) → P_7
  286. Intersect(i_{10}, t_{10}) → Q_7
  287. Polygon(P_7, Q_7, M_2, V_6) → poly42, p_{12}, q_{12}, m_{13}, v_6
  288. Intersect(f_{10}, i_{10}) → R_7
  289. Polygon(D_6, A_2, E_6, R_7) → poly43, d_{14}, a_{15}, e_{14}, r_{12}
  290. Intersect(i_{10}, p_{10}) → S_7
  291. Polygon(U_6, Z, F_6, S_7) → poly44, u_6, z, f_{14}, s_{12}
  292. Intersect(p_{10}, a_{11}) → T_7
  293. Intersect(a_{11}, i_{11}) → U_7
  294. Polygon(D_2, G_6, T_7, U_7) → poly45, d_{15}, g_{14}, t_{12}, u_7
  295. Intersect(s_{10}, a_{11}) → V_7
  296. Polygon(J_6, M_2, K_6, V_7) → poly46, j_{14}, m_{14}, k_{14}, v_7
  297. Intersect(i_{10}, s_{10}) → W_7
  298. Intersect(c_{10}, i_{10}) → Z_7
  299. Polygon(W_7, Z_7, A_6, I_6) → poly47, w_7, z_7, a_{16}, i_{14}
  300. Intersect(e_{11}, j_{11}) → A_8
  301. Polygon(T, M_6, A_8, S_6) → poly48, t_{13}, m_{15}, a_{17}, s_{13}
  302. Intersect(g_{11}, j_{11}) → B_8
  303. Polygon(P_6, D_2, Q_6, B_8) → poly49, p_{13}, d_{16}, q_{13}, b_{15}
  304. Intersect(j_{10}, a_{11}) → C_8
  305. Intersect(a_{11}, g_{11}) → D_8
  306. Polygon(C_8, Z, O_6, D_8) → poly50, c_{17}, z_8, o_{14}, d_{17}
  307. Intersect(a_{11}, e_{11}) → E_8
  308. Polygon(E_8, T_6, W, L_6) → poly51, e_{15}, t_{14}, w, l_{14}
  309. Segment(E, B_7) → n_{13}
  310. Segment(B_7, W) → r_{13}
  311. Segment(W, L_3) → h_{14}
  312. Segment(L_3, E) → n_{14}
  313. Segment(Q, D) → p_{14}
  314. Segment(C, T) → q_{14}
  315. Segment(W, B_7) → r_{14}
  316. Segment(C_6, H) → s_{14}
  317. Segment(K, A_6) → f_{15}
  318. Segment(Z, L) → g_{15}
  319. Segment(W, L_3) → h_{15}

AI Drawing Prompt Reference

Draw a right triangle with its sides labeled. Create two circles using the legs as diameters, connect their intersections to form new shapes. Continue constructing based on these intersections to create a complex tessellation pattern of triangles and circles, visually demonstrating the Pythagorean theorem.
The prompt above was compiled from the original drawing commands and is for reference only.