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Tessellation of a General Quadrilateral by Michael Borcherds

Tessellation of a General Quadrilateral by Michael Borcherds
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2013-01-28 21:16
PolygonAngleTransformationGeometric Construction
In collections Unit 7: Polygons and Quadrilaterals
This example demonstrates how to tessellate a general quadrilateral to showcase geometric patterns and spatial filling. It involves core mathematical concepts such as angles, area, congruence, quadrilateral properties, difference and slope, number calculations, intersections, and volume. The construction uses GeoGebra commands like Polygon, Vector, Translate, and Mirror. Suitable for high school students to explore the principles and applications of tessellations in mathematics and interest groups.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → poly1, a_1, b_1, c_1, d_1
  2. Vector(A, B) → u
  3. Translate(A, u) → A'
  4. Translate(B, u) → B'
  5. Translate(C, u) → C'
  6. Translate(D, u) → D'
  7. Polygon(A', B', C', D') → poly1', a', b', c', d'
  8. Mirror(A', A') → A''
  9. Mirror(B', A') → B''
  10. Mirror(C', A') → C''
  11. Mirror(D', A') → D''
  12. Polygon(A'', B'', C'', D'') → poly1'', a'', b'', c'', d''
  13. Polygon(I, J, K, L) → poly2, j, i, h, g
  14. Vector(I, J) → v
  15. Translate(I, v) → M
  16. Translate(J, v) → N
  17. Translate(K, v) → O
  18. Translate(L, v) → P
  19. Polygon(M, N, O, P) → poly4, r, s, p, q
  20. Mirror(M, M) → E
  21. Mirror(N, M) → F
  22. Mirror(O, M) → G
  23. Mirror(P, M) → H
  24. Polygon(E, F, G, H) → poly3, n, m, l, k
  25. Polygon(U, V, W, Z) → poly5, g_1, f_1, e_1, t
  26. Vector(U, V) → w
  27. Translate(U, w) → A_1
  28. Translate(V, w) → B_1
  29. Translate(W, w) → C_1
  30. Translate(Z, w) → D_1
  31. Polygon(A_1, B_1, C_1, D_1) → poly7, n_1, p_1, l_1, m_1
  32. Mirror(A_1, A_1) → Q
  33. Mirror(B_1, A_1) → R
  34. Mirror(C_1, A_1) → S
  35. Mirror(D_1, A_1) → T
  36. Polygon(Q, R, S, T) → poly6, k_1, j_1, i_1, h_1
  37. Polygon(I_1, J_1, K_1, L_1) → poly8, t_1, s_1, r_1, q_1
  38. Vector(I_1, J_1) → z
  39. Translate(I_1, z) → M_1
  40. Translate(J_1, z) → N_1
  41. Translate(K_1, z) → O_1
  42. Translate(L_1, z) → P_1
  43. Polygon(M_1, N_1, O_1, P_1) → poly10, g_2, h_2, e_2, f_2
  44. Mirror(M_1, M_1) → E_1
  45. Mirror(N_1, M_1) → F_1
  46. Mirror(O_1, M_1) → G_1
  47. Mirror(P_1, M_1) → H_1
  48. Polygon(E_1, F_1, G_1, H_1) → poly9, d_2, c_2, b_2, a_2
  49. Polygon(U_1, V_1, W_1, Z_1) → poly11, l_2, k_2, j_2, i_2
  50. Vector(U_1, V_1) → u_1
  51. Translate(U_1, u_1) → A_2
  52. Translate(V_1, u_1) → B_2
  53. Translate(W_1, u_1) → C_2
  54. Translate(Z_1, u_1) → D_2
  55. Polygon(A_2, B_2, C_2, D_2) → poly13, t_2, a_3, r_2, s_2
  56. Mirror(A_2, A_2) → Q_1
  57. Mirror(B_2, A_2) → R_1
  58. Mirror(C_2, A_2) → S_1
  59. Mirror(D_2, A_2) → T_1
  60. Polygon(Q_1, R_1, S_1, T_1) → poly12, q_2, p_2, n_2, m_2
  61. Polygon(I_2, J_2, K_2, L_2) → poly14, e_3, d_3, c_3, b_3
  62. Vector(I_2, J_2) → v_1
  63. Translate(I_2, v_1) → M_2
  64. Translate(J_2, v_1) → N_2
  65. Translate(K_2, v_1) → O_2
  66. Translate(L_2, v_1) → P_2
  67. Polygon(M_2, N_2, O_2, P_2) → poly16, l_3, m_3, j_3, k_3
  68. Mirror(M_2, M_2) → E_2
  69. Mirror(N_2, M_2) → F_2
  70. Mirror(O_2, M_2) → G_2
  71. Mirror(P_2, M_2) → H_2
  72. Polygon(E_2, F_2, G_2, H_2) → poly15, i_3, h_3, g_3, f_3
  73. Polygon(U_2, V_2, W_2, Z_2) → poly17, r_3, q_3, p_3, n_3
  74. Vector(U_2, V_2) → w_1
  75. Translate(U_2, w_1) → A_3
  76. Translate(V_2, w_1) → B_3
  77. Translate(W_2, w_1) → C_3
  78. Translate(Z_2, w_1) → D_3
  79. Polygon(A_3, B_3, C_3, D_3) → poly19, e_4, f_4, c_4, d_4
  80. Mirror(A_3, A_3) → Q_2
  81. Mirror(B_3, A_3) → R_2
  82. Mirror(C_3, A_3) → S_2
  83. Mirror(D_3, A_3) → T_2
  84. Polygon(Q_2, R_2, S_2, T_2) → poly18, b_4, a_4, t_3, s_3
  85. Polygon(I_3, J_3, K_3, L_3) → poly20, j_4, i_4, h_4, g_4
  86. Vector(I_3, J_3) → z_1
  87. Translate(I_3, z_1) → M_3
  88. Translate(J_3, z_1) → N_3
  89. Translate(K_3, z_1) → O_3
  90. Translate(L_3, z_1) → P_3
  91. Polygon(M_3, N_3, O_3, P_3) → poly22, r_4, s_4, p_4, q_4
  92. Mirror(M_3, M_3) → E_3
  93. Mirror(N_3, M_3) → F_3
  94. Mirror(O_3, M_3) → G_3
  95. Mirror(P_3, M_3) → H_3
  96. Polygon(E_3, F_3, G_3, H_3) → poly21, n_4, m_4, l_4, k_4
  97. Polygon(U_3, V_3, W_3, Z_3) → poly23, c_5, b_5, a_5, t_4
  98. Vector(U_3, V_3) → u_2
  99. Translate(U_3, u_2) → A_4
  100. Translate(V_3, u_2) → B_4
  101. Translate(W_3, u_2) → C_4
  102. Translate(Z_3, u_2) → D_4
  103. Polygon(A_4, B_4, C_4, D_4) → poly25, j_5, k_5, h_5, i_5
  104. Mirror(A_4, A_4) → Q_3
  105. Mirror(B_4, A_4) → R_3
  106. Mirror(C_4, A_4) → S_3
  107. Mirror(D_4, A_4) → T_3
  108. Polygon(Q_3, R_3, S_3, T_3) → poly24, g_5, f_5, e_5, d_5
  109. Polygon(I_4, J_4, K_4, L_4) → poly26, p_5, n_5, m_5, l_5
  110. Vector(I_4, J_4) → v_2
  111. Translate(I_4, v_2) → M_4
  112. Translate(J_4, v_2) → N_4
  113. Translate(K_4, v_2) → O_4
  114. Translate(L_4, v_2) → P_4
  115. Polygon(M_4, N_4, O_4, P_4) → poly28, c_6, d_6, a_6, b_6
  116. Mirror(M_4, M_4) → E_4
  117. Mirror(N_4, M_4) → F_4
  118. Mirror(O_4, M_4) → G_4
  119. Mirror(P_4, M_4) → H_4
  120. Polygon(E_4, F_4, G_4, H_4) → poly27, t_5, s_5, r_5, q_5
  121. Polygon(U_4, V_4, W_4, Z_4) → poly29, h_6, g_6, f_6, e_6
  122. Vector(U_4, V_4) → w_2
  123. Translate(U_4, w_2) → A_5
  124. Translate(V_4, w_2) → B_5
  125. Translate(W_4, w_2) → C_5
  126. Translate(Z_4, w_2) → D_5
  127. Polygon(A_5, B_5, C_5, D_5) → poly31, p_6, q_6, m_6, n_6
  128. Mirror(A_5, A_5) → Q_4
  129. Mirror(B_5, A_5) → R_4
  130. Mirror(C_5, A_5) → S_4
  131. Mirror(D_5, A_5) → T_4
  132. Polygon(Q_4, R_4, S_4, T_4) → poly30, l_6, k_6, j_6, i_6
  133. Polygon(I_5, J_5, K_5, L_5) → poly32, a_7, t_6, s_6, r_6
  134. Vector(I_5, J_5) → z_2
  135. Translate(I_5, z_2) → M_5
  136. Translate(J_5, z_2) → N_5
  137. Translate(K_5, z_2) → O_5
  138. Translate(L_5, z_2) → P_5
  139. Polygon(M_5, N_5, O_5, P_5) → poly34, h_7, i_7, f_7, g_7
  140. Mirror(M_5, M_5) → E_5
  141. Mirror(N_5, M_5) → F_5
  142. Mirror(O_5, M_5) → G_5
  143. Mirror(P_5, M_5) → H_5
  144. Polygon(E_5, F_5, G_5, H_5) → poly33, e_7, d_7, c_7, b_7
  145. Polygon(U_5, V_5, W_5, Z_5) → poly35, m_7, l_7, k_7, j_7
  146. Vector(U_5, V_5) → u_3
  147. Translate(U_5, u_3) → A_6
  148. Translate(V_5, u_3) → B_6
  149. Translate(W_5, u_3) → C_6
  150. Translate(Z_5, u_3) → D_6
  151. Polygon(A_6, B_6, C_6, D_6) → poly37, a_8, b_8, s_7, t_7
  152. Mirror(A_6, A_6) → Q_5
  153. Mirror(B_6, A_6) → R_5
  154. Mirror(C_6, A_6) → S_5
  155. Mirror(D_6, A_6) → T_5
  156. Polygon(Q_5, R_5, S_5, T_5) → poly36, r_7, q_7, p_7, n_7
  157. Polygon(I_6, J_6, K_6, L_6) → poly38, f_8, e_8, d_8, c_8
  158. Vector(I_6, J_6) → v_3
  159. Translate(I_6, v_3) → M_6
  160. Translate(J_6, v_3) → N_6
  161. Translate(K_6, v_3) → O_6
  162. Translate(L_6, v_3) → P_6
  163. Polygon(M_6, N_6, O_6, P_6) → poly40, m_8, n_8, k_8, l_8
  164. Mirror(M_6, M_6) → E_6
  165. Mirror(N_6, M_6) → F_6
  166. Mirror(O_6, M_6) → G_6
  167. Mirror(P_6, M_6) → H_6
  168. Polygon(E_6, F_6, G_6, H_6) → poly39, j_8, i_8, h_8, g_8
  169. Polygon(U_6, V_6, W_6, Z_6) → poly41, s_8, r_8, q_8, p_8
  170. Vector(U_6, V_6) → w_3
  171. Translate(U_6, w_3) → A_7
  172. Translate(V_6, w_3) → B_7
  173. Translate(W_6, w_3) → C_7
  174. Translate(Z_6, w_3) → D_7
  175. Polygon(A_7, B_7, C_7, D_7) → poly43, f_9, g_9, d_9, e_9
  176. Mirror(A_7, A_7) → Q_6
  177. Mirror(B_7, A_7) → R_6
  178. Mirror(C_7, A_7) → S_6
  179. Mirror(D_7, A_7) → T_6
  180. Polygon(Q_6, R_6, S_6, T_6) → poly42, c_9, b_9, a_9, t_8

AI Drawing Prompt Reference

Draw a tessellation pattern created by translating and reflecting a general quadrilateral, showcasing geometric transformations and spatial filling. Start by drawing a quadrilateral, then translate it along one of its sides to create another quadrilateral. Reflect both quadrilaterals, and combine the mirrored shapes to form a complex tessellated pattern, demonstrating the principles and aesthetics of geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.