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Two Right Tessellation

Two Right Tessellation
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2016-04-12 03:07
PolygonTrigonometryCoordinate GeometryTransformationGeometric Construction
This example demonstrates a tessellation using a special subset of cyclic quadrilaterals inspired by Christopher Danielson's kite tiles. It involves concepts like congruence, kites, quadrilaterals, intersections, variance, similar triangles, squares, and geometric distribution. The construction uses commands such as Semicircle, Point, LineBisector, Intersect, Rotate, Polygon, Line, Midpoint, and Mirror in GeoGebra. Suitable for high school mathematics education to explore geometric transformations and tessellation patterns.

GeoGebra Commands & Parameters

  1. Semicircle(A, B) → c
  2. Point(c) → C
  3. LineBisector(A, B) → b
  4. Semicircle(B, A) → d
  5. Intersect(d, b, 1) → E
  6. Rotate(B, (-(90°)), E) → C'
  7. Rotate(C, (-(90°)), E) → D'
  8. Polygon(B, A, 4) → poly1, e, f, g, h, F, G
  9. Point(d) → D
  10. Polygon(A, D, B, C) → poly2, a_1, h_1, c_1, d_1
  11. Rotate(A, (-(90°)), E) → A'
  12. Rotate(D, (-(90°)), E) → H'
  13. Rotate(B, (-(90°)), E) → C'_1
  14. Rotate(C, (-(90°)), E) → D'_1
  15. Polygon(A', H', C'_1, D'_1) → poly2', a', h', c'_1, d'_1
  16. Rotate(A', (-(90°)), E) → A''
  17. Rotate(H', (-(90°)), E) → H''
  18. Rotate(C'_1, (-(90°)), E) → I
  19. Rotate(D'_1, (-(90°)), E) → J
  20. Polygon(A'', H'', I, J) → poly2'', a'', h'', i, j
  21. Rotate(A'', (-(90°)), E) → A'''
  22. Rotate(H'', (-(90°)), E) → H'''
  23. Rotate(I, (-(90°)), E) → I'
  24. Rotate(J, (-(90°)), E) → J'
  25. Polygon(A''', H''', I', J') → poly2''', a''', h''', i', j'
  26. Line(D, B) → k
  27. Rotate(A', 180°, B) → A''_3
  28. Rotate(H', 180°, B) → H''_2
  29. Rotate(C'_1, 180°, B) → M
  30. Rotate(D'_1, 180°, B) → N
  31. Polygon(A''_3, H''_2, M, N) → poly2''_1, a''_3, h''_2, m, n
  32. Rotate(A, 180°, B) → A'_1
  33. Rotate(D, 180°, B) → H'_1
  34. Rotate(B, 180°, B) → C'_2
  35. Rotate(C, 180°, B) → D'_2
  36. Polygon(A'_1, H'_1, C'_2, D'_2) → poly2'_1, a'_1, h'_1, c'_2, d'_2
  37. Rotate(H''', 180°, B) → O
  38. Midpoint(H''_2, O) → P
  39. Rotate(A''_3, (-(90°)), P) → Q
  40. Rotate(H''_2, (-(90°)), P) → R
  41. Rotate(M, (-(90°)), P) → M'
  42. Rotate(N, (-(90°)), P) → N'
  43. Polygon(Q, R, M', N') → poly3, q, r, m', n'
  44. Rotate(A''_3, 180°, P) → S
  45. Rotate(H''_2, 180°, P) → T
  46. Rotate(M, 180°, P) → M'_1
  47. Rotate(N, 180°, P) → N'_1
  48. Polygon(S, T, M'_1, N'_1) → poly4, s, t, m'_1, n'_1
  49. Rotate(A'', 180°, F) → A'''_1
  50. Rotate(H'', 180°, F) → H'''_1
  51. Rotate(I, 180°, F) → I'_1
  52. Rotate(J, 180°, F) → J'_1
  53. Polygon(A'''_1, H'''_1, I'_1, J'_1) → poly2'''_1, a'''_1, h'''_1, i'_1, j'_1
  54. Rotate(A''', 180°, F) → U
  55. Rotate(H''', 180°, F) → V
  56. Rotate(I', 180°, F) → I''
  57. Rotate(J', 180°, F) → J''
  58. Polygon(U, V, I'', J'') → poly5, u, v, i'', j''
  59. Rotate(A', 180°, C') → A''_4
  60. Rotate(H', 180°, C') → H''_3
  61. Rotate(C'_1, 180°, C') → W
  62. Rotate(D'_1, 180°, C') → Z
  63. Polygon(A''_4, H''_3, W, Z) → poly2''_2, a''_4, h''_3, w, z_1
  64. Rotate(A'', 180°, C') → A'''_2
  65. Rotate(H'', 180°, C') → H'''_2
  66. Rotate(I, 180°, C') → I'_2
  67. Rotate(J, 180°, C') → J'_2
  68. Polygon(A'''_2, H'''_2, I'_2, J'_2) → poly2'''_2, a'''_2, h'''_2, i'_2, j'_2
  69. Rotate(A, 180°, A) → A'_2
  70. Rotate(D, 180°, A) → H'_2
  71. Rotate(B, 180°, A) → C'_3
  72. Rotate(C, 180°, A) → D'_3
  73. Polygon(A'_2, H'_2, C'_3, D'_3) → poly2'_2, a'_2, h'_2, c'_3, d'_3
  74. Rotate(A''', 180°, A) → A_1
  75. Rotate(H''', 180°, A) → B_1
  76. Rotate(I', 180°, A) → I''_1
  77. Rotate(J', 180°, A) → J''_1
  78. Polygon(A_1, B_1, I''_1, J''_1) → poly6, a_2, b_1, i''_1, j''_1
  79. Rotate(D, 180°, C') → H'_3
  80. Midpoint(H'_3, H'''_2) → C_1
  81. Rotate(A'''_2, 180°, C_1) → D_1
  82. Rotate(H'''_2, 180°, C_1) → E_1
  83. Rotate(I'_2, 180°, C_1) → F_1
  84. Rotate(J'_2, 180°, C_1) → G_1
  85. Polygon(D_1, E_1, F_1, G_1) → poly7, d_2, e_1, f_1, g_1
  86. Rotate(A''_4, 180°, C_1) → H_1
  87. Rotate(H''_3, 180°, C_1) → I_1
  88. Rotate(W, 180°, C_1) → W'
  89. Rotate(Z, 180°, C_1) → Z'
  90. Polygon(H_1, I_1, W', Z') → poly8, h_2, i_1, w', z'
  91. Rotate(H'', 180°, A) → H'''_3
  92. Midpoint(H'_2, H'''_3) → J_1
  93. Rotate(A'_2, 180°, J_1) → K_1
  94. Rotate(H'_2, 180°, J_1) → L_1
  95. Rotate(C'_3, 180°, J_1) → M_1
  96. Rotate(D'_3, 180°, J_1) → N_1
  97. Polygon(K_1, L_1, M_1, N_1) → poly9, k_1, l_1, m_1, n_1
  98. Rotate(A_1, 180°, J_1) → O_1
  99. Rotate(B_1, 180°, J_1) → P_1
  100. Rotate(I''_1, 180°, J_1) → Q_1
  101. Rotate(J''_1, 180°, J_1) → R_1
  102. Polygon(O_1, P_1, Q_1, R_1) → poly6', o_1, p_1, q_1, r_1
  103. Midpoint(D'_2, J'_2) → S_1
  104. Midpoint(J''_1, N) → T_1
  105. Midpoint(D'_3, J'_1) → U_1
  106. Rotate(A''', 180°, U_1) → V_1
  107. Rotate(H''', 180°, U_1) → W_1
  108. Rotate(I', 180°, U_1) → I''_2
  109. Rotate(J', 180°, U_1) → J''_2
  110. Polygon(V_1, W_1, I''_2, J''_2) → poly10, v_1, w_1, i''_2, j''_2
  111. Rotate(A', 180°, S_1) → A''_5
  112. Rotate(H', 180°, S_1) → H''_1
  113. Rotate(C'_1, 180°, S_1) → Z_1
  114. Rotate(D'_1, 180°, S_1) → A_2
  115. Polygon(A''_5, H''_1, Z_1, A_2) → poly2''_3, a''_5, h''_1, z_2, a_3
  116. Rotate(A, 180°, T_1) → A'_3
  117. Rotate(D, 180°, T_1) → H'_4
  118. Rotate(B, 180°, T_1) → C'_4
  119. Rotate(C, 180°, T_1) → D'_4
  120. Polygon(A'_3, H'_4, C'_4, D'_4) → poly2'_3, a'_3, h'_4, c'_4, d'_4
  121. Midpoint(J'', Z) → B_2
  122. Rotate(A'', 180°, B_2) → A'''_3
  123. Rotate(H'', 180°, B_2) → H'''_4
  124. Rotate(I, 180°, B_2) → I'_3
  125. Rotate(J, 180°, B_2) → J'_3
  126. Polygon(A'''_3, H'''_4, I'_3, J'_3) → poly2'''_3, a'''_3, h'''_4, i'_3, j'_3
  127. Mirror(C, Line[A, B]) → D_0
  128. Rotate(Q, 180°, E) → Q'
  129. Rotate(R, 180°, E) → R'
  130. Rotate(M', 180°, E) → M''
  131. Rotate(N', 180°, E) → N''
  132. Polygon(Q', R', M'', N'') → poly3', q', r', m'', n''
  133. Rotate(S, 180°, E) → S'
  134. Rotate(T, 180°, E) → T'
  135. Rotate(M'_1, 180°, E) → H
  136. Rotate(N'_1, 180°, E) → K
  137. Polygon(S', T', H, K) → poly4', s', t', h_3, k_2