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Kite Tessellation

Kite Tessellation
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2016-06-19 01:07
PolygonSequenceTransformationGeometric Construction
This GeoGebra example demonstrates a plane tessellation composed of kite-shaped quadrilaterals, created by applying rotations, reflections, and translations to fill the plane seamlessly. The core mathematical concepts include tessellation, geometric transformations (rotation, reflection, translation), and polygon symmetry. The construction uses GeoGebra commands such as Point, Mirror, Polygon, Rotate, Midpoint, and Sequence, as seen in the summary: a base kite polygon (poly1) is defined and then replicated via rotations and reflections to generate a periodic pattern. This is suitable for middle or high school geometry classes, art-mathematics integration, or tessellation workshops, helping students explore symmetry, transformations, and the mathematics of tiling.

GeoGebra Commands & Parameters

  1. Point(Ray[B, Rotate[A, 45°, B]]) → C
  2. Mirror(C, Line[A, B]) → C'
  3. Polygon(B, C, A, C') → poly1, b, c, a, c'
  4. Rotate(B, 90°, B) → B'
  5. Rotate(C, 90°, B) → C'_1
  6. Rotate(A, 90°, B) → A'
  7. Rotate(C', 90°, B) → C''
  8. Polygon(B', C'_1, A', C'') → poly1', b', c'_1, a', c''
  9. Rotate(B', 90°, B) → B''
  10. Rotate(C'_1, 90°, B) → D
  11. Rotate(A', 90°, B) → A''
  12. Rotate(C'', 90°, B) → C'''
  13. Polygon(B'', D, A'', C''') → poly1'', b'', d, a'', c'''
  14. Rotate(B'', 90°, B) → B'''
  15. Rotate(D, 90°, B) → D'
  16. Rotate(A'', 90°, B) → A'''
  17. Rotate(C''', 90°, B) → E
  18. Polygon(B''', D', A''', E) → poly1''', b''', d', a''', e
  19. Midpoint(c) → F
  20. Midpoint(a) → G
  21. Rotate(B, 180°, F) → B'_1
  22. Rotate(C, 180°, F) → C'_2
  23. Rotate(A, 180°, F) → A'_1
  24. Rotate(C', 180°, F) → C''_1
  25. Polygon(B'_1, C'_2, A'_1, C''_1) → poly1'_1, b'_1, c'_2, a'_1, c''_1
  26. Rotate(B, 180°, G) → B'_2
  27. Rotate(C, 180°, G) → C'_3
  28. Rotate(A, 180°, G) → A'_2
  29. Rotate(C', 180°, G) → C''_2
  30. Polygon(B'_2, C'_3, A'_2, C''_2) → poly1'_2, b'_2, c'_3, a'_2, c''_2
  31. Rotate(B'_1, 90°, B) → H
  32. Rotate(C'_2, 90°, B) → I
  33. Rotate(A'_1, 90°, B) → J
  34. Rotate(C''_1, 90°, B) → K
  35. Polygon(H, I, J, K) → poly2, h, i, j, k
  36. Rotate(B'_2, 90°, B) → L
  37. Rotate(C'_3, 90°, B) → M
  38. Rotate(A'_2, 90°, B) → N
  39. Rotate(C''_2, 90°, B) → O
  40. Polygon(L, M, N, O) → poly3, l, m, n, o
  41. Rotate(H, 90°, B) → H'
  42. Rotate(I, 90°, B) → I'
  43. Rotate(J, 90°, B) → J'
  44. Rotate(K, 90°, B) → K'
  45. Polygon(H', I', J', K') → poly2', h', i', j', k'
  46. Rotate(L, 90°, B) → L'
  47. Rotate(M, 90°, B) → M'
  48. Rotate(N, 90°, B) → N'
  49. Rotate(O, 90°, B) → O'
  50. Polygon(L', M', N', O') → poly3', l', m', n', o'
  51. Rotate(H', 90°, B) → H''
  52. Rotate(I', 90°, B) → I''
  53. Rotate(J', 90°, B) → J''
  54. Rotate(K', 90°, B) → K''
  55. Polygon(H'', I'', J'', K'') → poly2'', h'', i'', j'', k''
  56. Rotate(L', 90°, B) → L''
  57. Rotate(M', 90°, B) → M''
  58. Rotate(N', 90°, B) → N''
  59. Rotate(O', 90°, B) → O''
  60. Polygon(L'', M'', N'', O'') → poly3'', l'', m'', n'', o''
  61. Midpoint(B'_1, B'_2) → P
  62. Rotate(B'_2, 180°, P) → Q
  63. Rotate(C'_3, 180°, P) → R
  64. Rotate(A'_2, 180°, P) → S
  65. Rotate(C''_2, 180°, P) → T
  66. Polygon(Q, R, S, T) → poly4, q, r, s, t
  67. Rotate(B'_1, 180°, P) → U
  68. Rotate(C'_2, 180°, P) → V
  69. Rotate(A'_1, 180°, P) → W
  70. Rotate(C''_1, 180°, P) → Z
  71. Polygon(U, V, W, Z) → poly5, u, v, w, z_1
  72. Rotate(Q, 90°, B'_1) → Q'
  73. Rotate(R, 90°, B'_1) → R'
  74. Rotate(S, 90°, B'_1) → S'
  75. Rotate(T, 90°, B'_1) → T'
  76. Polygon(Q', R', S', T') → poly4', q', r', s', t'
  77. Rotate(B'_2, 90°, B'_2) → A_1
  78. Rotate(C'_3, 90°, B'_2) → B_1
  79. Rotate(A'_2, 90°, B'_2) → C_1
  80. Rotate(C''_2, 90°, B'_2) → D_1
  81. Polygon(A_1, B_1, C_1, D_1) → poly6, a_1, b_1, c_1, d_1
  82. Rotate(Q', 90°, B) → Q''
  83. Rotate(R', 90°, B) → R''
  84. Rotate(S', 90°, B) → S''
  85. Rotate(T', 90°, B) → T''
  86. Polygon(Q'', R'', S'', T'') → poly4'', q'', r'', s'', t''
  87. Rotate(Q, 90°, B) → Q'_1
  88. Rotate(R, 90°, B) → R'_1
  89. Rotate(S, 90°, B) → S'_1
  90. Rotate(T, 90°, B) → T'_1
  91. Polygon(Q'_1, R'_1, S'_1, T'_1) → poly4'_1, q'_1, r'_1, s'_1, t'_1
  92. Rotate(U, 90°, B) → U'
  93. Rotate(V, 90°, B) → V'
  94. Rotate(W, 90°, B) → W'
  95. Rotate(Z, 90°, B) → Z'
  96. Polygon(U', V', W', Z') → poly5', u', v', w', z'
  97. Rotate(A_1, 90°, B) → E_1
  98. Rotate(B_1, 90°, B) → F_1
  99. Rotate(C_1, 90°, B) → G_1
  100. Rotate(D_1, 90°, B) → H_1
  101. Polygon(E_1, F_1, G_1, H_1) → poly6', e_1, f_1, g_1, h_1
  102. Rotate(B, 180°, P) → B'_3
  103. Rotate(C, 180°, P) → C'_4
  104. Rotate(A, 180°, P) → A'_3
  105. Rotate(C', 180°, P) → C''_3
  106. Polygon(B'_3, C'_4, A'_3, C''_3) → poly1'_3, b'_3, c'_4, a'_3, c''_3
  107. Rotate(B'_3, 90°, B) → I_1
  108. Rotate(C'_4, 90°, B) → J_1
  109. Rotate(A'_3, 90°, B) → K_1
  110. Rotate(C''_3, 90°, B) → L_1
  111. Polygon(I_1, J_1, K_1, L_1) → poly7, i_1, j_1, k_1, l_1
  112. Rotate(I_1, 90°, B) → M_1
  113. Rotate(J_1, 90°, B) → N_1
  114. Rotate(K_1, 90°, B) → O_1
  115. Rotate(L_1, 90°, B) → P_1
  116. Polygon(M_1, N_1, O_1, P_1) → poly7', m_1, n_1, o_1, p_1
  117. Rotate(I_1, 90°, B) → Q_1
  118. Rotate(J_1, 90°, B) → R_1
  119. Rotate(K_1, 90°, B) → S_1
  120. Rotate(L_1, 90°, B) → T_1
  121. Rotate(I_1, 90°, B) → U_1
  122. Rotate(J_1, 90°, B) → V_1
  123. Rotate(K_1, 90°, B) → W_1
  124. Rotate(L_1, 90°, B) → Z_1
  125. Rotate(E_1, 90°, B) → A_2
  126. Rotate(F_1, 90°, B) → B_2
  127. Rotate(G_1, 90°, B) → C_2
  128. Rotate(H_1, 90°, B) → D_2
  129. Polygon(A_2, B_2, C_2, D_2) → poly6'', a_2, b_2, c_2, d_2
  130. Rotate(U', 90°, B) → U''
  131. Rotate(V', 90°, B) → V''
  132. Rotate(W', 90°, B) → W''
  133. Rotate(Z', 90°, B) → Z''
  134. Polygon(U'', V'', W'', Z'') → poly5'', u'', v'', w'', z''
  135. Rotate(Q'_1, 90°, B) → E_2
  136. Rotate(R'_1, 90°, B) → F_2
  137. Rotate(S'_1, 90°, B) → G_2
  138. Rotate(T'_1, 90°, B) → H_2
  139. Polygon(E_2, F_2, G_2, H_2) → poly8, e_2, f_2, g_2, h_2
  140. Rotate(Q'', 90°, B) → Q'''
  141. Rotate(R'', 90°, B) → R'''
  142. Rotate(S'', 90°, B) → S'''
  143. Rotate(T'', 90°, B) → T'''
  144. Polygon(Q''', R''', S''', T''') → poly4''', q''', r''', s''', t'''
  145. Rotate(A_2, 90°, B) → I_2
  146. Rotate(B_2, 90°, B) → J_2
  147. Rotate(C_2, 90°, B) → K_2
  148. Rotate(D_2, 90°, B) → L_2
  149. Polygon(I_2, J_2, K_2, L_2) → poly6''', i_2, j_2, k_2, l_2
  150. Rotate(U'', 90°, B) → U'''
  151. Rotate(V'', 90°, B) → V'''
  152. Rotate(W'', 90°, B) → W'''
  153. Rotate(Z'', 90°, B) → Z'''
  154. Polygon(U''', V''', W''', Z''') → poly5''', u''', v''', w''', z'''
  155. Rotate(E_2, 90°, B) → M_2
  156. Rotate(F_2, 90°, B) → N_2
  157. Rotate(G_2, 90°, B) → O_2
  158. Rotate(H_2, 90°, B) → P_2
  159. Polygon(M_2, N_2, O_2, P_2) → poly8', m_2, n_2, o_2, p_2
  160. Rotate(M_1, 90°, B) → Q_2
  161. Rotate(N_1, 90°, B) → R_2
  162. Rotate(O_1, 90°, B) → S_2
  163. Rotate(P_1, 90°, B) → T_2
  164. Polygon(Q_2, R_2, S_2, T_2) → poly7'', q_2, r_2, s_2, t_2
  165. Rotate(Q''', 90°, B) → U_2
  166. Rotate(R''', 90°, B) → V_2
  167. Rotate(S''', 90°, B) → W_2
  168. Rotate(T''', 90°, B) → Z_2
  169. Polygon(U_2, V_2, W_2, Z_2) → poly9, u_2, v_2, w_2, z_2
  170. Sequence(Rotate[Polygon[Sequence[Rotate[A, ((i * 90))°, P], i, 1, 4]], ((j * 90))°, B], j, 1, 4) → list1
  171. Sequence(Polygon[Rotate[C, ((j * 90))°, B], Rotate[M, ((j * 90))°, B], Rotate[S', ((j * 90))°, B], Rotate[R', ((j * 90))°, B]], j, 1, 4) → list2
  172. Sequence(Rotate[Rotate[listR, 180°, B'_3], ((i * 90))°, B], i, 1, 4) → list3
  173. Sequence(Rotate[Rotate[listY, 180°, B'_3], ((i * 90))°, B], i, 1, 4) → list4
  174. Sequence(Rotate[Rotate[list1, 180°, B'_3], ((i * 90))°, B], i, 1, 4) → list5
  175. Sequence(Rotate[Rotate[list2, 180°, B'_3], ((i * 90))°, B], i, 1, 4) → list6
  176. Sequence(Rotate[Rotate[listR, 180°, S'], ((i * 90))°, B], i, 1, 4) → list7
  177. Sequence(Rotate[Rotate[listY, 180°, S'], ((i * 90))°, B], i, 1, 4) → list8
  178. Sequence(Rotate[Rotate[list1, 180°, S'], ((i * 90))°, B], i, 1, 4) → list9
  179. Sequence(Rotate[Rotate[list2, 180°, S'], ((i * 90))°, B], i, 1, 4) → list10
  180. Sequence(Rotate[Sequence[Polygon[Rotate[C, ((i * 180))° + 90°, S'], Rotate[M, ((i * 180))° + 90°, S'], Rotate[S', ((i * 180))° + 90°, S'], Rotate[R', ((i * 180))° + 90°, S']], i, 1, 2], ((j * 90))°, B], j, 1, 4) → list11
  181. Sequence(Rotate[Rotate[list11, 180°, S'], ((i * 90))°, B], i, 1, 4) → list12

AI Drawing Prompt Reference

Draw a plane tessellation made of kite-shaped quadrilaterals. Start with a single symmetric kite defined by a base segment, a point rotated 45 degrees from one endpoint, and its mirror image across the base line. Then rotate this kite by 90, 180, and 270 degrees around one endpoint of the base, creating four kites around that center. Next, reflect the original kite by 180 degrees about the midpoints of its two opposite sides, producing two flipped kites. Finally, rotate these flipped kites by 90, 180, and 270 degrees around the same center point, filling the plane with a seamless periodic pattern of interlocking kites.
The prompt above was compiled from the original drawing commands and is for reference only.