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Islamic Geometric Pattern Division and Symmetry Construction

Islamic Geometric Pattern Division and Symmetry Construction
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2019-12-31 18:01
PolygonAngleTransformationGeometric Construction
① This example demonstrates how to construct an Islamic-style geometric pattern by dividing and combining basic polygons through symmetry, rotation, and reflection. ② Core mathematical concepts include symmetry, median lines, angle bisectors, rotational and mirror transformations, with potential use of the Pythagorean theorem in underlying measurements. ③ Construction highlights: using GeoGebra commands such as Polygon, Point, Segment, Circle, Ray, Line, Intersect, Rotate, Mirror, OrthogonalLine, AngularBisector, and Textfield, along with ConstructionStep to create an interactive step-by-step build. ④ Suitable for middle school to university geometry classes, especially for teaching symmetry, transformation geometry, and pattern design, as well as interdisciplinary projects linking mathematics and art.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. ConstructionStep() → stap
  3. Polygon(A, B, 4) → veelhoek1, f, g, h, i, C, D
  4. Point(i) → E
  5. Segment(A, E) → j
  6. Rotate(F, 90°, G) → F'
  7. Rotate(F, (-(90°)), G) → F'_1
  8. Mirror(F, G) → F'_2
  9. Segment(F, F'_1) → k
  10. Mirror(D, k) → D'
  11. Rotate(D', 90°, G) → D''
  12. Rotate(D', (-(90°)), G) → D''_1
  13. Mirror(D', G) → D''_2
  14. Segment(F, D') → l
  15. Segment(F'_1, D''_1) → m
  16. Segment(F'_2, D''_2) → n
  17. Segment(F', D'') → p
  18. Segment(F'_1, F'_2) → q
  19. Segment(F'_2, F') → r
  20. Segment(F', F) → s
  21. OrthogonalLine(D''_1, r) → t
  22. Intersect(t, r) → H
  23. Segment(D''_1, H) → d
  24. Segment(D''_2, D''_1) → a
  25. Textfield(F) → Invulvak1
  26. OrthogonalLine(D'', k) → b
  27. Intersect(b, k) → I
  28. Segment(I, D'') → c
  29. OrthogonalLine(D''_2, s) → e
  30. Intersect(e, s) → J
  31. Segment(D''_2, J) → f_1
  32. OrthogonalLine(D', q) → g_1
  33. Intersect(g_1, q) → K
  34. Segment(D', K) → h_1
  35. Segment(F, F'_1) → i_1
  36. Segment(F'_1, F'_2) → j_1
  37. Segment(F'_2, F') → k_1
  38. Segment(F', F) → l_1
  39. Segment(D''_1, H) → m_1
  40. Intersect(b, i) → L
  41. Segment(L, I) → n_1
  42. Intersect(e, f) → M
  43. Segment(J, M) → p_1
  44. Intersect(g_1, h) → N
  45. Segment(K, N) → q_1
  46. Intersect(t, g) → O
  47. Segment(H, O) → r_1
  48. Segment(L, F'_1) → s_1
  49. Circle(H, D''_1) → c_1
  50. Intersect(r, k_1) → P
  51. Intersect(r, k_1) → Q
  52. Intersect(r, k_1) → R
  53. Intersect(r, k_1) → S
  54. Intersect(c_1, k_1, 2) → T
  55. Circle(F', T) → d_1
  56. Intersect(d_1, p, 1) → U
  57. Segment(J, U) → t_1
  58. Intersect(d_1, f, 1) → V
  59. Segment(J, V) → a_1
  60. Intersect(c_1, n, 1) → W
  61. AngularBisector(U, D''_2, D''_1) → b_1
  62. AngularBisector(D''_1, H, T) → e_1
  63. Intersect(b_1, e_1) → Z
  64. Segment(D''_1, Z) → f_2
  65. Segment(Z, U) → g_2
  66. Segment(Z, T) → h_2
  67. Circle(K, N) → c_2
  68. Intersect(q, j_1) → A_1
  69. Intersect(c_2, j_1, 2) → B_1
  70. Circle(F'_2, B_1) → d_2
  71. Intersect(d_2, g, 2) → C_1
  72. AngularBisector(B_1, K, N) → i_2
  73. AngularBisector(C_1, F'_2, B_1) → j_2
  74. Intersect(i_2, j_2) → D_1
  75. Segment(B_1, D_1) → k_2
  76. Segment(D_1, C_1) → l_2
  77. Segment(D_1, N) → m_2
  78. AngularBisector(B_1, F'_2, W) → n_2
  79. AngularBisector(D', K, B_1) → p_2
  80. Intersect(n_2, p_2) → E_1
  81. Segment(D', E_1) → q_2
  82. Segment(E_1, B_1) → r_2
  83. Segment(E_1, W) → s_2
  84. Circle(I, D'') → e_2
  85. Circle(D', D'') → t_2
  86. Intersect(k, i_1) → F_1
  87. Intersect(k, i_1) → G_1
  88. Intersect(k, i_1) → H_1
  89. Intersect(k, i_1) → I_1
  90. Intersect(e_2, i_1, 2) → J_1
  91. AngularBisector(D'', I, J_1) → a_2
  92. Intersect(t_2, m, 1) → K_1
  93. AngularBisector(K_1, D', D'') → b_2
  94. Intersect(a_2, b_2) → L_1
  95. Segment(D'', L_1) → f_3
  96. Segment(L_1, J_1) → g_3
  97. Segment(L_1, K_1) → h_3
  98. Segment(K_1, K) → i_3
  99. Circle(K, K_1) → c_3
  100. Circle(D'', D''_2) → d_3
  101. Intersect(d_3, l, 1) → M_1
  102. Circle(F, M_1) → e_3
  103. Intersect(s, l_1) → N_1
  104. Intersect(s, l_1) → O_1
  105. Intersect(e_3, l_1, 1) → P_1
  106. AngularBisector(M_1, F, P_1) → j_3
  107. AngularBisector(D''_2, J, P_1) → k_3
  108. Intersect(k_3, j_3) → Q_1
  109. Segment(M_1, Q_1) → l_3
  110. Segment(Q_1, P_1) → m_3
  111. Segment(Q_1, D''_2) → n_3
  112. Segment(M_1, I) → p_3
  113. Circle(F, M_1) → q_3
  114. Intersect(q_3, i, 1) → R_1
  115. Segment(R_1, I) → r_3
  116. Circle(A, M) → s_3
  117. Intersect(i, j) → S_1
  118. Intersect(i, j) → T_1
  119. Intersect(s_3, i, 1) → U_1
  120. AngularBisector(M, A, U_1) → t_3
  121. AngularBisector(P_1, J, M) → a_3
  122. Intersect(t_3, a_3) → V_1
  123. Segment(U_1, V_1) → b_3
  124. Segment(V_1, P_1) → f_4
  125. Segment(V_1, M) → g_4
  126. AngularBisector(J_1, I, L) → h_4
  127. Intersect(h_4, s_1) → W_1
  128. Segment(W_1, J_1) → i_4
  129. Circle(D, L) → c_4
  130. Intersect(c_4, h, 1) → Z_1
  131. Segment(W_1, Z_1) → j_4
  132. AngularBisector(N, K, F'_1) → k_4
  133. Mirror(Z_1, F'_1) → A_2
  134. Segment(A_2, K) → l_4
  135. AngularBisector(C, B, A) → m_4
  136. Point(m_4) → B_2
  137. Circle(B, B_2) → d_4
  138. Segment(B, B_2 + Vector[B, B_2]) → b_4
  139. Intersect(d_4, f, 1) → C_2
  140. Circle(C_2, B) → e_4
  141. Intersect(e_4, f, 2) → D_2
  142. Ray(D_2, B_2) → n_4
  143. Intersect(g, n_4) → E_2
  144. Segment(D_2, E_2) → p_4
  145. Circle(E_2, ll) → q_4
  146. Intersect(q_4, g, 1) → F_2
  147. Circle(F_2, ll) → r_4
  148. Intersect(r_4, g, 2) → G_2
  149. Segment(D_2, G_2) → s_4
  150. Segment(H_2, F') → a_4
  151. Line(C, s_4) → t_4
  152. Polygon(F, F'_1, F'_2, LL) → vierhoek1, f_5, f'_1, f'_2, g_5

AI Drawing Prompt Reference

Start with a square centered at the origin. Take a point on one side and rotate it by plus and minus 90 degrees around the center, then mirror it across the center line, creating three symmetric points. Connect these points to form an inner star-like polygon. Then draw perpendicular lines from certain points to opposite sides, find their intersections, and connect these intersections to create a symmetric Islamic geometric pattern. The final figure consists of the square outline and internal symmetric line segments, emphasizing rotational and mirror symmetry.
The prompt above was compiled from the original drawing commands and is for reference only.