Curve Stitching the Trefoil Knot

Curve Stitching the Trefoil Knot
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2022-12-22 11:23
This example demonstrates how to stitch curves into a trefoil knot using geometric constructions. It involves concepts like locus, circle properties, and geometric transformations. The construction uses GeoGebra commands such as Polygon, Circle, Intersect, and LineBisector to create shapes and intersections. Suitable for high school students or college students learning geometry and curve properties.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. Distance(A, B) → d
  3. Segment(A, C) → i
  4. Circle(A, t) → c
  5. Circle(A, (t * 2)) → e
  6. Circle(A, (t * 3)) → k
  7. Circle(A, (t * 4)) → p
  8. Circle(A, (t * 5)) → q
  9. Circle(A, (t * 6)) → r
  10. Circle(A, (t * 7)) → s
  11. Circle(A, (t * 8)) → c_1
  12. Circle(A, (t * 9)) → d_1
  13. Circle(A, (t * 10)) → e_1
  14. Circle(A, (t * 11)) → f_1
  15. Circle(A, (t * 12)) → g_1
  16. Circle(A, (t * 13)) → h_1
  17. Circle(A, (t * 14)) → k_1
  18. Circle(A, (t * 15)) → p_1
  19. Circle(A, (t * 16)) → q_1
  20. Circle(A, (t * 17)) → r_1
  21. Circle(A, (t * 18)) → s_1
  22. Circle(A, (t * 19)) → t_1
  23. Circle(A, (t * 20)) → c_2
  24. Intersect(c, i, 1) → D
  25. Intersect(e, i, 1) → E
  26. Intersect(k, i, 1) → F
  27. Intersect(p, i, 1) → G
  28. Intersect(q, i, 1) → H
  29. Intersect(r, i, 1) → I
  30. Intersect(s, i, 1) → J
  31. Intersect(c_1, i, 1) → K
  32. Intersect(d_1, i, 1) → L
  33. Intersect(e_1, i, 1) → M
  34. Intersect(f_1, i, 1) → N
  35. Intersect(g_1, i, 1) → O
  36. Intersect(h_1, i, 1) → P
  37. Intersect(k_1, i, 1) → Q
  38. Intersect(p_1, i, 1) → R
  39. Intersect(q_1, i, 1) → S
  40. Intersect(r_1, i, 1) → T
  41. Intersect(s_1, i, 1) → U
  42. Intersect(t_1, i, 1) → V
  43. LineBisector(g) → j
  44. Mirror(V, j) → V'
  45. Mirror(U, j) → U'
  46. Mirror(T, j) → T'
  47. Mirror(S, j) → S'
  48. Mirror(R, j) → R'
  49. Mirror(Q, j) → Q'
  50. Mirror(P, j) → P'
  51. Mirror(O, j) → O'
  52. Mirror(N, j) → N'
  53. Mirror(M, j) → M'
  54. Mirror(L, j) → L'
  55. Mirror(K, j) → K'
  56. Mirror(J, j) → J'
  57. Mirror(I, j) → I'
  58. Mirror(H, j) → H'
  59. Mirror(G, j) → G'
  60. Mirror(F, j) → F'
  61. Mirror(E, j) → E'
  62. Mirror(D, j) → D'
  63. LineBisector(A, B) → l
  64. Mirror(V, l) → V'_{1}
  65. Mirror(U, l) → U'_{1}
  66. Mirror(T, l) → T'_{1}
  67. Mirror(S, l) → S'_{1}
  68. Mirror(R, l) → R'_{1}
  69. Mirror(Q, l) → Q'_{1}
  70. Mirror(P, l) → P'_{1}
  71. Mirror(O, l) → O'_{1}
  72. Mirror(O, l) → O'_{2}
  73. Mirror(N, l) → N'_{1}
  74. Mirror(L, l) → L'_{1}
  75. Mirror(M, l) → M'_{1}
  76. Mirror(K, l) → K'_{1}
  77. Mirror(J, l) → J'_{1}
  78. Mirror(I, l) → I'_{1}
  79. Mirror(H, l) → H'_{1}
  80. Mirror(G, l) → G'_{1}
  81. Mirror(F, l) → F'_{1}
  82. Mirror(E, l) → E'_{1}
  83. Mirror(D, l) → D'_{1}
  84. Segment(V, D') → m
  85. Segment(D', D'_{1}) → n
  86. Segment(D'_{1}, V) → a
  87. Segment(U, E') → b
  88. Segment(E', E'_{1}) → i_1
  89. Segment(E'_{1}, U) → j_1
  90. Segment(T, F') → l_1
  91. Segment(F', F'_{1}) → m_1
  92. Segment(F'_{1}, T) → n_1
  93. Segment(S, G') → a_1
  94. Segment(G', G'_{1}) → b_1
  95. Segment(G'_{1}, S) → f_2
  96. Segment(R, H') → g_2
  97. Segment(H', H'_{1}) → h_2
  98. Segment(H'_{1}, R) → i_2
  99. Segment(Q, I') → j_2
  100. Segment(I', I'_{1}) → k_2
  101. Segment(I'_{1}, Q) → l_2
  102. Segment(P, J') → m_2
  103. Segment(J', J'_{1}) → n_2
  104. Segment(J'_{1}, P) → p_2
  105. Segment(O, K') → q_2
  106. Segment(K', K'_{1}) → r_2
  107. Segment(K'_{1}, O) → s_2
  108. Segment(N, L') → t_2
  109. Segment(L', L'_{1}) → a_2
  110. Segment(L'_{1}, N) → b_2
  111. Segment(M, M') → d_2
  112. Segment(M', M'_{1}) → e_2
  113. Segment(M'_{1}, M) → f_3
  114. Segment(L, N') → g_3
  115. Segment(N', N'_{1}) → h_3
  116. Segment(L, N'_{1}) → i_3
  117. Segment(K, O') → j_3
  118. Segment(O', O'_{1}) → k_3
  119. Segment(O'_{1}, K) → l_3
  120. Segment(J, P') → m_3
  121. Segment(P', P'_{1}) → n_3
  122. Segment(P'_{1}, J) → p_3
  123. Segment(I, Q') → q_3
  124. Segment(Q', Q'_{1}) → r_3
  125. Segment(Q'_{1}, I) → s_3
  126. Segment(H, R') → t_3
  127. Segment(R', R'_{1}) → a_3
  128. Segment(R'_{1}, H) → b_3
  129. Segment(G, S') → c_3
  130. Segment(S', S'_{1}) → d_3
  131. Segment(S'_{1}, G) → e_3
  132. Segment(F, T') → f_4
  133. Segment(T', T'_{1}) → g_4
  134. Segment(T'_{1}, F) → h_4
  135. Segment(E, U') → i_4
  136. Segment(U', U'_{1}) → j_4
  137. Segment(U'_{1}, E) → k_4
  138. Segment(D, V') → l_4
  139. Segment(V', V'_{1}) → m_4
  140. Segment(D, V'_{1}) → n_4
  141. Segment(C, A) → p_4
  142. Segment(A, B) → q_4
  143. Segment(B, C) → r_4

AI Drawing Prompt Reference

Draw a trefoil knot formed by stitching curves. The figure should include a polygon and multiple circles that intersect with the polygon to create intersection points. Then reflect these points symmetrically using mirror operations. The overall construction should reflect geometric principles, showing how curves connect to form the knot shape.
The prompt above was compiled from the original drawing commands and is for reference only.