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Explore Frieze Symmetry

Explore Frieze Symmetry
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2020-02-01 11:14
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This example explores frieze symmetry by controlling the gyration point. It demonstrates the concepts of rotation and symmetry, using commands like Circle, Rotate, and Polygon in GeoGebra. Suitable for middle school and above students learning geometric transformations.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Rotate(B, α, A) → B'
  3. Rotate(B, (-α), A) → B'_1
  4. Polygon(A, B', B, B'_1) → q1, a, b', b, b'_1
  5. Midpoint(a) → C
  6. Midpoint(b) → D
  7. Midpoint(A, C) → E
  8. Midpoint(C, B') → F
  9. Midpoint(B'_1, D) → G
  10. Midpoint(D, B) → H
  11. Polygon(E, B'_1, G, H) → q2, e, f, g, h
  12. Rotate(A, 180°, I) → A'
  13. Rotate(B', 180°, I) → B''
  14. Rotate(B, 180°, I) → B'_2
  15. Rotate(B'_1, 180°, I) → J
  16. Polygon(A', B'', B'_2, J) → q1', a', b'', b'_2, j
  17. Rotate(E, 180°, I) → E'
  18. Rotate(B'_1, 180°, I) → K
  19. Rotate(G, 180°, I) → G'
  20. Rotate(H, 180°, I) → H'
  21. Polygon(E', K, G', H') → q2', e', k, g', h'
  22. Circle(A_1, B_1) → c_1
  23. Rotate(B_1, α_1, A_1) → B'_3
  24. Rotate(B_1, (-α_1), A_1) → B'_4
  25. Polygon(A_1, B'_3, B_1, B'_4) → q1_1, a_1, b'_3, b_1, b'_4
  26. Midpoint(a_1) → C_1
  27. Midpoint(b_1) → D_1
  28. Midpoint(A_1, C_1) → E_1
  29. Midpoint(C_1, B'_3) → F_1
  30. Midpoint(B'_4, D_1) → G_1
  31. Midpoint(D_1, B_1) → H_1
  32. Polygon(E_1, B'_4, G_1, H_1) → q2_1, e_1, f_1, g_1, h_1
  33. Rotate(A_1, 180°, I_1) → A'_1
  34. Rotate(B'_3, 180°, I_1) → B''_1
  35. Rotate(B_1, 180°, I_1) → B'_5
  36. Rotate(B'_4, 180°, I_1) → J_1
  37. Polygon(A'_1, B''_1, B'_5, J_1) → q1'_1, a'_1, b''_1, b'_5, j_1
  38. Rotate(E_1, 180°, I_1) → E'_1
  39. Rotate(B'_4, 180°, I_1) → K_1
  40. Rotate(G_1, 180°, I_1) → G'_1
  41. Rotate(H_1, 180°, I_1) → H'_1
  42. Polygon(E'_1, K_1, G'_1, H'_1) → q2'_1, e'_1, k_1, g'_1, h'_1
  43. Vector(A, A + (spacing, 0)) → u
  44. Translate(I, u) → I'
  45. Translate(I_1, u) → L
  46. Translate(A, u) → A'_2
  47. Translate(B', u) → B''_2
  48. Translate(B, u) → B'_6
  49. Translate(B'_1, u) → M
  50. Polygon(A'_2, B''_2, B'_6, M) → q1'_2, a'_2, b''_2, b'_6, m_2
  51. Translate(A', u) → A''
  52. Translate(B'', u) → B'''
  53. Translate(B'_2, u) → N
  54. Translate(J, u) → J'
  55. Polygon(A'', B''', N, J') → q1'', a'', b''', n_2, j'
  56. Translate(A_1, u) → O
  57. Translate(B'_3, u) → P
  58. Translate(B_1, u) → Q
  59. Translate(B'_4, u) → R
  60. Polygon(O, P, Q, R) → q3, o, p, q, r
  61. Translate(A'_1, u) → S
  62. Translate(B''_1, u) → T
  63. Translate(B'_5, u) → U
  64. Translate(J_1, u) → V
  65. Polygon(S, T, U, V) → q4, s, t, u_1, v
  66. Translate(E, u) → E'_2
  67. Translate(B'_1, u) → W
  68. Translate(G, u) → G'_2
  69. Translate(H, u) → H'_2
  70. Polygon(E'_2, W, G'_2, H'_2) → q2'_2, e'_2, w, g'_2, h'_2
  71. Translate(E', u) → E''
  72. Translate(K, u) → K'
  73. Translate(G', u) → G''
  74. Translate(H', u) → H''
  75. Polygon(E'', K', G'', H'') → q2'', e'', k', g'', h''
  76. Translate(E_1, u) → Z
  77. Translate(B'_4, u) → L_1
  78. Translate(G_1, u) → M_1
  79. Translate(H_1, u) → N_1
  80. Polygon(Z, L_1, M_1, N_1) → q5, z_1, l_1, i, l
  81. Translate(E'_1, u) → O_1
  82. Translate(K_1, u) → P_1
  83. Translate(G'_1, u) → Q_1
  84. Translate(H'_1, u) → R_1
  85. Polygon(O_1, P_1, Q_1, R_1) → q6, o_1, p_1, q_1, r_1