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Rotational Symmetry: Dynamic Geometric Demonstration

Rotational Symmetry: Dynamic Geometric Demonstration
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2018-01-16 21:13
CirclePolygonAngleTransformationGeometric Construction
① This GeoGebra example dynamically demonstrates a geometric figure with rotational symmetry, allowing users to observe how the shape coincides with itself after rotation about a center. ② Core mathematical concepts include rotational symmetry, symmetry transformations, angle measurement, and combinations of polygons and circles, involving rotation centers, rotation angles, and symmetry axes. ③ Construction highlights: Based on the construction summary, commands such as Segment, OrthogonalLine, Point, Intersect, Polygon, Circle, Line, Rotate, Mirror, and Angle are used, creating multiple points, lines, circles, and polygons (e.g., A, B, poly1, c_1) and building symmetric structures through rotation and reflection operations. ④ Suitable for middle and high school geometry classes, this example is ideal for teaching rotational symmetry, symmetry of regular polygons, and transformation geometry, and can serve as a mathematical experiment or inquiry activity to help students grasp the mathematical essence of symmetry.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. OrthogonalLine(A, a) → b
  3. OrthogonalLine(B, a) → c
  4. Point(b) → C
  5. OrthogonalLine(C, b) → d
  6. Intersect(c, d) → D
  7. Polygon(C, D, B, A) → poly1, c_1, d_1, b_1, a_1
  8. Segment(C, B) → e
  9. Segment(D, A) → f
  10. Intersect(e, f) → E
  11. Polygon(F, G, 4) → poly2, f_1, g, h, i, H, I
  12. Segment(I, G) → j
  13. Segment(H, F) → k
  14. Intersect(j, k) → J
  15. Segment(K, L) → l
  16. Circle(K, L) → p
  17. Point(p) → M
  18. Segment(K, M) → m
  19. Line(M, l) → n
  20. Line(L, m) → q
  21. Intersect(n, q) → N
  22. Segment(M, L) → r
  23. Segment(N, K) → s
  24. Intersect(r, s) → O
  25. Polygon(M, N, L, K) → poly3, m_1, n_1, l_1, k_1
  26. Rotate(C, (-α), E) → C'
  27. Rotate(D, (-α), E) → D'
  28. Rotate(B, (-α), E) → B'
  29. Rotate(A, (-α), E) → A'
  30. Polygon(C', D', B', A') → Rectangle, c', d', b', a'
  31. Rotate(M, (-α), O) → M'
  32. Rotate(N, (-α), O) → N'
  33. Rotate(L, (-α), O) → L'
  34. Rotate(K, (-α), O) → K'
  35. Polygon(M', N', L', K') → Rhombus, m', n', l', k'
  36. Segment(P, Q) → t
  37. Segment(P, R) → e_1
  38. Line(Q, e_1) → g_1
  39. Line(R, t) → h_1
  40. Intersect(g_1, h_1) → S
  41. Polygon(P, R, S, Q) → poly4, p_1, r_1, s_1, q_1
  42. Segment(R, Q) → i_1
  43. Segment(S, P) → j_1
  44. Intersect(i_1, j_1) → T
  45. Rotate(P, (-α), T) → P'
  46. Rotate(R, (-α), T) → R'
  47. Rotate(S, (-α), T) → S'
  48. Rotate(Q, (-α), T) → Q'
  49. Polygon(P', R', S', Q') → Parallelogram, p', r', s', q'
  50. Segment(U, V) → t_1
  51. Segment(U, W) → a_2
  52. Line(W, t_1) → b_2
  53. Point(b_2) → Z
  54. Segment(Z, V) → c_2
  55. Polygon(U, V, Z, W) → poly5, u, v, z, w
  56. Segment(W, V) → d_2
  57. Segment(Z, U) → e_2
  58. Intersect(d_2, e_2) → A_1
  59. Rotate(U, (-α), A_1) → U'
  60. Rotate(V, (-α), A_1) → V'
  61. Rotate(Z, (-α), A_1) → Z'
  62. Rotate(W, (-α), A_1) → W'
  63. Polygon(U', V', Z', W') → Trapezoid, u', v', z', w'
  64. Line(B_1, C_1) → f_2
  65. Mirror(D_1, f_2) → E_1
  66. Polygon(D_1, B_1, E_1, C_1) → poly6, d_3, b_3, e_3, c_3
  67. Segment(D_1, E_1) → g_2
  68. Intersect(f_2, g_2) → F_1
  69. Rotate(D_1, (-α), F_1) → G_1
  70. Rotate(B_1, (-α), F_1) → H_1
  71. Rotate(E_1, (-α), F_1) → I_1
  72. Rotate(C_1, (-α), F_1) → J_1
  73. Polygon(G_1, H_1, I_1, J_1) → Kite, g_3, h_2, i_2, j_2
  74. Rotate(poly2, (-α), J) → Square
  75. Line(K_1, L_1) → k_2
  76. Circle(M_1, N_1) → Circle
  77. Rotate(N_1, α, M_1) → O_1
  78. Angle(N_1, M_1, O_1) → β
  79. Polygon(P_1, Q_1, 5) → poly7, l_2, m_2, n_2, p_2, q_2, R_1, S_1, T_1
  80. Line(P_1, T_1) → r_2
  81. Line(R_1, S_1) → s_2
  82. Line(S_1, T_1) → t_2
  83. Line(Q_1, P_1) → f_3
  84. Line(R_1, Q_1) → h_3
  85. Intersect(s_2, r_2) → U_1
  86. Intersect(t_2, f_3) → V_1
  87. Intersect(r_2, h_3) → W_1
  88. Intersect(f_3, s_2) → Z_1
  89. Intersect(h_3, t_2) → A_2
  90. Intersect(r_2, t_2) → B_2
  91. Intersect(f_3, r_2) → C_2
  92. Intersect(h_3, f_3) → D_2
  93. Intersect(s_2, h_3) → E_2
  94. Intersect(s_2, t_2) → F_2
  95. Polygon(U_1, B_2, V_1, C_2, W_1, D_2, Z_1, E_2, G_2, F_2) → poly8, u_1, f_4, b_4, v_1, c_4, w_1, d_4, z_1, e_4, g_4
  96. Line(H_2, I_2) → i_3
  97. Mirror(J_2, i_3) → O_2
  98. Mirror(K_2, i_3) → P_2
  99. Mirror(L_2, i_3) → Q_2
  100. Mirror(M_2, i_3) → R_2
  101. Mirror(N_2, i_3) → S_2
  102. Polygon(J_2, O_2, P_2, Q_2, R_2, S_2, N_2, M_2, L_2, T_2) → poly9, j_3, t_3, o_2, p_3, q_3, r_3, s_3, n_3, m_3, l_3

AI Drawing Prompt Reference

Draw a geometric figure with rotational symmetry, including a quadrilateral ABCD and a rhombus EFGH. First, connect points A and B, B and C, C and D, D and A to form quadrilateral ABCD. Then, construct the symmetric figure A'B'C'D' of quadrilateral ABCD with rotation center E and rotation angle -α. Next, construct a rhombus EFGH with diagonals intersecting at point O, and then construct the symmetric figure E'F'G'H' of rhombus EFGH with rotation center O and rotation angle -α. Finally, construct a quadrilateral PQRS with diagonals intersecting at point T, and then construct the symmetric figure P'Q'R'S' of quadrilateral PQRS with rotation center T and rotation angle -α.
The prompt above was compiled from the original drawing commands and is for reference only.