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Exploring Pi

Exploring Pi
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2015-07-07 00:09
CircleTrigonometryGeometric ConstructionSlider AnimationGraphing
This example dynamically demonstrates the ratio of a circle's circumference to its diameter, visually showcasing the constancy and calculation of Pi. It involves core mathematical concepts like circle, rotation, dilation, Pythagorean theorem, combinatorics, and intersection calculations. Construction highlights the use of commands such as Circle, Point, Rotate, CircleArc, Intersect, Segment, Line, Vector, Translate, Polygon, and Sequence in GeoGebra. Suitable for high school mathematics education to understand basic circle properties and the practical application of Pi.

GeoGebra Commands & Parameters

  1. Circle(A, 1) → c
  2. Point(c) → B
  3. Rotate(B, (-a), A) → B'
  4. CircleArc(A, B, B') → d
  5. Intersect(xAxis, yAxis) → D
  6. Segment(D, B) → b
  7. Point(yAxis) → C
  8. Point(b) → E
  9. Circle(C, E) → e
  10. Point(c) → F
  11. Line(F, A) → f
  12. Point(f) → G
  13. Circle(A, 0.5) → g
  14. Segment(G, A) → h
  15. Rotate(f, (-a), A) → f'
  16. Intersect(g, f', 1) → H
  17. Intersect(g, f', 2) → I
  18. Segment(H, A) → i
  19. Segment(I, A) → j
  20. Line(H, xAxis) → k
  21. Line(I, xAxis) → l
  22. Point(k) → K
  23. Point(k) → M
  24. Point(l) → N
  25. Point(l) → O
  26. Segment(O, N) → m
  27. Segment(M, K) → n
  28. Circle(M, 1.2) → q
  29. Vector((0, 0), (a, 0)) → u
  30. Translate(L, u) → L'
  31. Circle(L', 1.2) → r
  32. Intersect(q, r, 2) → P
  33. Segment(M, P) → p
  34. Segment(L', P) → s
  35. Circle(O, 1.2) → t
  36. Intersect(r, t, 1) → Q
  37. Segment(O, Q) → a_1
  38. Segment(L', Q) → b_1
  39. Translate(R, u) → R'
  40. Circle(R', 0.2) → d_1
  41. Segment(R', L') → c_1
  42. Point(c_1) → S
  43. Translate(T, u) → T'
  44. Translate(U, u) → U'
  45. Segment(S, T') → e_1
  46. Segment(S, U') → f_1
  47. Translate(W, u) → W'
  48. Translate(V, u) → V'
  49. Segment(T', W') → g_1
  50. Segment(U', V') → h_1
  51. Rotate(S, a / 4, S) → S'
  52. Rotate(S', sin(a), S) → S''
  53. Polygon(S, W', T') → poly1, t', s_1, w'
  54. Rotate(S, sin((2 * a)), S'') → S'_1
  55. Rotate(S, sin(a), S'') → S'_2
  56. Rotate(W', sin(a), S'') → W''_1
  57. Rotate(T', sin(a), S'') → T''_1
  58. Polygon(S'_2, W''_1, T''_1) → poly1', t''_1, s'_2, w''_1
  59. Polygon(V', U', S) → poly2, s_2, v', u'
  60. Rotate(V', (-sin(a)), S'') → V''
  61. Rotate(U', (-sin(a)), S'') → U''
  62. Rotate(S, (-sin(a)), S'') → S'_3
  63. Polygon(V'', U'', S'_3) → poly2', s'_3, v'', u''
  64. Segment(W''_1, T''_1) → i_1
  65. Segment(T''_1, S) → j_1
  66. Segment(S, U'') → k_1
  67. Segment(V'', U'') → l_1
  68. Segment(U'', V'') → m_1
  69. Point(p) → Z
  70. Polygon(K, M, Z) → poly3, z_1, k_2, m_2
  71. Rotate(S, (0.5 * sin((2 * a))), S'') → S'_4
  72. Rotate(W', (0.5 * sin((2 * a))), S'') → W''
  73. Rotate(T', (0.5 * sin((2 * a))), S'') → T''
  74. Polygon(S'_4, W'', T'') → poly1'_1, t'', s'_4, w''
  75. Point(a_1) → A_1
  76. Polygon(O, N, A_1) → poly4, a_2, o, n_1
  77. Point(c) → B_1
  78. Sequence(Rotate[B_1, (π / 24 * n), A], n, 1, 49) → list1
  79. Sequence(Segment[Element[list1, n], A], n, 1, 49) → list2
  80. Sequence(Rotate[Element[list2, n], (-a), A], n, 1, 49) → list3
  81. Circle((0, 1), 1) → p_1
  82. Point(xAxis) → D_1
  83. Segment(C_1, D_1) → q_1
  84. Point(d_1) → E_1

AI Drawing Prompt Reference

Draw a circle and mark its center and diameter. Mark two points on the diameter, rotate one point, and draw the arc formed. Find the intersection point of the coordinate axes and connect it to one of the points on the circle. Mark another point on the circle, connect it to the center, and draw the perpendicular line. Draw a circle with the center as the origin and half the radius of the original circle. Find the intersection points of this circle with the perpendicular line. Connect the intersection points to the center. Draw a line segment between two points on the perpendicular line. Draw a larger circle with one of the intersection points as the center. Draw a vector from the original center and translate the original circle along this vector to draw a new circle. Find the intersection points of the two circles and connect each intersection point to its corresponding center. Repeat this process for the other intersection point. Draw a larger circle with one of the intersection points as the center. Find the intersection points of this new circle with the translated circle and connect each intersection point to its corresponding center. Repeat this process for the other intersection point.
The prompt above was compiled from the original drawing commands and is for reference only.