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Decagons and Pentagons in Islamic Geometry

Decagons and Pentagons in Islamic Geometry
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2019-12-31 16:06
PolygonTransformationGeometric Construction
This example demonstrates Islamic geometric patterns using decagons and pentagons. Key mathematical concepts include symmetry of polygons. It uses commands like rotation, intersection, ray, and polygon to construct the pattern, suitable for middle school and above students learning geometric construction.

GeoGebra Commands & Parameters

  1. ConstructionStep() → stap
  2. Intersect(xAxis, yAxis) → A
  3. Point(xAxis) → B
  4. Rotate(B, 54°) → BB
  5. Ray(A, BB) → f
  6. OrthogonalLine(B, xAxis) → g
  7. Intersect(g, f) → C
  8. Polygon(A, B, C, D) → vierhoek1, a, b, c, d
  9. Centroid(vierhoek1) → E
  10. Segment(A, C) → h
  11. Segment(D, B) → i
  12. Angle(E, A, D) → α
  13. Angle(B, A, E) → β
  14. Midpoint(A, E) → F
  15. OrthogonalLine(F, h) → j
  16. Circle(F, G) → e
  17. Intersect(e, j, 2) → H
  18. Intersect(e, j, 1) → I
  19. Segment(H, I) → k
  20. Intersect(d, k) → J
  21. AngularBisector(E, A, D) → l
  22. Point(l) → K
  23. Segment(A, K) → m
  24. Intersect(k, m) → L
  25. AngularBisector(A, L, F) → n
  26. Point(n) → M
  27. Segment(L, M) → p
  28. Intersect(h, p) → N
  29. Circle(A, N) → q
  30. Circle(L, N) → r
  31. Intersect(q, r, 2) → O
  32. Polygon(O, N, 5) → veelhoek1, s, t, f_1, g_1, h_1, P, Q, R
  33. Polygon(P, N, 5) → veelhoek2, i_1, j_1, k_1, l_1, m_1, S, T, U
  34. Polygon(T, S, 5) → veelhoek3, n_1, p_1, q_1, r_1, s_1, V, W, Z
  35. Mirror(q, x = x(E)) → q'
  36. Mirror(q, y = y(E)) → q'_1
  37. Mirror(q, E) → q'_2
  38. Mirror(veelhoek1, x = x(E)) → veelhoek1'
  39. Mirror(veelhoek2, x = x(E)) → veelhoek2'
  40. Mirror(veelhoek3, x = x(E)) → veelhoek3'
  41. Mirror(veelhoek1, hor) → veelhoek1'_1
  42. Mirror(veelhoek2, hor) → veelhoek2'_1
  43. Mirror(veelhoek3, hor) → veelhoek3'_1
  44. Mirror(veelhoek1', hor) → veelhoek1''
  45. Mirror(veelhoek2', hor) → veelhoek2''
  46. Mirror(veelhoek3', hor) → veelhoek3''
  47. Polygon(P, U, 10) → veelhoek4, t_1, i_2, A_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1, a_1, b_1, c_1, d_1, e_1, f_2, g_2, h_2
  48. Intersect(a, p_1) → I_1
  49. Intersect(a, xAxis) → J_1
  50. Intersect(a, xAxis) → K_1
  51. Mirror(I_1, ver) → L_1
  52. Mirror(I_1, hor) → M_1
  53. Mirror(L_1, hor) → N_1
  54. Intersect(a, xAxis) → O_1
  55. Mirror(Z, ver) → Z'
  56. Mirror(Z, hor) → Z'_1
  57. Mirror(Z', hor) → Z''
  58. Mirror(T, hor) → T'
  59. Mirror(S, ver) → S'
  60. Mirror(S', hor) → S''
  61. Mirror(S, hor) → S'_1
  62. Segment(S, I_1) → j_2
  63. Segment(S', L_1) → k_2
  64. Segment(S'_1, M_1) → l_2
  65. Segment(S'', N_1) → m_2