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Baravelle Spiral Printing

Baravelle Spiral Printing
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2018-12-27 21:35
CircleTransformationGeometric Construction
This example demonstrates the construction of a Baravelle spiral using dynamic geometry tools, featuring a sequence of circles and triangles that exhibit geometric transformations and rotational symmetry. Core math concepts include geometric transformations, rotation, similar triangles, sequences, and coordinate geometry. The construction utilizes GeoGebra commands like Intersect, Point, Polygon, and Circle. Suitable for high school and advanced learners to explore complex geometric patterns and enhance spatial reasoning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  4. Point(f) → E
  5. Circle(B, Distance[A, E]) → c
  6. Intersect(c, g, 1) → F
  7. Polygon(E, F, 4) → poly2, j, k, l, m, G, H
  8. Circle(E, (Distance[A, E] / Distance[A, B] * Distance[E, F])) → d
  9. Intersect(d, j, 1) → I
  10. Circle(F, Radius[d]) → e
  11. Intersect(e, k, 1) → J
  12. Polygon(I, J, 4) → poly3, n, p, q, r, K, L
  13. Circle(I, (Distance[A, E] / Distance[A, B] * Distance[I, J])) → s
  14. Intersect(poly1, poly2) → M, N, O, P, Q, R, S, T
  15. Intersect(s, n) → U,
  16. Circle(J, (Distance[A, E] / Distance[A, B] * Distance[I, J])) → t
  17. Intersect(t, p) → V,
  18. Polygon(U, V, 4) → poly4, a, b, f_1, g_1, W, Z
  19. Circle(U, (Distance[A, E] / Distance[A, B] * Distance[U, V])) → c_1
  20. Intersect(c_1, a) → A_1,
  21. Circle(V, Radius[c_1]) → d_1
  22. Intersect(d_1, b, 1) → B_1
  23. Polygon(A_1, B_1, 4) → poly5, h_1, i_1, j_1, k_1, C_1, D_1
  24. Circle(A_1, (Distance[A, E] / Distance[A, B] * Distance[A_1, B_1])) → e_1
  25. Intersect(e_1, h_1) → E_1,
  26. Circle(B_1, Radius[e_1]) → p_1
  27. Intersect(p_1, i_1, 1) → F_1
  28. Polygon(E_1, F_1, 4) → poly6, l_1, m_1, n_1, q_1, G_1, H_1
  29. Circle(E_1, (Distance[A, E] / Distance[A, B] * Distance[E_1, F_1])) → r_1
  30. Circle(F_1, Radius[r_1]) → s_1
  31. Intersect(r_1, m_1) → I_1,
  32. Intersect(s_1, m_1) → J_1,
  33. Intersect(r_1, l_1) → K_1,
  34. Polygon(K_1, J_1, 4) → poly7, t_1, a_1, b_1, f_2, L_1, M_1
  35. Circle(K_1, (Distance[A, E] / Distance[A, B] * Distance[K_1, J_1])) → c_2
  36. Circle(J_1, (Distance[A, E] / Distance[A, B] * Distance[K_1, J_1])) → d_2
  37. Intersect(c_2, t_1, 1) → O_1
  38. Intersect(d_2, a_1, 1) → P_1
  39. Polygon(O_1, P_1, 4) → poly8, g_2, h_2, i_2, j_2, Q_1, R_1
  40. Circle(O_1, (Distance[A, E] / Distance[A, B] * Distance[O_1, P_1])) → e_2
  41. Circle(P_1, (Distance[A, E] / Distance[A, B] * Distance[O_1, P_1])) → k_2
  42. Intersect(k_2, h_2) → N_1,
  43. Intersect(e_2, g_2) → S_1,
  44. Polygon(S_1, N_1, 4) → poly9, l_2, m_2, n_2, p_2, T_1, U_1
  45. Circle(S_1, (Distance[A, E] / Distance[A, B] * Distance[S_1, N_1])) → q_2
  46. Circle(N_1, (Distance[A, E] / Distance[A, B] * Distance[S_1, N_1])) → r_2
  47. Intersect(r_2, m_2) → V_1,
  48. Intersect(q_2, l_2) → W_1,
  49. Polygon(W_1, V_1, 4) → poly10, s_2, t_2, a_2, b_2, Z_1, A_2
  50. Polygon(E, I, N_1, P_1, J_1, F_1, B_1, V, J, F, B, U, A_1, E_1, K_1, O_1, S_1, W_1, V_1) → poly11, e_3, v_1, k_3, l_3, m_3, n_3, p_3, v, j_4, f_4, b_3, i_3, u, f_3, g_3, h_3, o_1, j_3, w_1