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Obtuse Triangle Fractal Construction

Obtuse Triangle Fractal Construction
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2020-06-08 18:33
TriangleLimitTransformationGeometric Construction
① This example demonstrates how to generate a self-similar fractal pattern by repeatedly rotating and scaling the sides of an obtuse triangle. ② Core mathematical concepts include fractal geometry, similarity transformations (rotation and scaling), limits, rational number ratios, and relationships between interior and exterior angles of triangles. ③ Construction highlights: Based on the construction summary, GeoGebra commands such as Intersect, Point, Rotate, Angle, Line, Polygon are used, along with multiple auxiliary objects like points A, B, C, angle markers α, β, γ, lines f, g, h, and triangles t1 through t4, creating nested triangle sequences through iterative rotation. ④ Suitable for high school or university students studying fractal geometry, geometric transformations, and limits, also useful for math labs or inquiry-based learning to visualize infinite iteration and self-similarity.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Rotate(B, 30°, A) → B'
  4. Angle(B, A, B') → α
  5. Rotate(A, (-(30°)), B) → A'
  6. Angle(A', B, A) → β
  7. Line(A, B') → f
  8. Line(B, A') → g
  9. Intersect(f, g) → C
  10. Polygon(A, B, C) → t1, c, a, b
  11. Rotate(B, (-(30°)), C) → B'_1
  12. Angle(B'_1, C, B) → γ
  13. Line(C, B'_1) → h
  14. Rotate(A, 30°, C) → A'_1
  15. Angle(A, C, A'_1) → δ
  16. Line(C, A'_1) → i
  17. Intersect(c, xAxis) → D
  18. Intersect(i, c) → E
  19. Intersect(h, t1) → F, G
  20. Rotate(G, (-(30°)), C) → G'
  21. Angle(G', C, G) → ε
  22. Line(C, G') → j
  23. Rotate(C, (-(30°)), E) → C'
  24. Angle(C', E, C) → ζ
  25. Line(E, C') → k
  26. Intersect(k, j) → H
  27. Polygon(C, E, H) → t2, h_1, c_1, e
  28. Polygon(C, H, G) → t3, g_1, c_2, h_2
  29. Polygon(B, C, G) → t4, g_2, b_1, c_3
  30. Polygon(C, A, E) → t5, e_1, c_4, a_1
  31. fraccc(B, C, G) → t6, I, J, t7, t8, t9, K
  32. fraccc(C, A, E) → t10, L, M, t11, t12, t13, N
  33. fraccc(C, E, H) → t14, O, P, t15, t16, t17, Q
  34. fraccc(G, C, H) → t18, R, S, t19, t20, t21, T
  35. fraccc(G, B, K) → t22, U, V, t23, t24, t25, W
  36. fraccc(C, G, J) → t26, Z, A_1, t27, t28, t29, B_1
  37. fraccc(A, E, M) → t30, C_1, D_1, t31, t32, t33, E_1
  38. fraccc(E, M, D_1) → t34, F_1, G_1, t35, t36, t37, H_1
  39. fraccc(A, M, E_1) → t38, I_1, J_1, t39, t40, t41, K_1
  40. fraccc(M, A, E_1) → t42, L_1, M_1, t43, t44, t45, N_1
  41. fraccc(E, C, N) → t46, O_1, P_1, t47, t48, t49, Q_1
  42. fraccc(E, G, H) → t50, R_1, S_1, t51, t52, t53, T_1
  43. fraccc(G, H, S_1) → t54, U_1, V_1, t55, t56, t57, W_1
  44. fraccc(H, E, T_1) → t58, Z_1, A_2, t59, t60, t61, B_2
  45. fraccc(E, N, L) → t62, C_2, D_2, t63, t64, t65, E_2
  46. fraccc(M, E, L) → t66, F_2, G_2, t67, t68, t69, H_2
  47. fraccc(N, M, L) → t70, I_2, J_2, t71, t72, t73, K_2
  48. fraccc(C, N, Q) → t74, L_2, M_2, t75, t76, t77, N_2
  49. fraccc(K, J, I) → t78, O_2, P_2, t79, t80, t81, Q_2
  50. fraccc(K, G, W) → t82, R_2, S_2, t83, t84, t85, T_2
  51. fraccc(G, K, I) → t86, U_2, V_2, t87, t88, t89, W_2
  52. fraccc(J, G, I) → t90, Z_2, A_3, t91, t92, t93, B_3
  53. fraccc(G, J, T) → t94, C_3, D_3, t95, t96, t97, E_3
  54. fraccc(C, H, S) → t98, F_3, G_3, t99, t100, t101, H_3
  55. fraccc(J, C, S) → t102, I_3, J_3, t103, t104, t105, K_3
  56. fraccc(B, K, V) → t106, L_3, M_3, t107, t108, t109, N_3
  57. fraccc(H, G, T) → t110, O_3, P_3, t111, t112, t113, Q_3
  58. fraccc(E, H, P) → t114, R_3, S_3, t115, t116, t117, T_3
  59. fraccc(N, E, P) → t118, U_3, V_3, t119, t120, t121, W_3
  60. fraccc(H, C, Q) → t122, Z_3, A_4, t123, t124, t125, B_4

AI Drawing Prompt Reference

Draw an obtuse triangle and generate a self-similar fractal pattern by repeatedly rotating and scaling its sides. Start by plotting a point at the intersection of two axes and draw a horizontal line from it. Rotate one endpoint of the line segment around the intersection point by 30 degrees to create a new point, then connect these two points to form an obtuse triangle. Repeat this process for each side of the triangle, rotating the new line segments by -30 or 30 degrees around their endpoints and connecting the corresponding points to build nested triangles, gradually creating a complex fractal structure.
The prompt above was compiled from the original drawing commands and is for reference only.