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Logarithmic Spiral in the Case of a Square

Logarithmic Spiral in the Case of a Square
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2020-08-03 03:48
FunctionTransformationParametric EquationGeometric ConstructionLocus
This example demonstrates the construction of a logarithmic spiral within a square framework, showing how the spiral emerges through rotations, reflections, and tangents. The core mathematical concepts include logarithmic and exponential functions, parametric equations of spirals, rotation and reflection transformations, as well as tangent lines and angle bisectors. The construction uses GeoGebra commands such as Intersect, Circle, Segment, Polygon, Rotate, CurveCartesian, LineBisector, Mirror, Point, Tangent, and Vector, involving objects like squares, circles, segments, points, tangents, and the spiral curve itself. It is suitable for high school or college students and teachers exploring logarithmic functions, spiral geometry, and dynamic geometric constructions, offering an interactive way to investigate the mathematical properties of logarithmic spirals and their transformations.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, a) → c
  3. Intersect(c, xAxis, 2) → B
  4. Circle(B, d) → e
  5. Intersect(e, c) → C, D, ,
  6. Segment(B, D) → f
  7. Segment(B, C) → g
  8. Circle(C, d) → h
  9. Intersect(h, c, 1) → E
  10. Segment(C, E) → i
  11. Segment(E, D) → j
  12. Polygon(D, E, C, B) → q1, d_1, e_1, c_1, b_1
  13. Rotate(D, (-(45°)), A) → D'
  14. Rotate(E, (-(45°)), A) → E'
  15. Rotate(C, (-(45°)), A) → C'
  16. Rotate(B, (-(45°)), A) → B'
  17. Polygon(D', E', C', B') → q1', d', e', c', b'
  18. CurveCartesian(((a * ℯ^((-θ))); θ), θ, 0, (4 * π)) → b
  19. Rotate(b, (-(45°)), A) → b'_1
  20. LineBisector(b') → k
  21. Mirror(b'_1, k) → l
  22. Mirror(l, p) → l'
  23. Mirror(l, k) → l'_1
  24. LineBisector(d') → m
  25. LineBisector(e') → n
  26. Mirror(l', m) → l''
  27. Mirror(l'', q) → l'''
  28. Mirror(n, m) → n'
  29. Mirror(Mirror[Mirror[l'', n], n'], n') → o
  30. Mirror(l''', q) → r
  31. Mirror(r, n') → r'
  32. Mirror(r, s) → r'_1
  33. Point(o) → F
  34. Point(b'_1) → G
  35. Point(l') → H
  36. Point(r'_1) → I
  37. Polygon(H, I, F, G) → q2, h_1, i_1, f_1, g_1
  38. Tangent(F, r') → t
  39. Tangent(G, b'_1) → j_1
  40. Tangent(H, l') → k_1
  41. Tangent(I, r'_1) → l_1
  42. Circle(F, 1) → p_1
  43. Intersect(p_1, t) → J, K
  44. Circle(G, 1) → q_1
  45. Circle(H, 1) → r_1
  46. Circle(I, 1) → s_1
  47. Intersect(s_1, l_1, 2) → L
  48. Intersect(r_1, k_1, 2) → P
  49. Intersect(q_1, j_1, 2) → M
  50. Vector(G, M) → v
  51. Vector(H, P) → w
  52. Vector(I, L) → u_1
  53. Vector(F, J) → u

AI Drawing Prompt Reference

Draw a logarithmic spiral within a square, formed by rotations, reflections, and tangents. Start with a square and two concentric circles, connecting their intersections. Rotate square vertices by 45 degrees and join them. Plot the logarithmic spiral. Use tangent lines and angle bisectors to reflect parts of the shape, creating a symmetric, intricate design that highlights the spiral's geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.