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Theodorus (Pythagorean) Spiral: Irrational Lengths from Repeated Right Triangles

Theodorus (Pythagorean) Spiral: Irrational Lengths from Repeated Right Triangles
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2020-08-15 10:36
TriangleCircleAngleGeometric ConstructionLocus
This example demonstrates the construction of the Theodorus spiral (also called the square-root spiral or Pythagorean spiral), built step-by-step by attaching right triangles. The main phenomenon is that each new triangle introduces new lengths, and many of these lengths are irrational square roots (e.g., √2, √3, √5, …), producing a clear spiral locus. Key math concepts include the Pythagorean theorem, geometric construction with right triangles, square roots as lengths, and the idea of a locus generated by iterative constraints; it can also motivate connections to quadratic equations via “root” lengths. In GeoGebra, the spiral is formed by: constructing segments with Segment; defining required angles using Angle and creating perpendicular lines with OrthogonalLine; drawing circles with Circle; and using Intersect to compute intersection points that determine the next vertex. The listed objects (numerous labeled points such as A, B, C, D, E, F, G, H, I, together with angles and constructed lines) reflect this iterative build process and the resulting spiral curve. Suitable for secondary-school advanced geometry, enrichment lessons, and teaching geometric construction leading to discovery of irrational values.

GeoGebra Commands & Parameters

  1. Segment(A, C) → a
  2. Segment(C, B) → b
  3. Segment(B, A) → c
  4. Angle(A, C, B) → α
  5. OrthogonalLine(B, c) → d
  6. Circle(B, a) → e
  7. Intersect(e, d) → D, E_1
  8. Segment(A, D) → f
  9. Segment(B, D) → g
  10. Angle(D, B, A) → β
  11. OrthogonalLine(D, f) → h
  12. Circle(D, 1) → k
  13. Intersect(k, h, 2) → E
  14. Segment(E, D) → i
  15. Segment(E, A) → j
  16. Angle(E, D, A) → γ
  17. OrthogonalLine(E, j) → l
  18. Circle(E, 1) → p
  19. Intersect(p, l, 1) → F
  20. Segment(A, F) → m
  21. Segment(F, E) → n
  22. Angle(F, E, A) → δ
  23. OrthogonalLine(F, m) → q
  24. Circle(F, E) → r
  25. Intersect(r, q, 2) → G
  26. Segment(G, F) → s
  27. Segment(G, A) → t
  28. Angle(G, F, A) → ε
  29. OrthogonalLine(G, t) → a_1
  30. Circle(G, F) → c_1
  31. Intersect(c_1, a_1, 1) → H
  32. Segment(H, A) → b_1
  33. Segment(H, G) → d_1
  34. OrthogonalLine(H, b_1) → e_1
  35. Circle(H, G) → f_1
  36. Intersect(f_1, e_1, 1) → I
  37. Segment(H, I) → g_1
  38. Segment(I, A) → h_1
  39. OrthogonalLine(I, h_1) → i_1
  40. Circle(I, H) → k_1
  41. Intersect(k_1, i_1, 1) → J
  42. Segment(J, A) → j_1
  43. Segment(J, I) → l_1
  44. OrthogonalLine(J, j_1) → m_1
  45. Circle(J, I) → p_1
  46. Intersect(p_1, m_1, 1) → K
  47. Segment(K, A) → n_1
  48. Segment(J, K) → q_1
  49. OrthogonalLine(K, n_1) → r_1
  50. Circle(K, J) → s_1
  51. Intersect(s_1, r_1, 1) → L
  52. Segment(L, A) → t_1
  53. Segment(K, L) → a_2
  54. Angle(H, G, A) → ζ
  55. Angle(I, H, A) → η
  56. Angle(J, I, A) → θ
  57. Angle(K, J, A) → ι
  58. Angle(L, K, A) → κ
  59. OrthogonalLine(L, t_1) → b_2
  60. Circle(L, K) → c_2
  61. Intersect(c_2, b_2, 1) → M
  62. Segment(L, M) → d_2
  63. Segment(M, A) → e_2
  64. Angle(A, L, M) → λ

AI Drawing Prompt Reference

Please draw a spiral formed by repeatedly attaching right triangles. Start with a right triangle with legs of length 1 and 1, and hypotenuse √2. Next, attach a new right triangle using the previous hypotenuse as one leg, creating a triangle with hypotenuse √3. Continue this process, where each new triangle's hypotenuse becomes a leg of the next triangle, generating a spiral curve with increasing leg and hypotenuse lengths. The key elements include the starting point, vertices of each new triangle, and the spiral's endpoint. Ensure the spiral extends outward smoothly.
The prompt above was compiled from the original drawing commands and is for reference only.