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Fibonacci Spiral 4: Constructing the Golden Spiral with Squares and Arcs

Fibonacci Spiral 4: Constructing the Golden Spiral with Squares and Arcs
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2021-10-10 02:54
CirclePolygonSequenceTransformationGeometric Construction
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① This example demonstrates how to construct a Fibonacci spiral (golden spiral) by arranging a sequence of squares with increasing side lengths and connecting them with quarter-circle arcs, visually linking the Fibonacci sequence to a geometric spiral. ② Core mathematical concepts include the Fibonacci sequence, golden ratio, spiral geometry, recursive side-length growth, and smooth arc continuity. ③ Construction details: using GeoGebra commands such as Kwadrat (square) to create squares, CircleArc to draw quarter-circle arcs between square vertices, and Point/Intersect to define key points and intersections. Objects include A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, auxiliary lines q1–q5, and arcs f, g, h, i, j, k, l, m, n, p, q, r, s, t, a, b, c, d, e, f_1. ④ Suitable for middle school to undergraduate mathematics, this example is ideal for exploring the intersection of sequences and geometry, and can serve as a basis for art, design, or computational drawing projects, helping learners visualize the Fibonacci sequence in a geometric form.

GeoGebra Commands & Parameters

  1. Kwadrat(A, B) → q1, f, g, h, i, C, D
  2. Kwadrat(A, E) → q2, j, k, l, m, F, G
  3. Kwadrat(F, H) → q3, n, p, q, r, I, J
  4. Kwadrat(I, K) → q4, s, t, a, b, L, M
  5. Kwadrat(E, L) → q5, c, d, e, f_1, N, O
  6. Kwadrat(H, O) → q6, g_1, h_1, i_1, j_1, P, Q
  7. Kwadrat(K, Q) → q7, k_1, l_1, m_1, n_1, R, S
  8. CircleArc(B, C, A) → c_1
  9. CircleArc(B, A, F) → d_1
  10. CircleArc(C, F, I) → e_1
  11. CircleArc(D, I, L) → p_1
  12. CircleArc(E, L, O) → q_1
  13. CircleArc(H, O, Q) → r_1
  14. CircleArc(K, Q, S) → s_1
  15. Point(d) → T
  16. Intersect(s_1, m_1, 1) → U
  17. Kwadrat(T, U) → q8, t_1, a_1, b_1, f_2, V, W
  18. CircleArc(T, U, W) → c_2
  19. Point(h_1) → Z
  20. Intersect(c_2, b_1, 1) → A_1
  21. Kwadrat(Z, A_1) → q9, g_2, h_2, i_2, j_2, B_1, C_1
  22. CircleArc(Z, A_1, C_1) → d_2

AI Drawing Prompt Reference

Draw a Fibonacci golden spiral by arranging a sequence of squares with side lengths following the Fibonacci sequence. Start with a small square, then add larger squares to its right, top, left, and continue outward in a spiral pattern. In each square, draw a quarter-circle arc connecting two adjacent vertices. These arcs connect seamlessly to form a smooth spiral that grows outward from the center, creating the classic golden spiral shape.
The prompt above was compiled from the original drawing commands and is for reference only.