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Fibonacci Spiral 3

Fibonacci Spiral 3
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2021-10-06 00:26
PolygonSequenceGeometric Construction
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This example demonstrates how to construct a Fibonacci spiral using squares and circular arcs. It involves core mathematical concepts such as the Fibonacci sequence, golden ratio, and geometric construction. Key GeoGebra commands/objects used include square, intersect, and circle arc. Suitable for high school mathematics teaching or exploration by math enthusiasts.

GeoGebra Commands & Parameters

  1. Kwadrat(A, B) → q1, f, g, h, i, C, D
  2. Kwadrat(B, E) → q2, j, k, l, m, F, G
  3. Kwadrat(C, G) → q3, n, p, q, r, H, I
  4. Intersect(i, f) → K
  5. Kwadrat(J, K) → q4, s, t, a, b, L, M
  6. Intersect(l, j) → N
  7. Kwadrat(N, J) → q5, c, d, e, f_1, O, P
  8. Intersect(q, p) → Q
  9. Kwadrat(Q, O) → q6, g_1, h_1, i_1, j_1, R, S
  10. Kwadrat(L, R) → q7, k_1, l_1, m_1, n_1, T, U
  11. Kwadrat(P, T) → q8, p_1, q_1, r_1, s_1, V, W
  12. Kwadrat(S, V) → q9, t_1, a_1, b_1, c_1, Z, A_1
  13. Kwadrat(U, Z) → q10, d_1, e_1, f_2, g_2, B_1, C_1
  14. Kwadrat(W, B_1) → q11, h_2, i_2, j_2, k_2, D_1, E_1
  15. Kwadrat(A_1, D_1) → q12, l_2, m_2, n_2, p_2, F_1, G_1
  16. CircleArc(D, C, B) → c_2
  17. CircleArc(D, B, G) → d_2
  18. CircleArc(C, G, H) → e_2
  19. CircleArc(A, H, J) → q_2
  20. CircleArc(E, J, O) → r_2
  21. CircleArc(I, O, R) → s_2
  22. CircleArc(L, R, T) → t_2
  23. CircleArc(P, T, V) → c_3
  24. CircleArc(S, V, Z) → d_3
  25. CircleArc(U, Z, B_1) → e_3
  26. CircleArc(W, B_1, D_1) → f_3
  27. CircleArc(A_1, D_1, F_1) → g_3

AI Drawing Prompt Reference

Start with two adjacent small squares, then successively construct a series of squares with increasing side lengths, each sharing a side with the previous one and using intersection points of neighboring squares to determine the next vertex, forming a spiral arrangement of squares. After that, draw quarter-circle arcs on the outer side of each square, centered at a square's vertex with radius equal to the side length, connecting the arcs end-to-end to create a smooth Fibonacci spiral. The final figure consists of nested squares and a continuous spiral curve, illustrating the golden ratio and the beauty of the sequence.
The prompt above was compiled from the original drawing commands and is for reference only.