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Fibonacci Numbers and the Golden Spiral

Fibonacci Numbers and the Golden Spiral
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2014-09-28 06:00
SequenceProbability & StatisticsGeometric Construction
In collections Logarithmic Spiral and Fibonacci Numbers
This applet demonstrates the Fibonacci sequence and the golden spiral, showcasing the beautiful connection between number sequences and geometric shapes. Key mathematical concepts include sequences and the golden ratio. Using GeoGebra commands such as Point, Polygon, and CircleArc, the Fibonacci squares and golden spiral are quickly constructed. Suitable for students in middle school and above to learn about sequences and geometric shapes.

GeoGebra Commands & Parameters

  1. Point(Circle[A, f]) → B
  2. Polygon(A, B, C, D) → poly1, a, b, c, d
  3. Polygon(Corner[1], (x(Corner[1]) + 0.3 * Distance[Corner[1], Corner[2]], y(Corner[1])), (x(Corner[4]) + 0.3 * Distance[Corner[1], Corner[2]], y(Corner[4])), Corner[4]) → poly10, i_2, j_2, k_2, l_2
  4. First(Fibonacci, n + 1) → list1
  5. Element(list1, n - 1) → last
  6. Element(list1, n) → beforeLast
  7. Polygon((-22, 3), (-22, 4), (-21, 4), (-21, 3)) → poly2, g, h, i, j
  8. CircleArc((-21, 4), (-22, 4), (-21, 3)) → e
  9. Polygon((-22, 4), (-21, 4), (-21, 5), (-22, 5)) → poly3, l, m, p, q
  10. CircleArc((-21, 4), (-21, 5), (-22, 4)) → k
  11. Polygon((-21, 5), (-21, 3), (-19, 3), (-19, 5)) → poly4, s, t, a_1, b_1
  12. CircleArc((-21, 3), (-19, 3), (-21, 5)) → r
  13. Polygon((-19, 3), (-22, 3.0000000000000004), (-22, 4.440892098500626E-16), (-19, 0)) → poly5, d_1, e_1, f_1, g_1
  14. CircleArc((-22, 3.0000000000000004), (-22, 4.440892098500626E-16), (-19, 3)) → c_1
  15. Polygon((-22, 0), (-22, 5), (-27, 5), (-27, 0)) → poly6, i_1, j_1, k_1, l_1
  16. CircleArc((-22, 5), (-27, 5), (-22, 0)) → h_1
  17. Polygon((-27, 5), (-19, 5), (-19, 13), (-27, 13)) → poly7, m_1, n_1, q_1, r_1
  18. CircleArc((-19, 5), (-19, 13), (-27, 5)) → p_1
  19. Polygon((-19, 13), (-19, 0), (-6, 0), (-6, 13)) → poly8, t_1, a_2, b_2, c_2
  20. CircleArc((-19, 0), (-6, 0), (-19, 13)) → s_1

AI Drawing Prompt Reference

Draw the Fibonacci sequence and the golden spiral. Start by drawing a series of squares with side lengths corresponding to the Fibonacci sequence. Then, draw arcs with the vertices of these squares as centers to form the golden spiral. Begin with a square of side length 1, and continue with squares of side lengths 1, 1, 2, 3, 5, 8, 13, and 21, drawing arcs at each vertex to create the spiral.
The prompt above was compiled from the original drawing commands and is for reference only.