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Sunflower Seed Arrangement and Special Numbers Simulation

Sunflower Seed Arrangement and Special Numbers Simulation
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2016-07-15 19:14
PolygonAngleSequenceSlider AnimationGraphing
In collections Logarithmic Spiral and Fibonacci Numbers
① This applet dynamically simulates the growth and arrangement of seeds in a sunflower head, vividly demonstrating the Fibonacci spiral patterns found in nature. ② Core mathematical concepts include the golden angle, Fibonacci sequences, arithmetic sequences, angular distribution on the unit circle, least common multiple (LCM) and greatest common divisor (GCD), as well as expected value and other special numbers with geometric significance. ③ Construction highlights: using GeoGebra's Sequence command to generate seed point sequences, Textfield and InputBox for interactive parameter input, Segment and Angle to construct connecting lines and angles, sliders (α, β, δ) to control rotation angles and growth speed, lists (list1, list2) for data storage, and polygons (poly3, poly4) to aid structural visualization. ④ Suitable for middle school to university students exploring the intersection of mathematics and biology, and ideal for math lab activities or introductory programming, helping learners understand how angles and sequences generate natural forms.

GeoGebra Commands & Parameters

  1. Sequence(Vector[(0, 0), (((k / 690 * ℯ^(k / 460)) * cos((α * k))), ((k / 690 * ℯ^(k / 460)) * sin((k * α))))], k, 1, 460) → vectors
  2. Sequence(Point[(0, 0), Vector[Element[vectors, k]]], k, 1, 460) → points
  3. Sequence(seed[gr, β, Vector[Element[vectors, k]]], k, 1, 460) → seeds
  4. Sequence(Segment[Element[points, (m * i)], Element[points, (m * (i + 1))]], i, 1, 460) → commect_m
  5. Textfield(m) → InputBox1
  6. Textfield(p) → InputBox2
  7. Textfield(α) → InputBox3
  8. Sequence(Sequence[Segment[Element[points, (m * i) - st], Element[points, (m * (i + 1)) - st]], i, 1, 60], st, 1, m) → list1
  9. Sequence(Segment[Element[points, (p * i) - kk], Element[points, (p * (i + 1)) - kk]], i, 1, 460) → connect_k
  10. Sequence(Sequence[Segment[Element[points, (p * i) - st], Element[points, (p * (i + 1)) - st]], i, 1, 60], st, 1, p) → list2
  11. seed(gr, β, Vector[Element[vectors, (34 * n)]]) → poly3
  12. seed(gr, β, Vector[Element[vectors, (34 * n) + 1]]) → poly4
  13. Segment(Element[points, (34 * n)], (0, 0)) → k
  14. Segment((0, 0), Element[points, (34 * n) + 1]) → l
  15. Angle(Element[points, (34 * n)], (0, 0), Element[points, (34 * n) + 1]) → δ

AI Drawing Prompt Reference

Create a dynamic simulation of sunflower seed arrangement. Generate a spiral of seed points from the center, where each seed's position is determined by a rotation angle and a radius that grows exponentially with the seed index. Connect consecutive seeds with line segments to form spiral arms, with adjustable step intervals via input fields. Additionally, highlight specific seed regions using polygons, and draw line segments from these special seeds to the center, showing the angle between them. The overall figure illustrates Fibonacci spirals and the golden angle.
The prompt above was compiled from the original drawing commands and is for reference only.