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Fibonacci Spiral

Fibonacci Spiral
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2020-12-10 22:58
TrigonometrySequenceCoordinate GeometryParametric EquationGraphing
This example demonstrates the Fibonacci spiral phenomenon commonly observed in plant growth, showcasing natural patterns through mathematical models. Core math concepts include parametric equations, trigonometric functions, sequences, and coordinate geometry. Construction highlights the use of GeoGebra commands like Sequence and CurveCartesian, along with objects such as a, b, c, A, t, r. Suitable for high school and above to understand the application of parametric equations in natural phenomena.

GeoGebra Commands & Parameters

  1. Sequence((x(A) + ((a * i) * cos(i)), y(A) + ((a * i) * sin(i))), i, 0, (b * 3.141592653589793), c) → liste1
  2. Sequence(Segment[Element[liste1, i], Element[liste1, i + 1]], i, 0, 200) → liste2
  3. Sequence((x(A) + ((a * i) * cos(i + 3.141592653589793 / 2)), y(A) + ((a * i) * sin(i + 3.141592653589793 / 2))), i, 0, (b * 3.141592653589793), c) → liste3
  4. Sequence((x(A) + ((a * i) * cos(i + (3 * 3.141592653589793 / 2))), y(A) + ((a * i) * sin(i + (3 * 3.141592653589793 / 2)))), i, 0, (b * 3.141592653589793), c) → liste4
  5. Sequence((x(A) + ((a * i) * cos(i + 3.141592653589793)), y(A) + ((a * i) * sin(i + 3.141592653589793))), i, 0, (b * 3.141592653589793), c) → liste5
  6. CurveCartesian(x(A) + ((a * i) * cos(i)), y(A) + ((a * i) * sin(i)), i, 0, (b * 3.141592653589793)) → d
  7. CurveCartesian(x(A) + ((a * i) * cos(i + 3.141592653589793)), y(A) + ((a * i) * sin(i + 3.141592653589793)), i, 0, (b * 3.141592653589793)) → e
  8. CurveCartesian(x(A) + ((a * i) * cos(i + 3.141592653589793 / 2)), y(A) + ((a * i) * sin(i + 3.141592653589793 / 2)), i, 0, (b * 3.141592653589793)) → f
  9. CurveCartesian(x(A) + ((a * i) * cos(i + (3 * 3.141592653589793 / 2))), y(A) + ((a * i) * sin(i + (3 * 3.141592653589793 / 2))), i, 0, (b * 3.141592653589793)) → g
  10. Sequence(Sphere[Element[liste1, i], (i * r / h)], i, 1, Length[liste1]) → l1
  11. Sequence(Sphere[Element[liste3, i], (i * r / h)], i, 1, Length[liste3]) → l2
  12. Sequence(Sphere[Element[liste4, i], (i * r / h)], i, 1, Length[liste4]) → l3
  13. Sequence(Sphere[Element[liste5, i], (i * r / h)], i, 1, Length[liste5]) → l4

AI Drawing Prompt Reference

Draw a Fibonacci spiral using parametric equations and trigonometric functions, starting from point A, with the radius gradually increasing to form a natural growth pattern. The spiral angles increase sequentially by π/2, 3π/2, π, and 3π, forming four spiral lines, each composed of a series of points whose size is proportional to their position on the spiral.
The prompt above was compiled from the original drawing commands and is for reference only.