MathPlotAI Logo MathPlotAI

Fibonacci Spiral: Geometric Beauty of the Golden Ratio

Fibonacci Spiral: Geometric Beauty of the Golden Ratio
Loading the math board and drawing, please wait…
2020-05-31 21:24
CircleSequenceTransformationGeometric ConstructionSlider Animation
① This example dynamically constructs the classic Fibonacci spiral through a series of rotated squares and quarter-circle arcs, visually revealing the golden ratio in geometry. ② Core mathematical concepts include the Fibonacci sequence, golden ratio, relationship between square side lengths and arcs, and iterative thinking in sequences and transformations. ③ Construction highlights: using GeoGebra commands Rotate to rotate squares, Sequence to generate successive squares, and CircleArc to draw quarter-circle arcs connecting square vertices; objects include points A, B, A1–A10, B1–B10 and many arcs, building each segment of the spiral iteratively. ④ Suitable for upper secondary to early university students, for exploring connections between sequences and geometry, as a teaching example in mathematical art or programming visualization, and for understanding recursive construction and limit concepts.

GeoGebra Commands & Parameters

  1. Rotate(B1, 1.5707963267948966, A1) → D1
  2. Rotate(B2, 1.5707963267948966, A2) → D2
  3. Rotate(B3, 1.5707963267948966, A3) → D3
  4. Rotate(B4, 1.5707963267948966, A4) → D4
  5. Rotate(B5, 1.5707963267948966, A5) → D5
  6. Rotate(B6, 1.5707963267948966, A6) → D6
  7. Rotate(B7, 1.5707963267948966, A7) → D7
  8. Rotate(B8, 1.5707963267948966, A8) → D8
  9. Rotate(B9, 1.5707963267948966, A9) → D9
  10. Rotate(B10, 1.5707963267948966, A10) → D10
  11. Rotate(B11, 1.5707963267948966, A11) → D11
  12. Rotate(B12, 1.5707963267948966, A12) → D12
  13. Rotate(B13, 1.5707963267948966, A13) → D13
  14. Rotate(B14, 1.5707963267948966, A14) → D14
  15. Rotate(B15, 1.5707963267948966, A15) → D15
  16. Rotate(B16, 1.5707963267948966, A16) → D16
  17. Rotate(B17, 1.5707963267948966, A17) → D17
  18. Rotate(B18, 1.5707963267948966, A18) → D18
  19. Rotate(B19, 1.5707963267948966, A19) → D19
  20. Rotate(B20, 1.5707963267948966, A20) → D20
  21. Rotate(B21, 1.5707963267949, A21) → D21
  22. Rotate(B22, 1.5707963267949, A22) → D22
  23. Rotate(B23, 1.5707963267949, A23) → D23
  24. Rotate(B24, 1.5707963267949, A24) → D24
  25. Rotate(B25, 1.5707963267949, A25) → D25
  26. Rotate(B26, 1.5707963267949, A26) → D26
  27. Rotate(B27, 1.5707963267949, A27) → D27
  28. Rotate(B28, 1.5707963267949, A28) → D28
  29. Rotate(B29, 1.5707963267949, A29) → D29
  30. Rotate(B30, 1.5707963267949, A30) → D30
  31. Sequence(Polygon[Element[A1:A30, k], Element[B1:B30, k], 4], k, 1, m) → l1
  32. Sequence(CircleArc[Element[A1:A30, k], Element[B1:B30, k], Element[D1:D30, k]], k, 2, m) → list1
  33. CircleArc(C1, D1, B1) → c

AI Drawing Prompt Reference

Draw a Fibonacci spiral composed of a sequence of squares with side lengths following the Fibonacci sequence, each rotated 90 degrees and placed adjacent to the previous one. Starting from the initial square, draw quarter-circle arcs connecting the outer vertices of consecutive squares, forming a continuous smooth spiral curve. Generate all squares and arcs iteratively in order, revealing the geometric beauty of the golden ratio.
The prompt above was compiled from the original drawing commands and is for reference only.