MathPlotAI Logo MathPlotAI

Square Spiral Construction

Square Spiral Construction
Loading the math board and drawing, please wait…
2018-05-12 05:56
PolygonTrigonometryCalculus
In collections Homothety, Rotation Symmetry -RO
This example demonstrates the dynamic generation of a square spiral polygon. It involves core mathematical concepts such as polygon construction, midpoint calculation, geometric transformations (rotation and dilation), unit circle, calculus, and trigonometry. The construction uses commands like Polygon, Midpoint, Dilate, and Rotate in GeoGebra. Suitable for high school and above students to explore geometric transformations, parametric equations, and the application of calculus in geometry.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. Midpoint(A, C) → E
  3. Dilate(poly1, 5 / 6, E) → poly1'
  4. Rotate(poly1', (-(12°)), E) → poly1''
  5. Dilate(poly1'', 5 / 6, E) → poly1'''
  6. Rotate(poly1''', (-(12°)), E) → q1
  7. Dilate(q1, 5 / 6, E) → q1'
  8. Rotate(q1', (-(12°)), E) → q1''
  9. Dilate(q1'', 5 / 6, E) → q1'''
  10. Rotate(q1''', (-(13°)), E) → q2
  11. Dilate(q2, 5 / 6, E) → q2'
  12. Rotate(q2', (-(12°)), E) → q2''
  13. Dilate(q2'', 5 / 6, E) → q2'''
  14. Rotate(q2''', -12°, E) → q3
  15. Dilate(q3, 5 / 6, E) → q3'
  16. Rotate(q3', (-(12°)), E) → q3''
  17. Dilate(q3'', 5 / 6, E) → q3'''
  18. Rotate(q3''', (-(12°)), E) → q4
  19. Dilate(q4, 5 / 6, E) → q4'
  20. Rotate(q4', (-(12°)), E) → q4''
  21. Dilate(q4'', 5 / 6, E) → q4'''
  22. Rotate(q4''', (-(12°)), E) → q5
  23. Dilate(q5, 5 / 6, E) → q5'
  24. Rotate(q5', (-(12°)), E) → q5''
  25. Dilate(q5'', 5 / 6, E) → q5'''
  26. Rotate(q5''', (-(12°)), E) → q6
  27. Dilate(q6, 5 / 6, E) → q6'
  28. Rotate(q6', (-(12°)), E) → q6''
  29. Dilate(q6'', 5 / 6, E) → q6'''
  30. Rotate(q6''', (-(12°)), E) → q7
  31. Dilate(q7, 5 / 6, E) → q7'
  32. Rotate(q7', (-(12°)), E) → q7''

AI Drawing Prompt Reference

Start with a square. Take the midpoint of one of its diagonals as a fixed center. Shrink the entire square by a constant ratio and rotate it counterclockwise by a small fixed angle around that center, producing a smaller rotated square. Repeat this process of shrinking and rotating, with slight variation in the rotation angle at certain steps, so that successive squares nest inside each other and gradually turn, forming an inward square spiral. The whole figure consists of a sequence of similar squares, each smaller and rotated by the same small angle relative to the previous one, creating a dynamic geometric pattern.
The prompt above was compiled from the original drawing commands and is for reference only.