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Square Spiral with Arithmetic Progression Side Lengths

Square Spiral with Arithmetic Progression Side Lengths
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2014-05-02 20:41
PolygonAngleSequenceTransformationGeometric Construction
① This example demonstrates a square spiral formed by segments whose lengths increase in an arithmetic progression, creating a nested spiral of polygons through successive rotations and extensions. ② Core mathematical concepts include arithmetic sequences, similarity transformations (rotation and translation), polygon construction, and proportional relationships of segment lengths, along with angles and vectors. ③ Construction highlights: Based on the construction summary, GeoGebra commands such as Intersect, Point, Segment, Rotate, Angle, Vector, Translate, and Polygon are used. Objects include points A, B, C, D, E and segments a, b, c, d, with rotation angles α, β, γ, etc., generating new vertices and segments to form the spiral. ④ Suitable for middle and high school students learning arithmetic sequences, geometric transformations, and dynamic geometry, as well as for exploring spiral patterns and their mathematical properties.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Segment(A, B) → a
  4. Point(xAxis) → C
  5. Segment(B, C) → b
  6. Point(xAxis) → D
  7. Segment(C, D) → c
  8. Point(xAxis) → E
  9. Segment(D, E) → d
  10. Rotate(A, (-α), B) → A'
  11. Angle(A, B, A') → β
  12. Rotate(B, 90° - α, C) → B'
  13. Angle(B', C, B) → γ
  14. Rotate(C, 180° - α, D) → C'
  15. Angle(C', D, C) → δ
  16. Rotate(D, 270° - α, E) → D'
  17. Angle(D, E, D') → ε
  18. Segment(C, B') → e
  19. Segment(B, A') → f
  20. Segment(D, C') → g
  21. Segment(E, D') → h
  22. Vector(B, B') → u
  23. Translate(A', u) → A''
  24. Segment(B', A'') → i
  25. Vector(C, C') → v
  26. Translate(B', v) → B''
  27. Translate(A'', v) → A'''
  28. Segment(C', B'') → j
  29. Segment(B'', A''') → k
  30. Vector(D, D') → w
  31. Translate(C', w) → C''
  32. Translate(B'', w) → B'''
  33. Translate(A''', w) → F
  34. Segment(D', C'') → l
  35. Segment(C'', B''') → m
  36. Segment(B''', F) → n
  37. Polygon(A', B, B', A'') → poly1, a', b_1, b', a''
  38. Polygon(C, C', B'', B') → poly2, c_1, c', b'', b'_1
  39. Polygon(C, A'', A''', C') → poly3, c_2, a''_1, a''', c'_1
  40. Polygon(D, D', C'', C') → poly4, d_1, d', c'', c'_2
  41. Polygon(D, D', B''', B'') → poly5, d_2, d'_1, b''', b''_1
  42. Polygon(D, D', F, A''') → poly6, d_3, d'_2, f_1, a'''_1

AI Drawing Prompt Reference

Draw a square spiral composed of segments whose lengths increase in an arithmetic progression, creating a nested spiral of polygons through successive rotations and extensions. Start from the intersection of the coordinate axes, draw a horizontal segment, then sequentially draw segments of lengths following an arithmetic progression. Rotate and extend these segments around their endpoints to form a spiral of nested polygons.
The prompt above was compiled from the original drawing commands and is for reference only.