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Square Knot

Square Knot
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2019-03-10 18:11
PolygonTransformationGeometric Construction
This example demonstrates the geometric construction of a square knot. It involves core math concepts like geometric construction, transformation, and polygon creation. Key GeoGebra commands/objects used include Circle, Rotate, Segment, Dilate, Intersect, Polygon, and Sequence. Suitable for high school mathematics teaching or exploration by geometry enthusiasts.

GeoGebra Commands & Parameters

  1. Circle((0, 0), abs(Z0)) → c
  2. Rotate(Z1, ((-π)) / 2, (0, 0)) → A
  3. Segment(Z1, A) → p1
  4. Dilate(Z0, sqrt(2) / 2, (0, 0)) → Z3
  5. Rotate(Z3, ((-π)) / 2, (0, 0)) → Z3Aux
  6. Segment(Z3, Z3Aux) → p2
  7. Intersect(p1, p2) → Z4
  8. Rotate(B, π / 2, (0, 0)) → C
  9. Rotate(Z1, π / 2, (0, 0)) → D
  10. Segment(Z0, D) → p3
  11. Segment(B, C) → p4
  12. Intersect(p3, p4) → Z5
  13. Polygon(Z0, Z1, Z4, Z3, Z5) → Shape, z0, z1, z4, z3, z5
  14. Rotate(Z0, t, (0, 0)) → Z0'
  15. Rotate(Z1, t, (0, 0)) → Z1'
  16. Rotate(Z4, t, (0, 0)) → Z4'
  17. Rotate(Z3, t, (0, 0)) → Z3'
  18. Rotate(Z5, t, (0, 0)) → Z5'
  19. Polygon(Z0', Z1', Z4', Z3', Z5') → Shape', z0', z1', z4', z3', z5'
  20. Sequence(Polygon[Rotate[Z0', ((2 * π / 4) * i), (0, 0)], Rotate[Z1', ((2 * π / 4) * i), (0, 0)], Rotate[Z4', ((2 * π / 4) * i), (0, 0)], Rotate[Z3', ((2 * π / 4) * i), (0, 0)], Rotate[Z5', ((2 * π / 4) * i), (0, 0)]], i, 0, 3) → L01
  21. Sequence(Rotate[Polygon[Rotate[Z0', π / 4, (0, 0)], Rotate[Z1', π / 4, (0, 0)], Rotate[Z4', π / 4, (0, 0)], Rotate[Z3', π / 4, (0, 0)], Rotate[Z5', π / 4, (0, 0)]], ((2 * π / 4) * i), (0, 0)], i, 0, 3) → L02