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Rotating Squares Tiling

Rotating Squares Tiling
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2014-08-08 08:41
PolygonTrigonometryCoordinate GeometryGeometric Construction
In collections Symmetry, Transformation and Tessellation
This example demonstrates the dynamic tilings generated by rotating squares, integrating core mathematical concepts like square construction, rotational transformations, sine function applications, and coordinate geometry relationships. Key GeoGebra commands and objects used in the construction include Segment, Line, Intersect, Polygon, Midpoint, Vector, Sequence, Rotate, and OrthogonalLine. It is suitable for high school students to understand geometric transformations, the connection between function graphs and actual shapes, and provides a foundation for introductory mathematical modeling.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a_2
  2. Segment(A, C) → b_2
  3. Line(B, b_2) → c_2
  4. Line(C, a_2) → d_2
  5. Intersect(c_2, d_2) → D
  6. Segment(C, D) → e_2
  7. Segment(B, D) → f
  8. Polygon(A, C, 4) → poly1, g, h, i, j, E, F
  9. Polygon(C, D, 4) → poly2, k, l, m, n, G, H
  10. Polygon(D, B, 4) → poly3, p, q, r, s, I, J
  11. Polygon(B, A, 4) → poly4, t, a_1, b_1, c_1, K, L
  12. Midpoint(F, C) → M
  13. Midpoint(G, C) → N
  14. Midpoint(J, B) → O
  15. Midpoint(K, B) → P
  16. Vector(E, L) → u
  17. Vector(K, J) → v
  18. Sequence(Sequence[Translate[tile1, Vector[(n * u) + (m * v)]], n, -3, 3], m, -3, 3) → list1
  19. Polygon(P, O, 4) → poly5, d_1, e_1, f_1, g_1, Q, R
  20. Rotate(poly5, 90*°, O) → poly5'
  21. Rotate(poly5, 180*°, O) → poly5'_1
  22. Rotate(poly5, 270*°, O) → poly5'_2
  23. OrthogonalLine(M, b_2) → h_1
  24. OrthogonalLine(P, a_2) → i_1
  25. Intersect(i_1, h_1) → S
  26. Segment(M, S) → a
  27. Segment(P, S) → b
  28. OrthogonalLine(N, e_2) → l_1
  29. OrthogonalLine(O, f) → m_1
  30. Intersect(l_1, m_1) → T
  31. Segment(N, T) → c
  32. Segment(O, T) → d
  33. Segment(S, T) → e
  34. Rotate(set1, 90*°, O) → set2
  35. Rotate(set1, 180*°, O) → set3
  36. Rotate(set1, 270*°, O) → set4
  37. Sequence(Sequence[Translate[tile2, Vector[(n * u) + (m * v)]], n, -3, 3], m, -3, 3) → list2
  38. Polygon(E, A, L) → poly6, l_2, e_3, a_3
  39. Polygon(B, K, J) → poly7, j_2, b_3, k_2