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Square and Parallelogram Tiling

Square and Parallelogram Tiling
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2014-08-05 18:31
PolygonVectorTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates how squares and parallelograms can tile a plane, forming a complex geometric pattern. It involves core mathematical concepts such as properties of geometric figures, vector operations, coordinate geometry, rotation, and transformation. Key GeoGebra commands used include Segment, Line, Intersect, Polygon, Midpoint, Vector, Sequence, Rotate, and OrthogonalLine. Suitable for high school mathematics education to understand the principles of spatial geometry and transformations.

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

AI Drawing Prompt Reference

Draw a complex geometric pattern tiled by squares and parallelograms, demonstrating how they fill a plane. Start by creating a square, then use vector operations and rotations to generate parallelograms, combining them into a continuous structure. Determine key points using coordinate geometry and vector calculations, construct new shapes with rotations and orthogonal lines, resulting in a symmetrical and layered plane tiling pattern.
The prompt above was compiled from the original drawing commands and is for reference only.