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Rotating Triangles Tiling

Rotating Triangles Tiling
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2014-08-08 09:24
TriangleCirclePolygonOptimization
In collections Symmetry, Transformation and Tessellation
This example demonstrates the geometric phenomenon of tiling a plane using rotating triangles. It involves core mathematical concepts such as triangles, unit circle, equations, curve sketching, incircle, and optimization problems. The construction utilizes GeoGebra commands and objects like Polygon, Circle, Center, Vector, Sequence, and Rotate. Suitable for high school and above students to understand geometric transformations, optimization, and plane figure constructions.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  2. Polygon(A, C, 3) → poly2, d, e, f, D
  3. Polygon(C, B, 3) → poly3, g, h, i, E
  4. Polygon(B, A, 3) → poly4, j, k, l, F
  5. Circle(F, A, B) → p
  6. Circle(A, D, C) → q
  7. Circle(C, E, B) → r
  8. Center(p) → G
  9. Center(q) → H
  10. Center(r) → I
  11. Vector(B, D) → u
  12. Vector(A, E) → v
  13. Vector(C, F) → w
  14. Sequence(Polygon[Translate[A, Vector[(n * u)]], Translate[B, Vector[(n * u)]], Translate[C, Vector[(n * u)]]], n, -10, 10) → list1
  15. Sequence(Translate[list1, Vector[(n * v)]], n, -10, 10) → list2
  16. Rotate(list2, 120*°, I) → list3
  17. Rotate(list2, 240*°, I) → list4
  18. Rotate(Translate[list2, Vector[(a * u) + (b * v) + (c * w)]], α, J) → triangles1
  19. Rotate(Translate[list3, Vector[(a * u) + (b * v) + (c * w)]], α, J) → triangles2
  20. Rotate(Translate[list4, Vector[(a * u) + (b * v) + (c * w)]], α, J) → triangles3
  21. Polygon(B, G, A, H, C, I) → poly5, b_2, g_1, a_2, h_1, c_2, i_1
  22. Sequence(Polygon[Translate[B, Vector[(n * u)]], Translate[G, Vector[(n * u)]], Translate[A, Vector[(n * u)]], Translate[H, Vector[(n * u)]], Translate[C, Vector[(n * u)]], Translate[I, Vector[(n * u)]]], n, -10, 10) → list5
  23. Sequence(Translate[list5, Vector[(n * v)]], n, -10, 10) → list6
  24. Rotate(list6, 120*°, I) → list7
  25. Rotate(list6, 240*°, I) → list8
  26. Rotate(Translate[list6, Vector[(a * u) + (b * v) + (c * w)]], α, J) → hexagons1
  27. Rotate(Translate[list7, Vector[(a * u) + (b * v) + (c * w)]], α, J) → hexagons2
  28. Rotate(Translate[list8, Vector[(a * u) + (b * v) + (c * w)]], α, J) → hexagons3

AI Drawing Prompt Reference

Draw a plane tiling pattern composed of rotating triangles, including multiple triangles and circles, demonstrating geometric transformations of rotation and translation. Start by drawing several basic triangles and then draw circles centered at their centers. Next, use vectors to perform translation and rotation operations, generating sets of triangles and circles. Finally, rotate these sets by 120° and 240° to form a complete plane tiling pattern.
The prompt above was compiled from the original drawing commands and is for reference only.