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Symmetry-Based Construction of Triangle & Rhombus Tessellations

Symmetry-Based Construction of Triangle & Rhombus Tessellations
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2020-12-30 17:18
CirclePolygonVectorTransformationGeometric Construction
This example demonstrates how to generate a periodic tessellation on the plane by using geometric transformations and circle-based constructions. It highlights key ideas such as geometric symmetry (reflection and mirror axes), tessellation by repeated polygonal tiles, and plane transformations like rotation and mirroring. The diagram also uses circle elements (including circle sectors) to control angles and ensure consistent edge matching across tiles. Construction workflow in GeoGebra: it builds a base tile with Polygon and circle sector/circle primitives (CircleSector, Circle), defines key points (e.g., Phase, H, D, etc.), and uses Vector to set direction and connectivity. Then Rotate and Mirror replicate the tile around specified points and lines to form neighboring copies, while Sequence helps generate repeated points/objects and organize periodic structure (including derived polygons like poly1', poly2', poly1'', poly2''). Suitable for secondary to early university geometry lessons, and useful for teaching transformations, symmetry, and tiling/tessellation construction.

GeoGebra Commands & Parameters

  1. Polygon(D, H, 4) → poly2, k, l, m, n, I, K
  2. CircleSector(A, E, F) → c
  3. CircleSector(A, F, E) → g
  4. Circle(A, E) → d
  5. Polygon(A, G, 4) → poly1, f, h, i, j, B, C
  6. Rotate(poly1, a, A) → poly1'
  7. Rotate(poly2, a + Phase, D) → poly2'
  8. Circle(D, H) → e
  9. Mirror(d, xAxis) → d'
  10. Mirror(e, xAxis) → e'
  11. Mirror(poly1', B) → poly1''
  12. Mirror(poly2', B) → poly2''
  13. Vector(H, I) → u
  14. Point(u) → M
  15. Vector(M, D) → v
  16. Sequence(Sequence[Translate[poly1', Vector[((4 * i) * u) + ((4 * j) * v)]], i, 0, o], j, 0, p) → l1
  17. Sequence(Sequence[Translate[d, Vector[((4 * i) * u) + ((4 * j) * v)]], i, 0, o], j, 0, p) → l2
  18. Sequence(Sequence[Translate[d', Vector[((4 * i) * u) + ((4 * j) * v)]], i, 0, o], j, 0, p) → l3
  19. Sequence(Sequence[Translate[e, Vector[((4 * i) * u) + ((4 * j) * v)]], i, 0, o], j, 0, p) → l4
  20. Sequence(Sequence[Translate[e', Vector[((4 * i) * u) + ((4 * j) * v)]], i, 0, o], j, 0, p) → l5
  21. Sequence(Sequence[Translate[poly1'', Vector[((4 * i) * u) + ((4 * j) * v)]], i, 0, o], j, 0, p) → l6
  22. Sequence(Sequence[Translate[poly2', Vector[((4 * i) * u) + ((4 * j) * v)]], i, 0, o], j, 0, p) → l7
  23. Sequence(Sequence[Translate[poly2'', Vector[((4 * i) * u) + ((4 * j) * v)]], i, 0, o], j, 0, p) → l8

AI Drawing Prompt Reference

Draw a periodic tessellation of triangles and rhombi on a plane. Start with a base rhombus and a base triangle sharing an edge, using circular arcs and circles to set angles and edge lengths for seamless tiling. Rotate the base shapes around a vertex, then mirror them across a horizontal axis. Further mirror the rotated shapes about another vertex to create new combined tiles. Finally, use two non-parallel translation vectors (along a rhombus side and a diagonal) to repeat all base and transformed tiles across the plane, forming a regular grid-like pattern.
The prompt above was compiled from the original drawing commands and is for reference only.