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Wallpaper Symmetry Group pmm

Wallpaper Symmetry Group pmm
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2017-11-01 21:26
VectorTransformationGeometric Construction
In collections Wallpaper Symmetry Groups
This example demonstrates the graphical properties of Wallpaper Symmetry Group pmm, involving core mathematical concepts such as symmetry, translation, and transformation. Construction highlights the use of GeoGebra commands like Intersect (intersection), Vector (vector), OrthogonalLine (orthogonal line), Point (point), Mirror (reflection), and Sequence (sequence); objects include A, B, u, f, C, v, pic1, g, h, pic1', pic1'_1, pic1'', L_1, L_2, L_3, L_4. Suitable for high school and above learning geometric transformations and symmetry.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Vector(A, B) → u
  3. OrthogonalLine(A, u) → f
  4. Point(f) → C
  5. Vector(A, C) → v
  6. OrthogonalLine(C, v) → g
  7. OrthogonalLine(B, u) → h
  8. Mirror(pic1, g) → pic1'
  9. Mirror(pic1, h) → pic1'_1
  10. Mirror(pic1', h) → pic1''
  11. Sequence(Sequence[Translate[pic1, Vector[((2 * a) * v) + ((2 * b) * u)]], a, -10, 10], b, -10, 10) → L_1
  12. Sequence(Sequence[Translate[pic1', Vector[((2 * a) * v) + ((2 * b) * u)]], a, -10, 10], b, -10, 10) → L_2
  13. Sequence(Sequence[Translate[pic1'', Vector[((2 * a) * v) + ((2 * b) * u)]], a, -10, 10], b, -10, 10) → L_3
  14. Sequence(Sequence[Translate[pic1'_1, Vector[((2 * a) * v) + ((2 * b) * u)]], a, -10, 10], b, -10, 10) → L_4

AI Drawing Prompt Reference

Create a wallpaper pattern with pmm symmetry. Start with a base motif defined by two perpendicular translation vectors. Reflect this motif across two perpendicular axes to generate three additional variants. Then translate the original motif and each reflected variant by all integer combinations of the two vectors, forming a periodic grid that covers the plane. The final pattern consists of four differently oriented copies of the base motif arranged in a rectangular lattice, exhibiting mirror symmetry along both axes and translational symmetry.
The prompt above was compiled from the original drawing commands and is for reference only.