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Transformations and Symmetry

Transformations and Symmetry. Transformations. Reflection Translation Glide Reflection Rotation Scaling (not topological) A transformation turns a geometric figure into another by one of the actions above. Reflection.

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Transformations and Symmetry

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  1. Transformations and Symmetry

  2. Transformations • Reflection • Translation • Glide Reflection • Rotation • Scaling (not topological) A transformation turns a geometric figure into another by one of the actions above.

  3. Reflection The blue triangle is reflected across line m (called the line of reflection).

  4. Translation Sliding a figure. A translation could be accomplished by several reflections.

  5. Glide Reflection The figure reflects across the line, and slides forward.

  6. Rotation The blue image is rotated about the point p.

  7. Scaling The blue star has been scaled upward by a multiplying factor. Scaling (changing size) is not a topological transformation.

  8. Symmetry • Our basic idea of symmetry is 2-fold symmetry • A figure whose halves are mirror images of each other over a fold line • We could think of this as being able to transform half of the image into the other half by reflection

  9. Which are the lines of symmetry? Only m

  10. Which are the lines of symmetry? L, M, N, O

  11. Which are the lines of symmetry? M, O

  12. Which are the lines of symmetry? T,R

  13. Symmetry (II) • More complicated types of symmetry arise if we consider objects obtained from reflection, PLUS other transformations such as rotation • Various patterns have been characterized as symmetry groups

  14. Conway Notation • Is one way mathematicians use to describe various symmetry groups • Can get very complicated!

  15. Kali • We’re going to look at some symmetry groups using the program Kali • Keep an eye out for two basic notations: • An integer (1,2,3) denotes a ROTATIONAL symmetry • An * denotes a reflection as well KALI

  16. The title picture … …was created in kali, then colored in with paint

  17. Symmetry in Nature Chinese Rose

  18. Symmetry in Nature Crab

  19. Symmetry in Nature Starfish

  20. Symmetry in Art Quilt

  21. Symmetry in Art Pennsylvania Dutch Hex Sign

  22. Symmetry in Art Persian Carpet

  23. Symmetry in Art Ukranian Painted Easter Eggs

  24. Symmetry in Art Mosaic Tile (Iran, 14th C.)

  25. Symmetry in Art A Kaliedotile

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