Sunday, April 12, 2009

Arabic Tiling or Tessellation

This was created in Geometer's Sketchpad, put could just as well be created with a compass and straight-edge.

1. Start with a circle. Draw lightly if using a pencil, because the circle will be erased later. Make certain the center point is marked.


2. Using the same diameter, draw circles around the original one, using the intersections as the center points. Each outer circle intersects the center of the original circle.



3. Continue until six circles are drawn around the original.

There are a couple of interesting asides:

Six circles, with a constant diameter, fit perfectly,
overlapping at the center points, around a center circle with that same diameter.

Also, six circles of the same diameter, fit perfectly
around a center circle with that diameter.




4. Draw sets of parallel lines through the intersection points. Again, if using a pencil, draw lightly because the lines will have to be erased.



5. Continue, drawing the parallel lines in the 3 possible directions.



6. Erase the original circle design. Continue drawing parallel lines in the three different possible directions. Use the intersection points as guides.



7. More of the same.



8. For my design, I continued until I could draw a hexagon, whose sides were the width of 6 rhombuses.



9. Using the lines, starting from the center, draw a pattern of stars and hexagons. I chose to put a star at the center, but it could easily have been a hexagon.



10. Erase (delete) the lines and start coloring the stars...



11. ... and hexagons.

12. Finished (points were also deleted).

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