1. Where Art and Geometry Meet!

2. In other words, shape can you put next to itself and NOT have any gaps? Circles? Octagons? Hexagons? Triangles?

3. When the shape used to tile the plane are the same regular polygons the tessellation is called regular. These are the only regular tessellations. Triangles Hexagons Squares

4. American Quilters have been using simple shapes to tile the plane for hundreds of years. Sometimes they repeat the same shape and use color to create a pattern.

5. Even the simplest of shapes, the triangle, can be used to create complex and beautiful patterns.

6. This quilt uses repeated right triangles. Notice that the big triangles are made up of the little ones. The pattern is called “Birds in Flight”. If you look at the white triangles, you see flocks of white birds flying south!

7. Just changing the orientation can change the way a tessellation is perceived. Here the “Birds in Flight” pattern is aligned vertically.

8. What do you see?, hexagons or triangles? What is the shape that is repeated to make this quilt?

9. The artist reversed the light and dark triangles to create different directions.

10. Rectangles and squares can also be used to create interesting designs.

11. Pattern: Log Cabin

12. Pattern: Resolution Square

13. Pattern: Courthouse Steps Border: Birds in the Air

14. Rectangles can even be used to “paint” a picture! Catch A Falling Star on A Hot August Night by Joen Wolfrom

15. If you vary the shapes you can still tile the plane. If the same group of regular polygons meet at every vertex, the tessellation is call Semi-Regular. There are only eight semi-regular tessellations. Can you name all eight semi-regular tessellations? Remember that the angles at each intersection must add up to 360. Use this chart to aid you in naming at least four of them. Regular PolygonMeasure of Interior Angle Triangle60⁰ Square90⁰ Hexagon120⁰ Octagon135⁰ Dodecagon150⁰

16. You name semi-regular tessellations by polygons (number of sides) going clockwise. Try to name the eight pictured tessellations. 3.4.6.4 4.8.8 3.12.12 3.3.3.3.6 3.3.3.4.4 3.3.4.3.4 3.6.3.6 4.6.12

17. Beautiful, interesting patterns emerge when the different polygons are repeated across the plane.

18. Classic 3-D

19. EXTREME 3-D! Do you see holes or hills?

21. Medieval Italian Mosaics

31. Escher created this lithograph to demonstrate how his tessellations evolve. 2 through 4 are rhombi. In 5 he starts his metamorphosis. By 7 the birds are formed. In 8, 9, and 10 he adds detail. Magically, in 11 and 12 the birds become fish!

35. Works Cited Slide 6Slide 6: Image from http://jenniferchiaverini.com. “Birds in the Air” by Jennifer Chiaverini.http://jenniferchiaverini.com Slide 7Slide 7: Image from http://jenniferchiaverini.com. “The Runaway Quilt” by Jennifer Chiaverini.http://jenniferchiaverini.com Slide 8Slide 8: Image from http://jenniferchiaverini.com. “Road to Triumph Ranch” Machine pieced by Heather Neidenbach, machine quilted by Sue Vollbrecht, 2006.http://jenniferchiaverini.com Slide 12Slide 12: Image from http://jenniferchiaverini.com. “The Giving Quilt” Gretchen Hartley.http://jenniferchiaverini.com Slide 13Slide 13: Image from http://jenniferchiaverini.com. “Joanna’s Freedom” Pieced by Geraldine Neidenbach and Heather Neidenbach. Quilted by Sue Vollbrecht.http://jenniferchiaverini.com Slide 11Slide 11: Image from http://jenniferchiaverini.com. “Gerda’s Log Cabin” by Jennifer Chiaverini.http://jenniferchiaverini.com Slide 9Slide 9: Image from http://jenniferchiaverini.com. “Eleanor’s Ocean Waves” Machine pieced by Geraldine Neidenbach and Heather Neidenbach, machine quilted by Sue Vollbrecht, 2003http://jenniferchiaverini.com Slide 14Slide 14: Image from http://joenwolfrom.com “Catch a Falling Star on a Hot August Night” by Joen Wolfrom.http://joenwolfrom.com

36. Slide 16Slide 16: Images from http://www.mathsisfun.com/geometry/tessellation.htmlhttp://www.mathsisfun.com/geometry/tessellation.html Works Cited, cont. Slide 21Slide 21: Images from http://www.csun.edu/~lmp99402/Math_Art/Tesselations/tesselations.html http://www.csun.edu/~lmp99402/Math_Art/Tesselations/tesselations.html