TesselationsWhere Art and Geometry Meet!

What shapes can tile the plane?In other words, shape can you put next to itself and NOT have any gaps?

Circles? Octagons? Hexagons?

Triangles?

Regular TessellationsWhen 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

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.

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

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

Semi-Regular TessellationsIf 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 Polygon Measure of Interior AngleTriangle 60⁰Square 90⁰

Hexagon 120⁰Octagon 135⁰

Dodecagon 150⁰

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

Beautiful, interesting patterns emerge when the different polygons are

repeated across the plane.

Even trompe l’oeil!

TROMPE L’OEIL FROM THE PAST

Medieval Italian Mosaics

STORM AT SEA

The Fantabulous Worlds of Escher!

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!

Works CitedSlide 6: Image from http://jenniferchiaverini.com. “Birds in the Air” by Jennifer Chiaverini. Slide 7: Image from http://jenniferchiaverini.com. “The Runaway Quilt” by Jennifer Chiaverini.

Slide 8: Image from http://jenniferchiaverini.com. “Road to Triumph Ranch” Machine pieced by Heather Neidenbach, machine quilted by Sue Vollbrecht, 2006.

Slide 12: Image from http://jenniferchiaverini.com. “The Giving Quilt” Gretchen Hartley .

Slide 13: Image from http://jenniferchiaverini.com. “Joanna’s Freedom” Pieced by Geraldine Neidenbach and Heather Neidenbach. Quilted by Sue Vollbrecht.

Slide 11: Image from http://jenniferchiaverini.com. “Gerda’s Log Cabin” by Jennifer Chiaverini.

Slide 9: Image from http://jenniferchiaverini.com. “Eleanor’s Ocean Waves” Machine pieced by Geraldine Neidenbach and Heather Neidenbach, machine quilted by Sue Vollbrecht, 2003

Slide 14: Image from http://joenwolfrom.com “Catch a Falling Star on a Hot August Night” by Joen Wolfrom.

Slide 16: Images from http://www.mathsisfun.com/geometry/tessellation.html

Works Cited, cont.

Slide 21: Images from http://www.csun.edu/~lmp99402/Math_Art/Tesselations/tesselations.html