Tessellations all around us Look for tessellations in walls, patios and pavements

Tessellations all around usLook for tessellations in walls, patios and pavements.

Tessellations all around us

Common shapes can be arranged in unusual ways

Sometimes an unusual shape will tessellate

Tessellations all around us

Sometimes 2 or more different shapes will tessellate.

Modern-day Tessellations

• Soccer balls• Bathroom floors• Wallpaper designs

• Brick walls are tessellations. The rectangular face of each brick is a tile on the wall.

Examples

• Chess and checkers are played on a tiling. Each colored square on the board is a tile, and the board is an example of a periodic tiling.

Alhambra

• The Alhambra, a Moor palace in Granada, Spain, is one of today’s finest examples of the mathematical art of 13th century Islamic artists.

• Which other regular polygons do you think can tile the plane?

Regular tiling

• Triangles?

Triangles

• The interior angle of every equilateral triangle is 60º. If we sum the angles around a vertex, we get 60º + 60º + 60º + 60º + 60º + 60º = 360º again!.

• Yep!

• How many triangles to make 1 complete rotation?

• Will pentagons work?

Pentagons

• The interior angle of a pentagon is 108º, and 108º + 108º + 108º = 324º.

• Hexagons?

Hexagons

• The interior angle is 120º, and 120º + 120º + 120º = 360º.

• How many hexagons to make 1 complete rotation?

• Heptagons? Octagons?

Heptagons

• Not without getting overlaps. In fact, all polygons with more than six sides will overlap.

• So, the only regular polygons that tessellate the plane are triangles, squares and hexagons.

• That was an easy game. Let’s make it a bit more rewarding.

Regular tiling

Tessellations by M.C. Escher

M. C. Escher, Cycle

Bulldog

(Tessellation 97)

Pegasus

(Tessellation 105)