The history of tessellations goes back to
Sumeria in about 4000 BC. The Sumerians
built their homes and temples using mosaic tiles
decorated with geometric patterns. Throughout
history, many civilizations such as the
Persians, Moors and Romans used these
decorative tiles extensively. The English word
tessellation comes from the Roman word for
tile: tessellae.
1
A Tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. Another word for a tessellation is a tiling.
* At any point where the vertices meet, the sum of the angles is 360 degrees. Ex:
A Regular Tessellation:
• when a tessellation is made up of congruent regular polygons
• Remember, a regular polygon is a simple closed figure that has all sides congruent and all angles congruent. • In a plane, only three regular polygons tessellate: triangles, squares or hexagons. (where the vertices meet, the sum of the angles is 360 degrees)
2
Interior angle measures of regular polygons: It is also possible to tessellate using combinations of regular polygons. . Sum of the interior angles of a regular polygon:
180o x (n­2)
where n represents the number of sides
3
Interior angle of regular polygons
=
Sum of angles ÷Number of angles
Example: Find the measure of each interior angle of the regular pentagon.
108o
108o
108o
108o
sum of angles = 180o x (n­2)
=180o x 3
= 540o
108o
Each interior angle = 540o÷5
= 108o
4
Examples: 5
Tesselations with Irregular Polygons: • all triangles and quadrilaterals tessellate. • polygons with more than six sides will not tessellate. • It is possible for combinations of shapes to tessellate.
6
Example 1: Does each shape tessellate?
Justify your answer.
7
8
9
Examples 1 and 2(page 464­466) illustrate that a shape that does not tessellate may be combined with one or more shapes to make a new shape that tessellates. This new shape is called a composite shape.
Do Pages 467­468
#6 ­ 9 10