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Polygons & Tessellations (page 1)
1. To tessellate (or tile) means to cover a flat surface with a repeated pattern,
with no gaps or overlaps. Tessellate comes from the Latin tessera, which
refers to a small block of stone, tile, or glass used in making a mosaic.
2. Tessellations often consist of a simple figure which is repeatedly translated,
rotated, and/or reflected into a pattern.
3. Tessellations occur all around us, in—
• nature (seeds, cracks in mud, honeycombs, rock formations)
• art (see examples in Geometry textbook p.667)
• everyday life (brick patios, tiled floors, wallpaper, tapestries).
4. Because the figures in a tessellation do not overlap or leave gaps, the sum of
the measures of the angles around any vertex must be 360°.
• If the angles around a vertex are congruent, then the measure of each
angle must be a factor of 360.
Example 1: Does a regular hexagon tessellate a plane?
Answer 1:
Step 1. The angle a of each vertex is given by the formula: a =
Step 2. Substitute 6 for n: a =
= 120
Step 3. Is 120 a factor of 360? Divide 360 by 120:
= 3.
Step 4. Since 3 is an integer, then 120 is a factor of 360; therefore a regular
hexagon will tessellate.
Example 2: Does a regular 18-gon tessellate a plane?
Answer 2:
Step 1. The angle a of each vertex is given by the formula: a =
Step 2. Substitute 18 for n: a =
= 160
Step 3. Is 160 a factor of 360? Divide 360 by 160:
= 2.25.
Step 4. Since 2.25 is not an integer, then 160 is not a factor of 360; therefore a
regular 18-gon will NOT tessellate.
5. Every triangle tessellates.
6. Every quadrilateral tessellates.
Polygons & Tessellations (page 2)
(from Prentice-Hall Geometry, p. 669.)
dwa 4/07/13