1. Investigation 12: Tessellation
Tessellations
Investigation 12: Tessellation

2. Investigation 12: Tessellation
Tessellations
A tessellation is a design or pattern in which a shape is use repeatedly to cover a plane with no gaps, overlaps, or empty spaces.
1. A regular tessellation is a pattern made with only one type of regular polygon.
2. The sum of the measures surrounding a point (or vertex) must be 360°.
3. Only regular polygons that have an interior angle which is a factor of 360 will tessellate.
Investigation 12: Tessellation

3. Can these figures form a regular tessellation?
No. Although this is a regular polygon, it has an interior angle = 135°, which is not a factor of 360
Yes. This is a regular polygon with a 90° interior angle which is a factor of 360.
Yes. This is a regular polygon with a 120° interior angle, which is a factor of 360.
Investigation 12: Tessellation

4. Can these figures form a regular tessellation?
Yes. This is a regular polygon with a 60° interior angle which is a factor of 360.
No. This is not a regular polygon. It can tessellate but not in a regular tessellation.
No. This is not a regular polygon. It can tessellate but not in a regular tessellation.
Investigation 12: Tessellation

5. Semi-regular Tessellations
If the same combination of regular polygons meet at each vertex, it is called a semi-regular tessellation.
Investigation 12: Tessellation

6. Irregular Tessellations
Other figures can make tessellations which are irregular. The figures used are irregular polygons and may be the same or different types.
Here is an irregular tessellation made with kites and one with trapezoids.
Investigation 12: Tessellation

7. Investigation 12: Tessellation
Lizard
(Tessellation 104)
Investigation 12: Tessellation

8. Investigation 12: Tessellation
Pegasus
(Tessellation 105)
Investigation 12: Tessellation

9. Tessellations by M.C. Escher
Investigation 12: Tessellation

10. Investigation 12: Tessellation
M. C. Escher, Cycle
Investigation 12: Tessellation

11. Investigation 12: Tessellation
Bulldog
(Tessellation 97)
Investigation 12: Tessellation

12. Special Notes on Tessellations
1. At each vertex of a tessellation, the sum of the measures of the angles must equal 360.
2. Any quadrilateral will tessellate.
3. Combinations of figures can be used to tessellate.
4. Only equilateral triangles, squares, and regular hexagons can make regular tessellations.
Investigation 12: Tessellation