1. TESSELLATIONS

2. Definition
A tessellation is a tiling of a plane with a shape or shapes without creating any gaps or overlaps

3. Periodic
A periodic tessellation is one on which you can outline a region that tiles the plane by translation, that is, by shifting the position of the region without rotating or reflecting

4. Periodic: Regular
A regular tessellation is a tiling in which only one shape is used
The shape must have congruent sides and congruent angles
There exists exactly three regular tessellations

5. Periodic: Semiregular
A semi-regular tessellation is when more than one regular shape is used to tile a plane
Every vertex has the same combination of shapes
There are only eight such tessellations

6. Periodic: Not Semiregular
The vertices do not have the same combination of shapes A B
A ( )
B ( )

7. Periodic: Irregular
An irregular tessellation is one in which nonregular shapes are used
An infinite number of shapes tile periodically
Any quadrilateral will tessellate

8. Escher
M.C. Escher, the Dutch artist, is famous for his many pictures of periodic tilings with shapes that resemble living things

9. Aperiodic
In an aperiodic (nonperiodic) tessellation translation alone is not used to cover the plane. Shapes are rotated and reflected as well.
An infinite number of shapes tile only periodically
An infinite number of shapes tile both periodically and aperiodically

10. Penrose
Roger Penrose found a pair of tiles that force aperiodicity. In other words, they can not tile by translation alone.
As the patterns expand, they seem to strive to repeat but as you continue out they are not the same

11. Tessellations Periodic Regular Semiregular Not Semiregular Irregular
Aperiodic

12. Credits Produced by Dottie Feinstein © 1997 (2007)
Conti, Shrines of Power, HBJ Press, 1978
Gardner, Penrose Tiles to Trapdoor Ciphers,
W.H. Freeman & Co., 1989
Tessellation Poster: Teaching Notes,
Dale Seymour, 1987

13. For more information see