1. Tessellations
Lindsay Stone
Fara Mandelbaum
Sara Fazio
Cori McGrail
Laura Welch
Jason Miller

2. Definition Tessellation – a careful juxtaposition of
elements into a coherent pattern sometimes
called a mosaic or tiling
                                                                              
Example:

3. Tessellations are different from patterns
because patterns usually do not have distinct
closed shapes
A closed shape is a shape that has a definite
interior and a definite exterior
                                                                              

4. History Mathematics Johannes Kepler 1619
Russian crystallographer E.S. Fedorov 1891
Science
-X-ray Crystallography
This picture is a transformation of
eight points in an array to make a
very small crystal lattice which
tessellates

5. Science Continued….
This image suggests the
relationship between
tessellations,symmetry,
and X-ray crystallography

6. M.C. Escher (1898 – 1972) -Created over 100 tessellated patterns
-Work involves topology, optical illusions,
hyperbolic
tessellations,
and other
advanced
mathematical topics
-Escher’s tilings
were designed
to resemble
recognizable
objects
-Escher’s work with tilings of the plane embodies
many ideas that scientists and mathematicians
discovered only after Escher did

7. Sun and Moon Uses birds to transform day into night
In this image the white birds bring forth the sun while the dark birds carry the moon and the stars. Day and night fight each other for attention but fit seemlessly together.

8. Symmetry Drawing No. 71
Symmetry Drawing No. 71 is one of his most complex with 12 different birds forming a rectangle in this image

9. Regular Tessellations
-A regular polygon tessellation is constructed from
regular polygons
-Regular polygons have equal sides and equal angles
-The regular polygons must fill the plane at each
vertex, with repeating patterns and no overlapping
pieces
Note: This pentagon does not fit
the vertex…therefore it is not a
regular tessellation

10. There are only 3 regular tessellations
One of triangles
One of squares
One of hexagons

11. This is NOT a regular polygon tessellation because…..
The plane is not filled at
the vertex because there
is a space left over
vertex
space
A regular polygon tessellation,
can be changed by using
“alterations” to the sides of the
polygon. These alterations
are called transformations

12. Three Common Transformations
1. Translation – which is a slide of one
side of the polygon, “move”
2. Reflection – flip or mirror image of
one side of the polygon
3. Rotation – turn of a side around one
vertex of a polygon

13. Translation – “slide”
this side
the alteration
moves here

14. Reflections – “flip”
the alteration
flips
here

15. Rotation – “turn”
the alteration
here
rotates around
this vertex

16. Steps to name an arrangement of regular polygons around a vertex
first find the regular polygon with the least number of sides.
2. Then find the longest consecutive run of this polygon, that is, two or
more repetitions of this polygon around the vertex.
3. Next, indicate the number of sides of this regular polygon. For
example, to name a triangle with 3 sides, we name it 3 and follow it
with a period (.). If you find more than one consecutive "run" of this
polygon, then name it twice, i.e., 3.3.
4. Proceeding in a clockwise or counterclockwise order, indicate the
number of sides of each polygon as you see them in the arrangement.
5. Do remember to start with the longest consecutive run of the
regular polygon with the shortest number of sides.
                                              

17. Semi-regular Tessellations
Definition – are tessellations of more than
one type of regular polygons such that the
polygon arrangement at each vertex is the
same

18. interior angle (degrees)
number of sides
interior angle (degrees) 3 60 4 90 5
108 6
120 7
128 8
135 9
140 10
144 11
150
... n
180(n-2)/n
In order for the semi-regular tessellation to work,
the interior angle sum must be equal to 360

19. Semi-Regular Tessellation’s
4.6.12
4.8.8

20. Semi-Regular Tessellation’s

21. Semi-Regular Tessellation’s

22. Demi-regular Tessellations
Definition – tessellations of regular polygons in which there are two or three different polygon arrangements
                                   

23. Duals and Vertex Configurations
Duals - connect the centers of the regular polygons
around a vertex creating a new shape
Vertex Configurations – connect the midpoints of
the sides of the regular polygons around a vertex
creating a new shape
6^3
3^6
4^4
3^6
6^3
4^4

27. In life tessellations appear all around us….
Mud Flats
Honeycombs
Hydrogen Peroxide
Checkers

28. Gallery

34. References
Totally Tessellated - ThinkQuest winner - great site, instruction, information.
Tessellations Tutorials - Math Forum site
- site for construction of tessellations great list of tessellation links
Math. Com - List of good tessellation links
World of Escher site - commerical site with gallery of Dutch artist,Escher who was famous for his tessellation art.
Science University’s Tilings Around Us Site.
Other links from Forum.