Tessellations: Properties and Periodicity
Gregory J. Clark (Dr. David Offner, advisor)
Background
A tessellation is collection of
plane figures that fill the plane with
no overlaps and no gaps.
Art
M.C. Esher used his visual and
intuitive understanding of
mathematics to
create art formed
from tessellations.
Mathematics
Johannes Kepler
wrote about the tessellations
formed by regular polygons
covering the plane. E. S. Fedorov
proved that all tessellations are
constructed in one of seventeen
different groups of isometries.
Architecture
Ancient architecture,
like this dining room
floor in a Chedworth
Roman villa, features
tessellations as a design.
Science
X-ray crystallography is used to
determine the atomic structure of
crystals. These structures
demonstrate the
symmetric properties
associated with
tessellations.
Example of Periodicity
Discussion
Problem: What tiles can tessellate the plane?
Two-Rectangle Tile
Tile Periodicity (p)
 A tile is formed by the union of two
Theorem:
rectangles each with width one in
The periodicity (p) of a T-R Tile is
consecutive columns.
LCM LCM
p=
+
 Notation:
n−d m+d
n is height of left rectangle
m is height of right rectangle
d is distance between topmost square
where
LCM [(n −d),(m+d)]
€ (n = 2, m = 3, d = 1)
e.g.
LCM [(2 −1),(3 +1)] = LCM[1,4] = 4
e.g.
n=2
m=3
d=1
p=
€
When does p = n + m?
Theorem:
€
p = n + m if and only if (n – d) and
(m + d) are relatively prime.
€
T-R Tile Tessellation
Theorem:
Every T-R Tile
tessellates the
plane.
Periodic Lattice Pattern
Previous Work:
4 4
+ = 4 +1 = 5
1 4
LCM [n,m]
When d = 0
nm nm
= nm p = n + m = m + n
When d != 0
Let
€
Wijshoff and van Leeuwen proved that
tessellations formed
by translation
tile periodically.
LCM [n',m'] = n'm'
and
m'= m+d
n' m' n' m'
p=
+
= m'+n'
n'
m'
e.g. (n = 2, m = 3, d = 1)
€ n'= n − d€= 2 −1=1 € and m'= m+d = 3+1= 4
p = m'+n' = 4 +1 = 5
€
€
n'= n −d
€
p = m + n = 3+2 = 5
€
€
€
Future Work
 Investigate the relationship
between tiles and lattice
patterns
 Investigate different Categories
of tiles
Acknowledgements
 Dr. David Offner, for his advice
and guidance during the
research process
 The Drinko Center for
Excellence in Teaching, for
approving a travel grant
 My parents, for their continuous
support
Selected Bibliography
 Harry A. G. Wijshoff, Jan van Leeuwen:
Arbitrary versus Periodic Storage
Schemes and Tessellations of the Plane
Using One Type of Polyomino Information
and Control 62(1): 1-25 (1984)
 Oracle ThinkQuest: Totally Tessellated