Daniel McNeil
April 3, 2007
Math 371
What is a tiling?
• A tiling, or tessellation, refers to a
collection of figures that cover a plane with
no gaps and no overlaps.
• Tessella is Latin term describing a piece of
clay or stone used to make a mosaic
Tiling on the Euclidean Plane
Regular Tilings
Are there any others?
Semiregular Tilings
(3,12,12)
(3,6,3,6)
(4,4,3,3,3)
(4,6,12)
(3,4,6,4)
(3,3,3,3,6)
(8,8,4)
(3,3,4,3,4)
Tilings and Patterns
• Book written in 1986
by Branko Grünbaum
and G.C. Shepherd.
• Remains most
extensive collection of
work to date.
• Took particular
interest in periodic
and aperiodic tilings.
Periodic vs Aperiodic
• Periodic tilings display
translational
symmetry in two nonparallel directions.
• Aperiodic tilings do
not display this
translational
symmetry.
Is there a polygon that tiles the
plane but cannot do so
periodically?
From Old and New Unsolved Problems in
Plane Geometry and Number Theory
Penrose Tilings
Roger Penrose
Penrose Tilings
• Discovered by Roger Penrose in 1973
• Most prevalent form of aperiodic tilings
• No translational symmetry, so never
repeats exactly, but does have identical
parts
• In 1984, Israeli engineer Dany Schectman
discovered that aluminum manganese had
a penrose crystal structure.
• In a Penrose tiling,
Nkite/Ndart = Φ
• Given a region of
diameter d, an
identical region can
always be found
within d(Φ+½).
Other Geometric Applications
Topologically Equivalent Tilings
Euler Characteristic
a=average number of sides per polygon
b=average number of sides meeting at a vertex
F=number of faces
V=number of vertices
Hyperbolic Tilings
Regular Tilings
• In Euclidean we saw that the angle of a
regular n-gon depends on n.
• What about Hyperbolic geometry?
• In Hyperbolic, the angle depends on both
n and the length of each side.
• 0<θ<(n-2)180o/n
Regular Tilings
• In Euclidean we could construct a regular
tiling with 4 squares at each vertex.
• Now in Hyperbolic we need 5 or more.
• In general, we have regular hyperbolic
tilings of k n-gons whenever 1/n+1/k<1/2
• Result: Infinitely many regular hyperbolic
tilings
1/n+1/k = 1/4+1/6 = 10/24 < 1/2
4,5
4,8
4,7
4,10
Semiregular Tilings
• Just like in Euclidean,
there are also
semiregular tilings in
Hyperbolic.
• This example shows
a square and 5
triangles at each
vertex.
Poincaré Upper Half Plane
• The vertical distance
between two points is
ln(y2/y1).
• Faces are all of equal
non-Euclidean size.
• Image can be
transformed from
Poincaré Disc to
PUHP.
Poincaré Disc vs PUHP
Poincaré Disc vs PUHP
Tilings in Art and Architecture
Tilings in Nature
• Abelson, Harold and DiSessa, Andrea. 1981. Turtle
Geometry. Cambridge: MIT Press
• Baragar, Arthur. 2001 A Survey of Classical and
Modern Geometries: With Computer Activities.
New Jersey: Prentice Hall
• Klee, Victor and Wagon Stan. 1991. Old and New
Unsolved Problems in Plane Geometry and
Number Geometry. New York: The Mathematical
Association of America
• Livio, Mario. 2002 The Golden Ratio. New York:
Broadway Books
• Stillwell, John. 2005. The Four Pillars of Geometry.
New York: Springer
• www.wikipedia.org
• www.mathworld.wolfram.com