Kaleidoscopic Tilings on Surfaces, this
Time with the Groups
S. Allen Broughton
Rose-Hulman Institute of Technology
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Credits
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Some of this work has been done and will
be done with numerous undergraduates.
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Outline
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Repeated patterns, structure, groups, and
semigroups
Groups and group actions
Talk 1: fundamental domains
Talk 2: homology, group algebras, and
separability
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Outline of Talk 1
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Tilings of surfaces and the plane– examples,
definition
Kaleidoscopic tilings of the hyperbolic plane
Hyperbolic geometry
Tiling groups G* and G
Riemann-Hurwitz equation
Fundamental homomorphism
Fundamental domain via a homomorphism
Examples via Matlab
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Patterns
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Many objects have repeated patterns, structures,
or other properties due to the underlying
presence of a symmetry group or semigroup
Semigroup examples
• fractal structures in an iterated function system
• convolution properties of Laplace, and z-transforms
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Group examples
• Tilings in 2 or more dimensions
• Crystallographic structures
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Groups and group action
Definition for these talks
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A group G is set of transformations of a set or space X,
such that G is closed under composition and taking of
inverses.
If g,h ε G then gh =g◦h ε G and g-1 ε G
For x ε X and g ε G define the action of g on x by
gx=g(x).
Note that (gh)x =g◦h(x) =g(hx).
Typically a group will preserve some property of the space
– e.g., distance preserving or rigid motion.
Subgroups are subsets of G closed under compositions
and taking inverses.
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Groups and group actions
Examples
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X = {1,2,3…,n} and G is a group of permutations
of X
X is a regular n-gon and G = Dn is the dihedral
group of symmetries of X
X is euclidian or hyperbolic space and group G is
the set of rigid motions of X
Any subgroup of the above examples mapping a
structure on X to itself
G acts on subsets of X, e.g. k-subsets, vertices,
edges, lines, or circles
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Icosahedral-Dodecahedral tiling
(2,3,5) – tiling – soccer ball
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(2,4,4) -tiling of the torus
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(2,2,2,2) -tiling of the torus
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(3,3,4) -tiling of the hyperbolic plane
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(2,3,7) -tiling of the hyperbolic plane
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Tiling: definition
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Let S be a surface of genus σ .
Tiling: Covering by polygons “without
gaps and overlaps”
Kaleidoscopic: Symmetric via reflections
in edges.
Geodesic: Edges in tiles extend to
geodesics in both directions
terminology: (l,m,n)-triangle
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Hyperbolic geometry
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Refer to tiling pictures
Points, lines and angles
Reflections
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The tiling group - 1
Q
c
_
π
n
p
r
∆0
_
π
R
_
π
a l
m
q
b
P
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The tiling group - 2
Full Tiling Group for triangle (a finite group)
G =< p , q , r >
*
Group Relations
p
2
= q
2
= r
( pq ) = ( qr )
l
2
m
= 1.
= ( rp )
n
= 1.
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Riemann Hurwitz equation - triangles
Let S be a surface of genus
number of triangles:
σ
and |G*| the
1 1 1
4σ − 4
= 1− − −
| G* |
l m n
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Tiling theorem - triangles
A surface S of genus
tiling group:
σ
has a tiling with
G = < p, q , r >
*
if and only if
• the group relations hold, and
• the Riemann Hurwitz equation holds.
Therefore tiling problems can be solved via
group computation
Start with the group then make the surface
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The Tiling Group
Observe/define:
a = pq,b = qr ,c = rp
Tiling Group:
G = < p, q , r >
*
Orientation Preserving Tiling Group:
G = < a ,b, c >
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Group Relations (simple geometric and
group theoretic proofs)
p
2
= q
2
= r
2
= 1.
a = b = c = 1,
a b c = 1, ( p q q r r p = 1)
l
m
n
θ (a ) = qaq
−1
θ (b ) = q b q
−1
= qpqq = qp = a
= qqrq = rq = b
−1
−1
,
.
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Universal Cover and Fundamental Domain
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Euclidean example (on board)
The covering group and the fundamental
homomorphism
The covering group is defined on the universal of
the surface
A “fundamental domain” comes from the
homomorphism
Surface constructed by identifying boundaries on
fundamental domain
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Covering group and the homomorphism
~ ~
~
~
~
~
Λ =< p , q , r > , Λ =< a , b , c >
*
Γ → Λ → G
~p → p , q~ → q , ~
r → r
*
*
Γ → Λ → G
~
~
~
a → a,b → b, c → c
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Fundamental domain from the
homomorphism
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Pick exactly one tile in the cover for each
group element in G*.
Matlab examples
Problem find a nice way to move in the
surface to create nice fundamental domains
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