Polygons that Tessellate the Plane
When Re‡ected in Their Edges
Ron Umble
Millersville University of PA
EMU Math Chat
November 12, 2009
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Tessellations
A tessellation (or tiling) of the plane is a collection of plane …gures
that …lls the plane with no overlaps and no gaps.
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Tessellations
A tessellation (or tiling) of the plane is a collection of plane …gures
that …lls the plane with no overlaps and no gaps.
A regular tessellation of the plane has (non-trivial) translational
symmetries in two independent directions.
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Tessellations
A tessellation (or tiling) of the plane is a collection of plane …gures
that …lls the plane with no overlaps and no gaps.
A regular tessellation of the plane has (non-trivial) translational
symmetries in two independent directions.
Tessellations of the plane generated by re‡ecting a polygon in its
edges are regular.
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Main Result
Theorem 1. Exactly eight polygons tessellate the plane under re‡ections:
Most symmetric examples of Laves tilings (Grunbaum & Shephard,
Tiling and Patterns, p. 96)
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Admissible Interior Angles
An n-center of a tessellation is the center of a group of n rotational
symmetries.
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Admissible Interior Angles
An n-center of a tessellation is the center of a group of n rotational
symmetries.
Crystallographic Restriction: If P is an n-center of a regular
tessellation, then n = 2, 3, 4, 6.
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Admissible Interior Angles
An n-center of a tessellation is the center of a group of n rotational
symmetries.
Crystallographic Restriction: If P is an n-center of a regular
tessellation, then n = 2, 3, 4, 6.
If V is a vertex of a polygon G that generates a tessellation under
re‡ections, then θ = m\V < 180 .
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Admissible Interior Angles
If G 0 is the re‡ection of G in an edge containing V , the interior angle
of G 0 at V has measure θ; inductively, every interior angle at V has
measure θ.
V
θ
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Admissible Interior Angles
If G 0 is the re‡ection of G in an edge containing V , the interior angle
of G 0 at V has measure θ; inductively, every interior angle at V has
measure θ.
V
θ
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θ
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Admissible Interior Angles
If G 0 is the re‡ection of G in an edge containing V , the interior angle
of G 0 at V has measure θ; inductively, every interior angle at V has
measure θ.
V
θ
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θ
θ
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Admissible Interior Angles
If G 0 is the re‡ection of G in an edge containing V , the interior angle
of G 0 at V has measure θ; inductively, every interior angle at V has
measure θ.
V
θ
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θ
θ
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Admissible Interior Angles
Since successively re‡ecting in the edges of G that share vertex V is a
rotational symmetry, V is an n-center for some n = 2, 3, 4, 6.
If G 0 is the image of G under a rotational symmetry at V , the number k
of copies of G sharing vertex V is the order of the rotational subgroup at
V . Thus k = 3, 4, 6 and θ = 60 , 90 , 120 .
G'
G
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Admissible Interior Angles
Otherwise, the number k of copies of G sharing vertex V is twice the
order of the rotational subgroup at V . Thus k = 4, 6, 8, 12 and
θ = 30 , 45 , 60 , 90 .
G G'
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Finding Generating Polygons
Conclusion: If a polygon G generates a tessellation of the plane
under re‡ections, its interior angles measure 30 , 45 , 60 , 90 , 120 .
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Finding Generating Polygons
Conclusion: If a polygon G generates a tessellation of the plane
under re‡ections, its interior angles measure 30 , 45 , 60 , 90 , 120 .
Combinatorial data is given by a system of linear equations.
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Finding Generating Polygons
Conclusion: If a polygon G generates a tessellation of the plane
under re‡ections, its interior angles measure 30 , 45 , 60 , 90 , 120 .
Combinatorial data is given by a system of linear equations.
For triangles:
30a + 45b + 60c
a + b + c
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+ 90d + 120e = 180
+ d +
e
= 3
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Finding Generating Polygons
Conclusion: If a polygon G generates a tessellation of the plane
under re‡ections, its interior angles measure 30 , 45 , 60 , 90 , 120 .
Combinatorial data is given by a system of linear equations.
For triangles:
30a + 45b + 60c
a + b + c
+ 90d + 120e = 180
+ d +
e
= 3
Using the combinatorial data, construct all possible n-gons and check
which ones tessellate.
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Periodic Trajectories and Orbits
A periodic orbit of a billiard ball in motion on a frictionless table Ω
bounded by a polygon G is a piecewise-linear constant speed curve
α : R !Ω such that α (a) = α (b ) for some a < b and
α0 (a + t ) = α0 (b + t ) for almost all t 2 R.
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Periodic Trajectories and Orbits
A periodic orbit α retraces the same path n
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Polygons that Tessellate the Plane
1 times
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Periodic Trajectories and Orbits
A periodic orbit α retraces the same path n
1 times
If n = 1, then α is primitive
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Periodic Trajectories and Orbits
A periodic orbit α retraces the same path n
1 times
If n = 1, then α is primitive
If n > 1, then α is an n-fold iterate
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Periodic Trajectories and Orbits
A periodic orbit α retraces the same path n
1 times
If n = 1, then α is primitive
If n > 1, then α is an n-fold iterate
If α is primitive, αn denotes its n-fold iterate
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Periodic Trajectories and Orbits
A periodic orbit α retraces the same path n
1 times
If n = 1, then α is primitive
If n > 1, then α is an n-fold iterate
If α is primitive, αn denotes its n-fold iterate
The period of a periodic orbit α is the number of times the ball
strikes a bumper
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Periodic Trajectories and Orbits
A periodic orbit α retraces the same path n
1 times
If n = 1, then α is primitive
If n > 1, then α is an n-fold iterate
If α is primitive, αn denotes its n-fold iterate
The period of a periodic orbit α is the number of times the ball
strikes a bumper
If α is primitive of period k, then αn has period kn
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Fagnano’s Period 3 Orbit on Acute Triangles
The orthic triangle is the inscribed triangle of least perimeter
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Fagnano’s Period 3 Orbit on Acute Triangles
The orthic triangle is the inscribed triangle of least perimeter
Vertices are feet of the altitudes
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Fagnano’s Period 3 Orbit on Acute Triangles
The orthic triangle is the inscribed triangle of least perimeter
Vertices are feet of the altitudes
Altitudes bisect the interior angles
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Unfolding Orbits on Equilateral Triangles
This technique is due to H. A. Schwarz
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Unfolding Orbits on Equilateral Triangles
This technique is due to H. A. Schwarz
Let P be the initial position of the ball
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Unfolding Orbits on Equilateral Triangles
This technique is due to H. A. Schwarz
Let P be the initial position of the ball
Release the ball with velocity !
v
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Unfolding Orbits on Equilateral Triangles
This technique is due to H. A. Schwarz
Let P be the initial position of the ball
Release the ball with velocity !
v
Re‡ect the triangle in the side the ball will strike next
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Unfolding Orbits on Equilateral Triangles
This technique is due to H. A. Schwarz
Let P be the initial position of the ball
Release the ball with velocity !
v
Re‡ect the triangle in the side the ball will strike next
Repeat until the ball arrives at some image Q of P with velocity !
v
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Unfolding Orbits on Equilateral Triangles
This technique is due to H. A. Schwarz
Let P be the initial position of the ball
Release the ball with velocity !
v
Re‡ect the triangle in the side the ball will strike next
Repeat until the ball arrives at some image Q of P with velocity !
v
PQ is an unfolding of the orbit
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B
C
C
A
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A
B
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Unfolding Orbits on Equilateral Triangles
Let T be the tessellation generated by repeatedly re‡ecting 4ABC
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Unfolding Orbits on Equilateral Triangles
Let H be the group generated by re‡ections in the lines of T .
The action of H on an edge of a basic triangle generates a tessellation H
by hexagons.
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Unfolding Orbits on Equilateral Triangles
Theorem 2. Let α be an orbit and let PQ be an unfolding. Then
(i) α has even period i¤ Q lies on a horizontal edge of H.
(ii) α has odd period i¤ α = γ2k
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1
for some k
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1.
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Equivalent Orbits
Periodic orbits α and β are equivalent if there exist respective
unfoldings PQ and RS and a horizontal translation τ such that
RS = τ PQ .
Q
S
P
R
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Equivalent Orbits
Periodic orbits α and β are equivalent if there exist respective
unfoldings PQ and RS and a horizontal translation τ such that
RS = τ PQ .
Q
S
P
R
Two unfoldings that terminate on the same horizontal edge of H are
equivalent.
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Equivalent Orbits
Remark. Sequences of incidence angles in equivalent unfoldings di¤er
by a permutation. One could consider a …ner relation in which
sequences di¤er by a cyclic permutation.
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Equivalent Orbits
Remark. Sequences of incidence angles in equivalent unfoldings di¤er
by a permutation. One could consider a …ner relation in which
sequences di¤er by a cyclic permutation.
The period of a class is the period of its elements
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Equivalent Orbits
Remark. Sequences of incidence angles in equivalent unfoldings di¤er
by a permutation. One could consider a …ner relation in which
sequences di¤er by a cyclic permutation.
The period of a class is the period of its elements
Even (period) classes have cardinality c
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Equivalent Orbits
Remark. Sequences of incidence angles in equivalent unfoldings di¤er
by a permutation. One could consider a …ner relation in which
sequences di¤er by a cyclic permutation.
The period of a class is the period of its elements
Even (period) classes have cardinality c
Odd classes are singletons γ2k
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1
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Equivalent Orbits
Remark. Sequences of incidence angles in equivalent unfoldings di¤er
by a permutation. One could consider a …ner relation in which
sequences di¤er by a cyclic permutation.
The period of a class is the period of its elements
Even (period) classes have cardinality c
Odd classes are singletons γ2k
1
Since H has countably many edges, there are countably many even
classes
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Rhombic Coordinates
T is a tessellation of the plane with equilateral triangles
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Rhombic Coordinates
T is a tessellation of the plane with equilateral triangles
P is a point on the horizontal edge AB of some basic triangle
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Rhombic Coordinates
T is a tessellation of the plane with equilateral triangles
P is a point on the horizontal edge AB of some basic triangle
s is the union of all line segments from P to its images
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Rhombic Coordinates
T is a tessellation of the plane with equilateral triangles
P is a point on the horizontal edge AB of some basic triangle
s is the union of all line segments from P to its images
τ is a horizontal translation such that τ (s ) misses the vertices of T
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Rhombic Coordinates
T is a tessellation of the plane with equilateral triangles
P is a point on the horizontal edge AB of some basic triangle
s is the union of all line segments from P to its images
τ is a horizontal translation such that τ (s ) misses the vertices of T
O = τ (P ) is the origin
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Rhombic Coordinates
T is a tessellation of the plane with equilateral triangles
P is a point on the horizontal edge AB of some basic triangle
s is the union of all line segments from P to its images
τ is a horizontal translation such that τ (s ) misses the vertices of T
O = τ (P ) is the origin
AB is the unit of length
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Rhombic Coordinates
T is a tessellation of the plane with equilateral triangles
P is a point on the horizontal edge AB of some basic triangle
s is the union of all line segments from P to its images
τ is a horizontal translation such that τ (s ) misses the vertices of T
O = τ (P ) is the origin
AB is the unit of length
!
AB is the x-axis
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Rhombic Coordinates
T is a tessellation of the plane with equilateral triangles
P is a point on the horizontal edge AB of some basic triangle
s is the union of all line segments from P to its images
τ is a horizontal translation such that τ (s ) misses the vertices of T
O = τ (P ) is the origin
AB is the unit of length
!
AB is the x-axis
The line through O with inclination 60 is the y -axis
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Rhombic Coordinates
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Periodic Orbits and Rhombic Coordinates
Software for …nding periodic orbits:
“Orbit Tracer 2” by Stephen Weaver at
http://www.millersville.edu/~rumble/StudentProjects/Weaver/Tessellation_v1.1.exe
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The Fundamental Region
Proposition 1. Every periodic orbit strikes some side of 4ABC with an
incidence angle in the range 30
θ 60 .
Every periodic orbit has an unfolding in ΓO = f(x, y ) j 0
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x
yg
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Classi…cation of Even Orbits
Let x, y 2 Z
(x, y ) lies on a horizontal edge of H i¤ x
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Polygons that Tessellate the Plane
y (mod 3) .
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Classi…cation of Even Orbits
Let x, y 2 Z
(x, y ) lies on a horizontal edge of H i¤ x
y (mod 3) .
Theorem 3. There is a bijection
f(x, y ) 2 ΓO j x
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y (mod 3)g $ forbits with even periodg
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Classi…cation of Even Orbits
Let x, y 2 Z
(x, y ) lies on a horizontal edge of H i¤ x
y (mod 3) .
Theorem 3. There is a bijection
f(x, y ) 2 ΓO j x
y (mod 3)g $ forbits with even periodg
Theorem 4. (Classi…cation)
(i) α has odd period i¤ α = γ2k
1
(ii) α $ (x, y ) has period 2n i¤ x
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for some k
1.
y (mod 3) and x + y = n.
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Counting Classes of Periodic Orbits
Distinct classes may have the same period:
(1,10)
(4,7)
C
AO
B
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Counting Classes of Periodic Orbits
Theorem 5. (Counting Formulas)
(i) The number of classes with period 2n is exactly
O (n ) =
n+2
2
n+2
.
3
(ii) The number of primitive classes with period 2n is exactly
P (n) = ∑ µ (d ) O (n/d ) ,
where µ (d ) =
8
<
d jn
1,
d =1
( 1)r , d = p1 p2
:
0,
otherwise.
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pr for distinct primes pi
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Conclusions
There are no orbits of period 2
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Conclusions
There are no orbits of period 2
There is at least one class of period 2n for each n
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2
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Conclusions
There are no orbits of period 2
There is at least one class of period 2n for each n
2
The only primitive odd orbit is Fagnano’s period 3
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Conclusions
There are no orbits of period 2
There is at least one class of period 2n for each n
2
The only primitive odd orbit is Fagnano’s period 3
There are no primitive orbits of period 8, 12, or 20
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Conclusions
There are no orbits of period 2
There is at least one class of period 2n for each n
2
The only primitive odd orbit is Fagnano’s period 3
There are no primitive orbits of period 8, 12, or 20
All classes of period 2n are primitive i¤ n is prime
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Project Proposal
Given a polygon G that tessellates the plane under re‡ections, …nd,
classify and count the classes of periodic orbits on G .
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The End
Thank you!
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