Wallpaper
Groups
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Geometric Transformations and Wallpaper
Groups
Wallpaper Groups
Lance Drager
Department of Mathematics and Statistics
Texas Tech University
Lubbock, Texas
2010 Math Camp
Wallpaper
Groups
Outline
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
1 Wallpaper Groups
Introduction
The Structure of Wallpaper Groups
Classification of Lattices
Names for the Wallpaper Groups
How to Classify Wallpaper Groups
Patterns to Classify
Wallpaper
Groups
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Introduction to Wallpaper Groups
• A Wallpaper Group is a discrete group of isometries of
the plane that contains noncollinear translations. These
are the symmetries of wallpaper patterns.
Wallpaper
Groups
The Lattice
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
• Let G be a wallpaper group.
• Choose a 6= 0 in the lattice L of G so that kak is a small
as possible.
• Choose b 6= 0 that is skew to a so that kbk is as small as
possible.
• kak ≤ kbk.
Theorem
The lattice L of G is
L = {ma + nb | m, n ∈ Z}.
Wallpaper
Groups
Lance Drager
Picture of a Lattice
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
b
a
A Lattice
Wallpaper
Groups
The Trace
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Definition
Here’s some standard linear algebra. If A is the matrix
a b
c d
we define tr(A), the trace of A, by
tr(A) = a + d
Exercises
All matrices here are 2 × 2.
1
2
If A and B are matrices, show tr(AB) = tr(BA). Just use
brute force.
If P is an invertible matrix, show tr(P −1 AP) = tr(A).
Wallpaper
Groups
The Crystallograhic Restriction 1
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Theorem (The Crystallographic Restriction)
The possible orders for a rotation in a wallpaper group are 2, 3,
4 and 6.
Proof.
Let R be a rotation in the point group K (G ). Then RL ⊆ L.
This means that Ra must be in the lattice, so Ra = ma + nb
for integers m, n. Similarily Rb = pa + qb for some p, q ∈ Z.
These equations can be written in matrix form as
m p
R[a | b] = [Ra | Rb] = [a | b]
.
n q
Let
P = [a | b],
m p
,
M=
n q
so RP = PM.
Wallpaper
Groups
Lance Drager
The Crystallograhic Restriction 2
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Proof Continued.
Since a and b are not colinear, P = [a | b] is invertible. so
R = PMP −1 . But then
tr(R) = tr(PMP −1 ) = tr(M) = m + q ∈ Z,
i.e., the trace of R is an integer. But
cos(θ) − sin(θ)
R = R(θ) =
=⇒ tr(R) = 2 cos(θ)
sin(θ) cos(θ)
Thus, cos(θ) must be a half-integer. Since −1 ≤ cos(θ) ≤ 1,
the possiblities are −1, −1/2, 0, 1/2, 1.
Wallpaper
Groups
Lance Drager
Crystallographic Restiction 3
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Proof Continued.
From Trig, we can figure out the corresponding angles:
cos(θ) = −1
cos(θ) = −1/2
cos(θ) = 0
cos(θ) = 1/2
cos(θ) = 1
=⇒
=⇒
=⇒
=⇒
=⇒
θ
θ
θ
θ
θ
= 180◦
= 120◦
= 90◦
= 60◦
=0
=⇒
=⇒
=⇒
=⇒
=⇒
o(R) = 2,
o(R) = 3,
o(R) = 4,
o(R) = 6,
R = I.
Wallpaper
Groups
Picture of the Angles
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
• Here’s the picture of the angles.
y
(0, 1)
√
(−1/2, 3/2)
√
(1/2, 3/2)
90◦
120◦
60◦
(−1, 0)
180
◦
0
◦
(1, 0)
x
Wallpaper
Groups
Classification of Lattices 1
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
• The lattices of wallpaper groups can be divided into 5
classes.
• Since a ± b are skew to a, kbk ≤ ka + bk, ka − bk.
• We can arrange ka − bk ≤ ka + bk by replacing b by −b,
if necessary. Then
kak ≤ kbk ≤ ka − bk ≤ ka + bk.
• We get 8 cases by considering = or < for each of the ≤’s
above.
Wallpaper
Groups
Classification of Lattices 2
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Case
1
2
3
4
5
6
7
8
Inequality
kak = kbk = ka − bk = ka + bk
kak = kbk = ka − bk < ka + bk
kak = kbk < ka − bk = ka + bk
kak = kbk < ka − bk < ka + bk
kak < kbk = ka − bk = ka + bk
kak < kbk = ka − bk < ka + bk
kak < kbk < ka − bk = ka + bk
kak < kbk < ka − bk < ka + bk
• Let’s look at the pictures.
Lattice
Impossible
Hexagonal
Square
Centered Rect.
Impossible
Centered Rect.
Rectangular
Oblique
Wallpaper
Groups
Lance Drager
Case 8, Oblique Lattice
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
b
a
Wallpaper
Groups
Lance Drager
Case 7, Rectangular Lattice
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
b
a
Wallpaper
Groups
Lance Drager
Case 3, Square Lattice
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
b
a
Wallpaper
Groups
Lance Drager
Case 2, Hexagonal Lattice
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
b
a
Wallpaper
Groups
Lance Drager
Case 6, Centered Rectangular
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
b
a
Wallpaper
Groups
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Case 4, Another Centered
Rectangular
b
a
Wallpaper
Groups
Names for the Wallpaper Groups
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Theorem
The are 17 geometric isomophism classes of wallpaper groups.
• Conway notation
• 2, 3, 4, 6 indicate a class of rotocenters of the given order.
• ∗ indicates a mirror. Digits after the star mean the
rotocenters are on a mirror line.
• x indicates a glide reflection (that does not come from a
pure reflection).
• 1 means there are only translations in the group.
• Crystallographic Notation (abbreviations from a longer,
logical system)
• p or c mean a primative or centered cell in the lattice.
• 2, 3, 4, 6 denotes the highest order of a rotation in the
group.
• m for mirror, stands for reflection, g stands for a glide.
Wallpaper
Groups
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Flowchart for Classification
Wallpaper
Groups
Lance Drager
Patterns to Classify
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
• Classify the following patterns. Don’t look at the answers
until you’ve given it a try!
Wallpaper
Groups
Wallpaper Problem 1
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Go to Answer
Wallpaper
Groups
Lance Drager
Wallpaper Problem 2
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Go to Answer
Wallpaper
Groups
Wallpaper Problem 3
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Go to Answer
Wallpaper
Groups
Wallpaper Problem 4
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Go to Answer
Wallpaper
Groups
Wallpaper Problem 5
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
Go to Answer
Wallpaper
Groups
Lance Drager
Wallpaper
Groups
Introduction
The Structure of
Wallpaper
Groups
Classification of
Lattices
Names for the
Wallpaper
Groups
How to Classify
Wallpaper
Groups
Patterns to
Classify
End of Lectures
Wallpaper
Groups
Answer to Wallpaper Problem 1
Lance Drager
∗632
Back to Problem
Wallpaper
Groups
Lance Drager
Answer to Wallpaper Problem 2
Back to Problem
3∗3
Wallpaper
Groups
Answer to Wallpaper Problem 3
Lance Drager
∗2222
Back to Problem
Wallpaper
Groups
Answer to Wallpaper Problem 4
Lance Drager
22∗
Back to Problem
Wallpaper
Groups
Lance Drager
Answer to Wallpaper Problem 5
Back to Problem
∗333