Planar Euclidean Crystallographic Groups

PLANAR EUCLIDEAN CRYSTALLOGRAPHIC GROUPS AND PARABOLIC ORBIFOLDS
CASEY LYNN KELLEHER
Abstract. We demonstrate the existence the 17 Euclidean planar crystallographic groups through focusing on the Euclidean Group on R2 and its actions on lattices generated by two linearly independent
vectors. We present two common notations for the groups, intuitive descriptions of each group, and
corresponding images of the groups’ actions and their corresponding orbifolds. We conclude with a group
theoretic proof of uniqueness of the groups, historical examples, and references.
Introduction
Wallpaper patterns have been prominent in artwork and design for over 3,000 years, so an interesting task
for a mathematician is study in internal structure of such pieces. In particular, the focus of this paper is to
demonstrate a means of catagorizing wallpaper images. Our solution is to classify wallpapers by structure
of the actions which generated such patterns through a group theoretic approach. We will also demonstrate
their unique associated topological structures: the parabolic orbifolds. Unlike conventional presentations
of these groups, we introduce them simultaneously with their orbifold with the intention of presenting the
reader with a more dynamic and memorable way to understand these objects.
Our first task will be to discuss a set of particular functions which act on the plane and form the Euclidean
Group. Taking two linearly independent vectors of R2 , we consider the all generated lattice types and deduce
the corresponding elements of the Euclidean Group which preserve each of these structures. Finally, through
a case by case analysis, we determine which combinations of elements of the Euclidean group, combined
with a given lattice, will generate groups satisfying our proposed definition of planar crystallographic group.
During this analysis, we provide images and intuitive discussions of each group and corresponding orbifolds.
We end with a proof of uniqueness of the 17 crystallographic groups and a collection of examples of the
groups from Japanese artwork.
Actions on the Plane
The Euclidean Group. Given metric spaces (Y, dY ) and (X, dX ) an isometry is a function T from the
metric space (X, dX ) to (Y, dY ) where T is bijective and has the property that dX (a, b) = dY (T(a), T(b))
for all a, b ∈ X. To state it simply, T is a function which preserves the distance between any two points.
We will consider when (X, dX ) = (Y, dY ) = (R2 , dE ), where dE denotes the standard Euclidean metric. As
mentioned, the interactions of the isometries over R2 are relevant to our study. The collection of all such
isometries together form the Euclidean group under function composition which acts on R2 .
The Euclidean Group (E2 , ◦) is the collection of all isometries P : R2 → R2 together with
the binary operation of function composition ◦.
Definition.
We will discuss the action of the Euclidean Group on the plane by examining two subgroups which
generate E2 . The first is the translational group, the collection of all translations of the plane, which we
denote by T2 . The group of translations T2 is isomorphic to R2 under addition, where for every vector
τ ∈ R2 we denote its corresponding translation Tτ ∈ T2 , where action of Tτ on p ∈ R2 is a shift by
τ given by Tτ (p) = τ + p. The second subgroup is the orthogonal group, the collection of orthogonal
transformations on R2 , which we will denote O2 . Recall that an orthogonal transformation R preserves the
inner product on a vector space, so in our setting, given p, q ∈ R2 , hp, qi = p · q = O(p) · O(q). This implies
0
Advised by Dr. Daryl Cooper from University of California at Santa Barbara
Key words and phrases. orbifolds, quotient topology, Euclidean group, crystallographic group, lattices.
1
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CASEY LYNN KELLEHER
that an orthogonal transformation is an angle preserving isometry, or equivalently, an isometry which fixes
the origin. A result of this additional property is that in R2 an orthogonal transformation is one of two
types of actions: a rotation by angle θ ∈ [0, 2π) about the origin, denoted Rθ , or a reflection about a
line passing through origin generated by some vector τ ∈ R2 , denoted Rτ . An additional element of the
Euclidean Group we will consider is a glide reflection, which is a reflection through a line τ ∈ R2 followed
by a translation by τ , written Gτ .
∼ T2 o O2 , the semidirect product induced by
These groups O2 and T2 together generate E2 , where E2 =
the action on O2 on T2 through conjugation. The group operation on T2 oO2 in this case is multiplication,
where for all R, Q ∈ O2 and α, β ∈ R2 , we have that (Tα , Q)(Tβ , R) = (Tα + QTβ , QR). This implies
that any element P of E2 may be expressed uniquely in the form P(p) = ρ + R(p) for some ρ ∈ R2 and
R ∈ O2 . For simplicity, we will create a mapping Φ : E2 → T2 o O2 where given P as defined above,
Φ(P) = (Tρ , R).
Φ decomposes elements of E2 as ordered pairs, thus presenting the elements in an unambiguous manner.
Given the normality of T2 , it is natural to consider the canonical projection Π into O2 given by Π :
T2 o O2 → O2 where Π(Tρ , R) = R for Tρ ∈ T2 and R ∈ O2 . This consideration, which we address in
the next section, is benificial since it allows us to ignore the location of an action and focus on whether
it is reversing an orientation of points through some reflection or preserving it through rotations. We may
subdivide E2 into four types of planar isometries, which we present without proof in the following theorem.
(the proof may be found in ? ]).
Theorem. Let α, p ∈ R2 . Elements of E2 may be catagorized as one of the four symmetries: a
translation by α from a point p, a rotation by θ about p, a reflection through an line generated by α
emanating from p, and a glide reflection based at p in the direction of the line generated by α.
Lattices of R2 . Given any subgroup G of E2 , we define its point group to be the set OG = Π(Φ(G)),
the collection of orthogonal portions of any action in G. Finite point groups combined with two linearly
independent translations generate our objects of focus: the planar crystallographic groups.
A collection K ⊂ E2 is a planar crystallographic group of R2 if K is generated by a finite
point group OK ⊂ O2 and two linearly independent translations τ1 , τ2 ∈ T2 , that is K = hOK , τ1 , τ2 i.
Definition.
Considering the orbit of the zero vector under the action of the translational subgroup of K is extremely
beneficial in creating and determining a group classification. By identifying the small collection of elements
of O2 which will harmoniously preserve the structure of the orbit we will ensure the point group is finite.
This intuitively relates to ensuring that the images tesselated in the production of a wallpaper pattern will
not overlap. We will pay particular attention to the preservation of the orbit of two linearly independent
vectors.
Let τ1 , τ2 ∈ R2 and linearly independent. The lattice generated by τ1 and τ2 is L = {ρ : ρ =
nτ1 + mτ2 for m, n ∈ Z}.
Definition.
Equivalently, this is the orbit of the zero vector under the actions of Tτ1 and Tτ2 . The lengths of the two
generating vectors, together with the lengths of their sum and difference, determine which of the 5 possible
lattice formations they will induce. Prior to naming these lattices, we will assert the convention that τ1 and
τ2 form an acute or right angle between each other, that is, ||τ1 – τ2 || ≤ ||τ1 + τ2 ||. If it is the case that we
choose two linearly independent vectors τ1 and τ2 that do not meet such specifications, we only need to
replace τ2 with –τ2 . Given these conventions, through a case by case analysis, we will deduce what sort of
orthogonal transformation preserve a given lattice L . In the appendix, we have included an essential part
of the proof which answers the following: ”Given an orthogonal transformation which preserves L , what
are the possible images of τ1 and τ2 ?". Assuming such results, we prove the following theorem.
Theorem. The only orthogonal transformations which preserve each the 5 lattice types of R2 are as
follows:
An oblique lattice is preserved by hRπ i.
A rectangular lattice is preserved by hRπ i ∪ {Rτ1 , Rτ2 }.
PLANAR EUCLIDEAN CRYSTALLOGRAPHIC GROUPS AND PARABOLIC ORBIFOLDS
3
A square lattice is preserved by hR π i ∪ {Rτ1 , Rτ2 , Rτ1 +τ2 , Rτ1 –τ2 }.
4
A centered lattice is preserved by hRπ i ∪ {Rτ1 , R2τ2 –τ1 }.
A hexagonal lattice is preserved by hR π i ∪ { Rτ1 , Rτ2 , Rτ1 –τ2 }.
3
Proof. Rather than go into technical detail in this proof, we will informally interpret the results of the
appendix. For the first three lattices, we would like to emphasize that the vectors τ1 and τ2 ’form’ the
sides of the polygon from which the lattice name is derived. For the last two, the relationship is not
as apparent, but may be easily understood with the figures displayed to the right as each is discussed.
Each figures feature a light green arrow for and a τ1 and dark green arrow for τ2 . The two purple
arrows are τ1 + τ2 and τ1 – τ2 . An important convention of our notation of ordered pairs below is as
follows: (±p, ±q) = {(p, q), (–p, –q)}, and (±p, ∓q) = {(p, –q), (–p, q)} (that is, the symbols ∓ and ±
are controlled together between pair components).
Oblique Lattice ||τ1 || < ||τ2 || < ||τ1 – τ2 || < ||τ1 + τ2 ||
As a consequence of the distinct norms and inequalities of the above vectors, this lattice exhibits a network of parallelograms. If an orthogonal
operation R preserves an oblique lattice generated by τ1 and τ2 , then
it follows that (R(τ1 ), R(τ2 )) ∈ {(±τ1 , ±τ2 )}. The pair (τ1 , τ2 ) corresponds to the action of the identity I and the pair (–τ1 , –τ2 ) corresponds
to a rotation by π, Rπ . The only rigid symmetries of a parallelogram are
rotations by multiples of π, so our deduced actions are intuitive.
Rectangular Lattice ||τ1 || < ||τ2 || < ||τ1 – τ2 || = ||τ1 + τ2 ||
A rectangular lattice is a network of rectangles formed by replacing the
latter inequality with equality and thus removing the tilt from the parallelogram. Recall that given a rectangle, the rigid motions are generated by a horizontal and vertical reflections and rotations by multiples
of π. If R ∈ O2 preserves a rectangular lattice, then R is such that
(R(τ1 ), R(τ2 )) ∈ {(±τ1 , ±τ2 )}. Thus the only orthogonal transformations which preserve this lattice are I, Rπ , and reflections in the directions
of both τ1 and τ2 , where the pair (τ1 , –τ2 ) is the action of Rτ1 and the
pair (–τ1 , τ2 ) is the action of Rτ2 .
Square Lattice ||τ1 || = ||τ2 || < ||τ1 – τ2 || = ||τ1 + τ2 ||
A square lattice can be created from the rectangular lattice by replacing the first inequality with equality, and thus forcing the side
lengths (||τ1 || and ||τ2 ||) to be equal. Since there are more rigid
motions which preserve squares than rectangles, there are consequently more transformations which preserve this particular lattice.
Given R ∈ O2 which preserves a square lattice, we have that
(R(τ1 ), R(τ2 )) ∈ {(±τ1 , ±τ2 ), (∓τ1 , ±τ2 ), (±τ2 , ±τ1 ), (∓τ2 , ±τ1 )}.
Since τ1 and τ2 are orthogonal, half-turns preserve the lattice since
for an 0 ≤ n ≤ 3 such that n is an integer (R nπ (τ1 ), R nπ (τ2 )) ∈
2
2
{(τ1 , τ2 ), (τ2 , –τ1 ), (–τ1 , –τ2 ), (–τ2 , τ1 )}. The remaining elements are reflections where Rτ1 and Rτ2 are expressed by (±τ1 , ∓τ2 ), and Rτ1 +τ2 and
Rτ1 –τ2 are expressed by the pairs (τ2 , τ1 ) and (–τ2 , –τ1 ) respectively.
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CASEY LYNN KELLEHER
Centered Lattice ||τ1 || < ||τ2 || = ||τ1 – τ2 || < ||τ1 + τ2 ||
A centered lattice exhibits a structure similar to that of a rectangular
structure. where the difference hinges on the fact that a central point
within the rectangles appears for the centered lattice since the generating
vector τ2 points to the center of the rectangle rather than trace out a
side. Although the symmetries are the same as that of the rectangular
lattice, they are described differently because of the angle between τ1
and τ2 . Now the vector 2τ2 – τ1 is orthogonal to τ1 and traces out
the side of the rectangle which induces the symmetry of this particular
lattice. If R ∈ O2 preserves a centered lattice, then (R(τ1 ), R(τ2 )) ∈
{(±τ1 , ±τ2 ), (±τ1 , ∓τ2 ), (±τ1 , ±(τ1 – τ2 ))}. Using the same argument
as for an rectangular lattice, we can quickly conclude that the operations
which preserve this lattice are the identity I, Rπ expressed by the pair
(–τ1 , –τ2 ), the reflection Rτ1 given by (τ1 , τ1 – τ2 ) and the reflection in
the perpendicular direction R2τ2 –τ1 , expressed by the pair (–τ1 , τ2 – τ1 ).
Hexagonal Lattice ||τ1 || = ||τ2 || = ||τ1 – τ2 || < ||τ1 + τ2 ||
A hexagonal lattice has generating vectors which trace two sides of
one of the six triangles which tile a hexagon. If R ∈ O2 preserves
a hexagonal lattice generated by τ1 and τ2 , then (R(τ1 ), R(τ2 )) ∈
{(±τ1 , ±τ2 ), (±τ2 , ±τ1 ), (±(τ1 – τ2 ), ±τ1 ), (±τ1 , ±(τ1 – τ2 )), (±(τ1 –
τ2 ), ∓τ2 ), (∓τ2 , ±(τ1 – τ2 ))}. Rotations by mutiples of π3 are given by
(R nπ (τ1 ), R nπ (τ2 )) ∈ {(±τ1 , ±τ2 ), (±τ2 , ±(τ2 – τ1 )), (±(τ2 – τ1 ), ∓τ1 )}
3
3
for integers n such that 0 ≤ n ≤ 5. The reflections are given in the remaining pairs, where Rτ1 corresponds to (τ1 , τ1 – τ2 ), Rτ2 to (τ2 – τ1 , τ2 ),
and Rτ1 –τ2 to (–τ2 , –τ1 ). 2
Now that we have affirmed what orthogonal transformations preserve for each lattice, we have significantly limited the number of potential point groups, and thus crystallographic groups to consider.
Planar Crystallographic Groups
Quotienting by the Groups. Rather than describe only the planar crystallographic groups, we will introduce them simultaneously with their corresponding topological structures, the parabolic orbfolds.
Given a planar crystallographic group K, we define the corresponding orbifold R2 /K, to be
the quotient topology formed by R2 modded out by the equivalence relation ∼, where p ∼ q if P(p) = q
for some P ∈ K.
Definition.
Roughly speaking, a parabolic orbifold is created by "folding the plane" with respect to the actions of
a given group K. Orbifolds are not only topological spaces, but are metric spaces where the metrics are
inherited by the standard metric on R2 (one can study this further in [6]). This quotienting process does not
necessarily result in a smooth surface, and the potential singularities are a means of relating such structures
to a given crystallographic group. There are three types of singularities which will be presented on the
parabolic orbifolds. First, we say that a point po ∈ W is an n-cone point if for each point of its preimage
p ∈ R2 , there exists some rotation Rθ ∈ K about p which rotates by θ and has order precisely n = 2π
θ .
The point p is called a rotocenter of order n, and it’s image po is located on a region of the orbifold locally
homeomorphic to a cone. In fact, po is the point of such a cone which has an internal angle of 2π
n . Next,
we say that a point qo ∈ W is a boundary point if its preimage points q ∈ R2 are located on one mirror
line about which a reflection acts on R2 . If the preimage q of qo is located at the intersection of n mirror
PLANAR EUCLIDEAN CRYSTALLOGRAPHIC GROUPS AND PARABOLIC ORBIFOLDS
5
lines, we say that qo is an n-corner point at which the orbifold displays a corner on a boundary with angle
precisely πn .
Notation. Now, we shall briefly discuss the notation which we will use for the groups. We include both
Conway’s orbifold notation as well as the more widely known standard notation.
Orbifold. This notation emphasizes the connection of the orbifold and crystallographic group where each
symbol incorporates double meaning. Every planar orbifold may be established through topological manipulations of the sphere S2 (puncturing, adding a handle, or creating a ’cross cap’) in addition to including
the cone points and corner points described during our discussion of modding. All group names will be
strings of the form
◦ · · · ◦ ABC · · · ? abc · · · xyz ? αβ · · · × · · · ×
Where A, B, C, · · · , a, b, c, · · · , and α, β, · · · are positive integers greater that 1 where A ≥ B ≥ C ≥ · · · ,
a ≥ b ≥ c ≥ · · · , and α ≥ β · · · . Note that for a particular group, specific numbers and symbols (◦,
×, and ?) may be excluded. ◦ symbolizes that the tesselations are generated solely by translations, and
thus void of any action by orthogonal operation. This corresponds to attaching a handle to S2 (connect
summation with a torus). × pertains to the ability to create a path between a generating tile and image
about a reflection line without intersecting a mirror line. This causes the image and its reflection to be
unified during the modding procedure without being "deflated" by a mirror. Their unification results in a
strange twisting cap on the surface, called a crosscap (connect summation of with the real projective plane).
? indicates the presence of a reflection(s), and corresponds to a boundary in the orbifold. Each number
following ? indicates the presence of the intersection of reflective lines, which is unique up to translation)
and how many lines intersect at a given point. They correspond to the number and order of corner points
on the boundary of the orbifold. A number prior to a ? pertains to the highest order of rotations which do
not act on reflective boundaries (if they do act on boundaries, they are instead converted into corner points
during the modding procedure). In the orbifold, these numbers represent orders of cone points. One may
refer to [3] for additional information.
Standard. With this more widely known notation groups are denoted by 4 letter strings of the following
format:
(cell type)(highest rotation order)(main reflection type)(secondary reflection type).
For "cell type" we consider the cell, or a compact region of positive area which is repeated throughout
the plane. There are two types: p is a primitive cell is the minimal region that is repeated by translation,
often formed with sides τ1 and τ2 , and contains no interior lattice points. We classify the marjority of the
planar Euclidean crystallographic groups with respect to these cells. Type c, for "centered", is where the
cell of the contains a lattice point in the center. We will only see those of type c for the centered lattice.
For the highest rotation number, we simply observe all possible rotations on the lattice, and choose the
number of the highest order. As we saw in the previous section, the set of numbers which will be presented
are {1, 2, 3, 4, 6}. For the following two slots, a ”main" direction is designated for a reflection (m) or glide
reflection (g) (e.g. in the direction of τ1 ), or nothing (1). The second indicates the presence of a secondary
reflection type operation in an alternate direction.
The Groups and Their Orbifolds. We will now use cases to discuss the possible crystallographic groups.
With this in mind, we will now consider each lattice type, and deduce possible combinations of orthogonal
operations which preserve such lattice and form point group structures. In addition, images will be provided
to display the operations of the group, as well as the corresponding parabolic orbifold resulting from modding
R by the given crystallographic group. The internal interactions of the elements of the point group as well
as their combinations with the translational group induce orthogonal actions not initially introduced in the
generating set. Additionally, some orthogonal operations are induced that are not elements of the point
group, such as a rotation induced by nonparallel reflections which occur between lattice points.
First, let us discuss notational and image conventions. We will display groups titles in orbifold notation
[2] followed by standard notation [1] within parantheses. For the images, we have the following conventions:
a light green arrow represents τ1 and its corresponding image, a dark green arrow constitutes τ2 and
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CASEY LYNN KELLEHER
its corresponding image. A red line constitutes a reflective symmetry (either a glide reflection or strict
reflection axis). The intersection of the red axes will create corner points. For the images of orbifolds, we
shall make the order of an n- corner point with a red 2n (since the angle of such corner point is πn = 2π
2n ).
A blue curve or blue region constitutes a rotational symmetry. For the images of orbifolds, we shall also
mark the order of a n-cone point with a blue n on the orbifold (since the angle of such cone point is 2π
n ).
A purple region or designates an nonunique example of a tile, that is, a portion of the pattern which
generates the rest. Some of the induced actions are difficult to see or describe, so we will be provide the
corresponding action that induces them for the reader’s assistance in both the description and figure color
coding.
Theorem. There exists exactly 17 planar crystallographic groups which are unique up to isomorphism.
Proof Our method for deducing the groups is as follows: first, we begin with a lattice (oblique, square,
centered, rectangular, hexagonal). We then consider all actions which preserve such lattice as described
in the previous section. Then, we consider one ’base’ point group element and introduce all possible
combinations of compatible additions to the point group. We shall exclude our computations for what
elements are ’compatible’; this relies on ensuring that orientation of elements are preserved and do not
contradict each other at every point on the plane. Our choices for base elements proceeds in the following
order: rotations (beginning from lowest order and increasing), reflections, then glide reflections.
We will now discuss the planar Euclidean crystallographic groups, beginning with the one group which
preserves all lattices, ◦, which we shall mention first rather than repeat in each case.
◦ (p1) This is the simplest of the wallpaper groups, and has trivial
point group O◦ = {I}. This implies that the entire group, ◦ is
generated solely by the translations τ1 and τ2 . The corresponding orbifold is a torus, a handle glued to S2 , which is a smooth
surface with no singularities or boundaries. The smoothness of
the surface alludes to the fact that R2 is the universal covering
space for the torus, with corresponding covering map given by
quotienting R2 by the actions of ◦.
Oblique Lattice Groups Only I and Rπ preserve an oblique lattice, giving us only one additional group to
introduce.
2
2
2
2
2222 (p2) The point group of 2222 consists only of rotations
by multiples of π, giving us O2222 = hRπ i, which is isomorphic to Z2 . Now, if we consider the entire group 2222, composition with translations yield that given any m, n ∈ Z, not
only do rotations about nτ1 + mτ2 preserve this lattice, but
also any rotation about n2 τ1 + m
2 τ2 , given by the composition
T(mτ1 +nτ2 ) Rπ . We have marked four unique rotocenters to
the right (any other gyration is a translation of those marked).
Each of these points corresponds to a distinct 2-cone point on
the surface of the sphere, resulting in the name, four point
pillow.
PLANAR EUCLIDEAN CRYSTALLOGRAPHIC GROUPS AND PARABOLIC ORBIFOLDS
7
Rectangular Lattice Groups The orthogonal transformations which preserve a rectangular lattice are I,Rπ ,
Rτ1 , and Rτ2 .
?? (pm) The point group of ?? consists of a reflection about
the line generated by either τ1 or τ2 , but not both. The choice
of the direction of the reflection is irrelevant since the realized crystallographic groups given by choices of τ1 and τ2 are
isomorphic. We will assume that, as in the figure, the reflection is Rτ2 , so we have O?? = hRτ2 i which is isomorphic to
Z2 . As depicted to the left, the reflections on axes emanating
from each point of the lattice generate parallel axes of reflection halfway in between, through the points m
2 τ1 + nτ2 . This
is created by translating the reflection through the point 21 τ2 ,
given by Rτ2 (Tτ1 )–2 . The resulting orbifold is an annulus, written symbolically as S1 × [0, 1] where S1 denotes the unit circle
within R2 . The two unique reflection axes in the direction of
τ2 contributes to loss of dimension of the sphere and the two
boundary curves (internal and external) of the annulus.
×× (pg) Similar to ??, the point group of ×× contains a reflection either in the direction of τ1 or τ2 , and the choice of
the direction of the reflection is irrelevant. Again, we choose
τ2 to demonstrate, giving us O×× = hRτ2 i. However, this
crystallographic group is differentiated from ??, since a glide
reflection in the direction of τ2 , Gτ2 replaces the reflection Rτ2
and the translation Tτ2 . Similar to the reflections of ??, the
glide reflection lines through lattice points generate another line
of glide reflections midway. These glide reflections are created
by translations of the glide reflection emanating from point 21 τ1
in the direction of τ2 by Gτ2 (Tτ1 )–2 . The orbifold for ×× is the
Klein bottle. It is worth noting that R2 is the covering space
for the Klein bottle, and the covering map may be realized by
this modding procedure.
4
4
4
4
?2222 (p2mm) Another variation of the point group of ?× includes both possible reflections, and allow them both to be
realized in the crystallographic group. This gives us O?2222 =
hRτ1 , Rτ2 i. Reflecting in both directions induces halfturn rotan
tions at lattice points as well as at positions of m
2 τ1 + 2 τ2 for
all m, n. Although this may not be initially clear, it is easy to
see with a quick sketch that Rτ2 Rτ1 = Rτ1 Rτ2 = Rπ . The corresponding orbifold is a square, since the points of rotation all
are on reflection axes, none can be realized as a cone point on
the orbifold and are instead corner points. This group is very
similar to 2222, except that the reflection boundaries ”deflate"
the four point pillow.
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CASEY LYNN KELLEHER
2
2
22? (p2mg) This group has the same point group as ?2222,
that is, O22? = hRτ1 , Rτ2 i. However, one of the reflections is
expressed through a glide reflection instead. The interaction of
Rτ1 and Rτ2 induce half turn symmetries once again, but these
become more sparse due to the spacing created by the glide
reflection. The position of such rotational points are m
2 τ1 +nτ2
for odd m only, which are lying off of strict reflection axes and
on glide reflection axes. This allows two distinct cone points of
order 2 to be obtained in the resulting orbifold, an open four
pillow.
22× (p2gg) This group shares the same point group as 22?
and 22× given by O22× = hRτ1 , Rτ2 i. This group is the final
variation of those containing such a point group, since both Rτ1
and Rτ2 are realized only as glide reflections. Similar to 22?
and ?2222, the two glide reflections Gτ1 , and Gτ2 generate the
rotation of π degrees. The corresponding orbifold is the Real
Projective Plane, which is formed by the two distinct twists
in the directions of τ1 and τ2 induced by the glide reflections.
Since it is nonembeddable in R2 , we display instead an inversion
of this surface.
1
Centered Lattice Groups The orthogonal transformations which preserve this lattice are I, Rπ , Rτ1 , Rτ2 – 2 τ1 ,
and Rπ . The distinction between the crystallographic groups of the centered lattice and the rectangular
lattice stems from the position of τ2 . The slant allows actions like glide reflections to develop independently
from a distinct translation and reflection component within the group.
?× (cm) This point group is generated only by one reflection
in the direction of τ1 which is realized in ?×, so we have that
O?× = hRτ1 i. The fact that the translation in the direction
of τ2 is not perpendicular to τ1 induces an axis of glide reflection acting between the horizontal rows of lattice points.
The resulting orbifold is a M obius Strip, given symbolically by
e where the symbol ×
e indicates the twist prior to gluing.
S1 ×I,
This twist was induced by the glide reflections acting on axes
independent of strict reflections, and the one boundary of the
strip is a consequence of the reflection axis. If we had chosen
instead to begin our group with a reflection in the direction per1
pendicular to τ1 , given by Rτ2 – 2 τ1 , (which goes in the direction
of the side length of the rectangle) we would create a group
isomorphic to that which we mentioned above. We also note
that by beginning with a glide reflection along lattice points
and instead having a reflection axis for Rτ1 through the centers
of the rectangles would ultimately result in a group isomorphic
to ours that has been translated in the direction of τ2 , and is
thus the "same" crystallographic group up to isomorphism.
PLANAR EUCLIDEAN CRYSTALLOGRAPHIC GROUPS AND PARABOLIC ORBIFOLDS
2 ? 22 (c2mm) The point group of 2 ? 22 contains the reflection
in the direction of τ1 as well as one perpendicular to τ1 , in
the direction of 2τ2 – τ1 , so O2?22 = hRτ1 , R(2τ2 –τ1 ) i which
is isomorphic to Z2 × Z2 . Both reflections are realized in the
crystallographic group itself, and they together generates glide
reflection halfway between reflection axes in both directions,
similar to the horizontal case above. The interaction of the
two reflections Rτ1 and R2τ2 –τ1 create halfturns, as in the case
of ?×. The corresponding orbifold is a cap, consisting of a disk
with two corner points of order 2 and a gyration point on top
of order 2.
2
4
4
9
Square Lattice Groups The orthogonal operations which preserve a square lattice are Rτ1 , Rτ2 , Rτ1 +τ2 ,
Rτ1 –τ2 , and R nπ for n ∈ Z.
2
442 (p4) The point group of 442 is generated soley by quarter
turns, so we have that O442 = hR π i. The two unique quarter
2
turns (up to translation) generate a half turn by composition
as seen to the left, where the upper right and left rotation are
quarter turns while the lower right is a half turn. The resulting
orbifold is an isoceles triangular pillow, where the angles are determined by the orders of the distinct cone points; two of order
4 and one of order 2. We write this orbifold as as S2 (4, 4, 2).
4
2
4
8
8
4
?442 (p4mm) This group has a point group which is that of
442 with the addition of a reflection about τ1 , so O?442 =
hRτ1 , R π i. This reflection is realized in ?442, and combined
2
with the quarter turn immediately generates Rτ2 . The reflections Rτ1 and Rτ2 together generate the reflection Rτ1 +τ2 and
Rτ1 –τ2 . We see that the point group exhibits, and is in fact isomorphic to, the dihedral group D8 . Since none of the rotations
act off of a location of a reflection axis, the orbifold does not
exhibit cone points and is an isoceles triangle. We denote this
orbifold as B2 (8, 8, 4).
10
CASEY LYNN KELLEHER
4 ? 2 (p4gm) This group has the same point group as ?442, given
by O4?2 = hRτ1 , R π i. The group is distinct from that above
2
because the reflection Rτ1 is replaced with a glide reflection,
which results in greater placement between reflection axes and
also isolation of a rotocenter from a reflection axis. Results of
the group’s structure on the orbifold, with respect to that of
?442 is a new cone point of order 4 and a decrease of the order
of the points contributing to the corner points of order 2 on
the boundary of the orbifold. We call the orbifold of 4 ? 2 an
open three pillow.
4
4
4
Hexagonal Lattice Groups The orthogonal transformations which preserve this particular lattice are Rτ1 ,
Rτ2 , Rτ2 –τ1 and R nπ for n ∈ Z.
3
333 (p3) This point group is generated by only rotations of
order three, given by O333 = hR 2π i. The resulting crystal3
lographic group exhibits only three unique rotations (up to
translation), and thus its orbifold exhibits three distinct 3-cone
points. We call this an equilateral three point pillow, and write
it as S2 (3, 3, 3).
3
3
3
?333 (p3m1) This point group generated with a rotation of order
three and the reflection through the line generated by τ1 + τ2 ,
giving us that O?333 = hR 2π , Rτ1 +τ2 i. The interaction of the
3
Rτ1 +τ2 and R 2π generates of 6 distinct reflection axes, as seen
3
to the left, all which intersect with rotocenters. As a result, this
orbifold is the ”deflated" the three point pillow, an equilaterial
triangle.
6
6
6
PLANAR EUCLIDEAN CRYSTALLOGRAPHIC GROUPS AND PARABOLIC ORBIFOLDS
11
3 ? 3 (p31m) This point group generated with a rotation of order
three and the reflection through the line generated by τ1 , giving
us that O3?3 = hR 2π , Rτ1 i. In comparison with ?333, the since
3
the reflection axis of Rτ1 and the direction of τ1 coincide, less
reflection axes are generated. The exposed rotocenter results
in one 3-cone point on the orbifold, and the sparse reflection
intersections result in one 3-corner point. This orbifold is an
open equilateral three point pillow.
3
6
632 (p6) This point group is generated by rotations by π6 , giving
us O632 = hR π i. Certainly, since rotations are orientation
6
preserving, it is impossible for them to generate any reflections,
so we find that three distinct rotocenters up to translation are
produced. As a consequence, quotienting R2 by this group
creates a regular 3 point pillow, featuring a one cone point of
order 6, of order 3, and of order 2.
6
2
3
12
6
?632 (p6mm) This point group is generated by R π and a re3
flection in the direction of τ1 (choosing τ2 instead generates
an isomorphic outcome), so we have O?632 = hRτ1 , R π i. The
3
axes of the generated reflection lines in the directions of τ1 ,
τ2 , and τ2 – τ1 together bound a triangle which forms a copy
of the orbifold, the ’deflated’ version of 632 given by a regular
triangle with three distinct n-corner points where n ∈ {6, 3, 2}.
4
Uniqueness. Since we have shown through case by case exhaustion of each lattice type that there are at
most 17 possible crystallographic groups, it remains to show that none of these 17 are isomorphic. We appeal
to the proofs given in Armstrong [1]. Recall in our description of the crystallographic groups that there are
a set of 8 possible groups to which the point groups are isomorphic: {e, Z2 , Z4 , Z6 , Z2 × Z2 , D6 , D8 , D12 }.
We will sort the crystallographic groups into types and see that within these types each of the members
are unique. When addressing these, it is important to emphasize that images of rotations, reflections, and
translations are preserved under isomorphism. This is the result of the intrinsic qualities of each action,
such as how reflections always have order 2 and how reflections may generate rotations but to reverse is
not necessarily true.
Type e, Z6 , and D6 For groups of type e, Z3 , Z4 , Z6 , and D6 , since there is only one member of each
type (◦, 333, 2222, 632, and ?632 respectively) we conclude that these groups are clearly unique from all
12
CASEY LYNN KELLEHER
others.
Type Z2 The members of this type are (2222, ?×, ??, and ××). Unlike the other groups, 2222 contains
a rotation, and is thus unique. Of the remaining, ×× is the only group without any reflection since being
generated by a glide reflection and translation disallows any reflection to be generated. Between ?? and
?×, glide reflections define their difference. Although both ?? and ?× consist of reflections being generated in perpendicular directions, the orthogonality of the generating vectors τ1 and τ2 of the lattice of ??
guarantees that a glide reflection is derived from reflections and translations, and therefore shifts by units
expected (that is, only in integer quantities). This is not the case for ?×; as seen in the figure, the glide
reflections generated between the two reflection axes only shifts by 21 τ1 , which means that the translational
component of the glide reflection, T 1 τ is not an element of ?×. This inability to decompose implies that
2
?× is not isomorphic to ??, so we conclude that thefour members of this type are all unique.
Type Z4 The members of this type are ?442 and 4 ? 2. Since ?442 contains both perpendicular reflections,
the group generated by the set of reflections contains every rotation within the crystallographic group. In
4 ? 2 however, since a reflection is replaced with a glide, the collection of pure reflections in 4 ? 2 cannot
generate all possible rotations. We conclude that ?442 and 4 ? 2 are distinct.
Type D12 The members of this type are ?333 and 3 ? 3. Within 3 ? 3, any reflection of order 3 (R 2π based
3
at arbitrary points) may be decomposed into a composition of reflections about various lines within 3 ? 3.
This is not the case for ?333. Thus, the two groups are not isomorphic.
We have thus shown the existance and uniqueness of the 17 planar crystallographic groups. 2
PLANAR EUCLIDEAN CRYSTALLOGRAPHIC GROUPS AND PARABOLIC ORBIFOLDS
13
Wallpaper Pattern Examples from Japan. Below we display a collection of patterns found throughout
Japanese history, each which exhibit the action of one of the 17 wallpaper patterns. We include names of
such example designs from [7] in italics and a translation of the name into English. These common designs
◦ bingata
crimson shape
2222 ariso-donsu
stormy shore-damask
?? yakata-mon
mansion-design
×× hana-usagi-kinran
flower-rabbit-gold
thread
?2222 yoshiwara-tsunagi 22? shikan-jima
Yoshiwara1-connector
Shikan2-stripes
22× fundo-tsunagi
counter
weightconnector
?× seikaiha
blue ocean waves
2 ? 22 takeda-bishi
Takeda3-rhombus
?442 shippo-tsunagi
7 treasures5-connector
442 rokuyata-koshi
Rokuyata4-lattice
?333
kasane-kikkou-ni
wa
4 ? 2 sayagata
333 onaga-mitsudomoe repeat-tortoise
shellsilk containing patterns long tail-mitsudomoe6 loop
3 ? 3 bishamon-kikkou
Bishamon7-tortoise
shell
632 kagome
basket hole
?632 asanoha
hemp leaf
from artwork across the Japan were collected by Tohsuke Urabe from the Department of Mathematical
Sciences in Ibaraki University, Japan and are displayed on Geometry of Wall Paper Patterns .
1
Yoshiwara downtown region of Edo (old Tokyo)
Nakamura Shikan famous kabuki actor
3
Takeda Shingen Fuedal Japanese Lord
4
Okabe Rokuyata Kabuki character
5
The 7 Buddhist treasures gold, silver, pearls, agate, crystal, coral, lapis lazuli
6
Mitsudomoe Traditional family crest with three (mitsu) comma marks
7
Bishamon (Vaisravana) a guardian god of Buddhism
2
14
CASEY LYNN KELLEHER
Appendix
Lattice Preserving Elements of O2 . Let the lattice L be generated by linearly independent vectors τ1
and τ2 as defined in the ”Lattices of R2 " section. We deduce the possible images of τ1 and τ2 under a
lattice preserving orthogonal transformation. We will write h·, ·i to denote the metric induced inner product,
and thus in the context of this paper we mean the Euclidean dot product. Also, recall our convention for
ordered pairs is as follows: (±p, ±q) = {(p, q), (–p, –q)}, and (±p, ∓q) = {(p, –q), (–p, q)} (that is, the
symbols ∓ and ± are controlled together between pair components). By determining p, q, m, n ∈ Z such
that ||τ1 || = ||pτ1 + qτ2 || and ||τ2 || = ||mτ1 + nτ2 ||, we are determining all possible images of τ1 and τ2
under a lattice preserving orthogonal transformation. Before we begin such proofs, we will first produce a
simple identity which we will utilize throughout this section. Let m, n ∈ Z, and observe that
||mτ1 + nτ2 ||2
=
hmτ1 + nτ2 , mτ1 + n~τ2 i
=
hmτ1 , mτ1 i + 2 hmτ1 , nτ2 i + hnτ2 , nτ2 i
=
m2 hτ1 , τ1 i + 2mn hτ1 , τ2 i + n2 hτ2 , τ2 i
=
m2 ||τ1 ||2 + 2mn hτ1 , τ2 i + n2 ||τ2 ||2 .
We will refer to this identity by †. We now address each lattice type and restate the restrictions on the
generating vector lengths as described in the ”Lattices of R2 " section.
Hexagonal Lattice.
||τ1 || = ||τ2 || = ||τ1 – τ2 || < ||τ1 + τ2 || .
Since it is clear that ||τi || < ||mτj || for i, j ∈ {1, 2}, where m ∈ Z\{0, ±1}, then we shall consider nontrivial
linear combinations (m 6= 0and n 6= 0). The property that ||τ1 || = ||τ2 || and identity † together imply that
||mτ1 + nτ2 ||2 = m2 + n2 ||τ2 ||2 + 2mn hτ1 , τ2 i. Thus, if we suppose that ||mτ1 + nτ2 || = ||τ1 || (which
||mτ1 + nτ2 || is equal to ||τ2 || and ||τ2 – τ1 ||) then ||mτ1 + nτ2 ||2 = ||τ1 ||2
is, inadvertently assuming that
and equivalently, m2 + n2 ||τ2 ||2 + 2mn hτ1 , τ2 i = ||τ1 ||2 by our above work, which implies that ||τ1 ||2 =
–2mn hτ , τ i. Further, note that
m2 +n2 –1 1 2
||τ1 – τ2 ||2
=
||τ1 ||2 – 2 hτ1 , τ2 i + ||τ2 ||2
=
||τ1 ||2 – 2 hτ1 , τ2 i + ||τ1 – τ2 ||2 since ||τ2 ||2 = ||τ1 – τ2 ||2 .
It follows that ||τ1 ||2 = 2 hτ1 , τ2 i, which shows clearly that hτ1 , τ2 i =
6 0. These facts imply that
||mτ1 + nτ2 ||2 = ||τ1 ||2 if and only if ||τ1 ||2 = 2 hτ1 , τ2 i = m2–2mn
hτ
, τ i. Thus 2 = m2–2mn
, or
+n2 –1 1 2
+n2 –1
equivalently, (m, n) ∈ {(1, –1), (–1, 1)}. Therefore, the only linear combinations that can have the same
norm as τ1 , τ2 , and (τ1 – τ2 ) are ±τ1 , ±τ2 ,and ±(τ1 – τ2 ). Because ||τ1 – τ2 || < ||τ1 + τ2 ||, it follows
that the angles between τ1 , τ2 , and the angles between their images under any orthogonal transformation
must be acute. Trivial calculations reveal that the only possible images of the pair (τ1 , τ2 ) under an orthogonal transformation R ∈ O2 which preserves a hexagonal lattice are (±τ1 , ±τ2 ), (±τ2 , ±τ1 ), (±(τ1 –
τ2 ), ±τ1 ), (±τ1 , ±(τ1 – τ2 )), (±(τ1 – τ2 ), ∓τ2 ), (∓τ2 , ±(τ1 – τ2 )).
Square and Rectangular Lattice.
||τ1 || ≤ ||τ2 || < ||τ1 – τ2 || = ||τ1 + τ2 ||
Where = is imposed by square identity, and < imposed by rectangular. For this particular lattice, we first
claim that hτ1 , τ2 i = 0. Observe that since ||τ1 – τ2 || = ||τ1 + τ2 ||, in addition, note that ||τ1 ± τ2 ||2 =
||τ1 ||2 ± 2 hτ1 , τ2 i + ||τ2 ||2 , so since ||τ1 – τ2 || = ||τ1 + τ2 ||, then it follows that hτ1 , τ2 i = – hτ1 , τ2 i or
equivalently, that hτ1 , τ2 i = 0, as desired.
With this in mind, we suppose that ||mτ1 + nτ2 || = ||τ1 ||. This implies that
||τ1 || =
=
m2 ||τ1 ||2 + 2mn hτ1 , τ2 i + n2 ||τ2 ||2
m2 ||τ1 ||2 + n2 ||τ2 ||2 since 2mn hτ1 , τ2 i = 0.
PLANAR EUCLIDEAN CRYSTALLOGRAPHIC GROUPS AND PARABOLIC ORBIFOLDS
15
In the square case, note that since ||τ1 || = ||τ2 ||, we are considering the case when ||τ1 + τ2 || = ||τi ||
for i ∈ {1,
2}. For convenience, we will display the case of i = 1, from which we have that ||τ1 || =
m2 + n2 ||τ1 ||2 , so rearrangment of terms gives us that
0 =
m2 + n2 – 1 ||τ1 ||2 .
The above is true if and only if m2 + n2 = 1, i.e. for the solution pairs (m, n) ∈ {(±1, 0) , (0, ±1)}.
We conclude that τ1 and τ2 may be mapped to only ±τ1 and ±τ2 . Note, however, that the pairs
(±τ1 , ∓τ1 ) and (±τ2 , ∓τ2 ) are impossible outcomes since these pairs are not orthogonal vectors, and
thus fail our deduced inner product value (that is hτ1 , τ2 i =
6 h±τi , ∓τi i for i ∈ {1, 2} ). We conclude that the possible image pairs for R ∈ O2 where R preserves a square lattice are (R(τ1 ), R(τ2 )) =
{(±τ1 , ±τ2 ), (∓τ1 , ±τ2 ), (±τ2 , ±τ1 ), (∓τ2 , ±τ1 )}.
Now we will approach the rectangular case, we revert to the line prior to assuming the square case.
Note that we must address the cases when ||τ1 + τ2 || is equal to ||τ1 || and is equal to ||τ2 || separately.
Assuming first equivalence to the magnitude of τ1 , this particular lattice’s identities yield that ||τ1 ||2 =
m2 ||τ1 ||2 + n2 ||τ2 ||2 since 2mn hτ1 , τ2 i = 0, and thus that
1 – m2
=
n2
||τ2 ||2
||τ1 ||2
≥ n2 .
Note that equality holds if and only if n = 0, in which case m2 = 1 and m = ±1. If n 6= 0 then
m2 + n2 < 1, which holds only if m = n = 0, a contradiction. Thus the only possible images for τ1 are
±τ1 . Similarly, if we assume that ||mτ1 + nτ2 || = ||τ2 ||, then we have that ||τ2 ||2 = m2 ||τ1 ||2 + n2 ||τ2 ||2
since 2mn hτ1 , τ2 i = 0, so
–m2
=
||τ ||2
2
2
≥
n
–
1
n2 – 1
||τ1 ||2
This implies that 1 > m2 + n2 . For this to hold, we have either (m, n) ∈ {(±1, 0), (0, ±1)}. Certainly,
if n = ±1 and m = 0, then we have that ||mτ1 + nτ2 || = || ± τ2 || = ||τ2 ||. The latter pair gives that
||mτ1 + nτ2 || = || ± τ1 || 6= ||τ2 ||, resulting in a contradiction. We conclude that τ1 may only be mapped to
±τ1 and τ2 may only be mapped to ±τ2 . We conclude that if R ∈ O2 preserves a rectangular lattice, then
it must be that (R(τ1 )R(τ2 )) ∈ {(±τ1 , ±τ2 ), (±τ1 , ∓τ2 )}.
Centered Lattice.
||τ1 || < ||τ2 || = ||τ1 – τ2 || < ||τ1 + τ2 ||
Given the above, observe that expanding hτ1 – τ2 , τ1 – τ2 i and the equality ||τ2 || = ||τ1 – τ2 || yields that
hτ1 – τ2 , τ1 – τ2 i = ||τ1 ||2 – 2 hτ1 , τ2 i + ||τ2 ||2 , which implies that 2 hτ1 , τ2 i = ||τ1 ||2 . We first assume that
||mτ1 + nτ2 || = ||τ1 ||, so it follows from † that since ||τ1 ||2 = m2 ||τ1 ||2 + 2mn hτ1 , τ2 i + n2 ||τ2 ||2 , we have
that
0 =
m2 – 1 ||τ1 ||2 + 2mn hτ1 , τ2 i + n2 ||τ2 ||2
=
m2 – 1 ||τ1 ||2 + mn ||τ1 ||2 + n2 ||τ2 ||2 by above, which gives that
n2
=
(1 – m2 – mn)
||τ1 ||2
||τ2 ||2
≤ (1 – m2 – mn).
If n = 0, then 1 – m2 = 0, so m ∈ {1, –1}. Otherwise if n 6= 0, then 1 – m2 – mn > n2 which we rewrite as
+ mn + n2 < 1. If |m| ≥ |n|, then mn + m2 ≥ 0, so n2 < 1. But we assumed that n 6= 0, so |m| < |n|.
Given that, it must be that n2 + mn > 0, so m2 < 1. This implies that m = 0, yielding n2 < 1, again a
contradiction. Therefore the only possible image for τ1 is ±τ1 .
m2
16
CASEY LYNN KELLEHER
Next, we assume that ||mτ1 + nτ2 || = ||τ2 ||. This case is symmetric to the previous proof. After
manipulations of † given this equality we come to the conclusion that.
n2 – 1
=
(–m2 – mn)
||τ1 ||2
||τ2 ||2
2
≤ (–m – mn)
n2
If m = 0, then
– 1 = 0, so n ∈ {1, –1}. In this case, we have that m = 0 or m2 = –mn, which gives
us that m = –n. Otherwise if m 6= 0, then –m2 – mn > n2 – 1 which we rewrite as m2 + mn + n2 < 1. If
|n| ≥ |m|, then mn + n2 ≥ 0, so m2 < 1. But we assumed that m 6= 0, so |n| < |m|. Given that, it must be
that m2 + mn > 0, so n2 < 1. This implies that n = 0, yielding m2 < 1, again a contradiction. Therefore
the only possible images for τ2 is ±τ2 . We conclude that if R ∈ O2 preserves a centered lattice, then it
must be the case that (R(τ1 ), R(τ2 )) ∈ {(±τ1 , ±τ2 ), (±τ1 , ∓τ2 ), (±τ1 , ±(τ1 – τ2 ))}.
Oblique Lattice.
||τ1 || < ||τ2 || < ||τ1 – τ2 || < ||τ1 + τ2 || .
We first show that hτ1 , τ2 i > 0. Since ||τ1 ± τ2 ||2 = ||τ1 ||2 ±2 hτ1 , τ2 i+||τ2 ||2 , then taking the difference
of the two yields that 4 hτ1 , τ2 i = ||τ1 + τ2 ||2 – ||τ1 – τ2 ||2 > 0, thus hτ1 , τ2 i > 0. For the remaining proof
it is favorable to address it in cases concerning the parity of m and n.
First, we assume that either m, n ≥ 0, or m, n ≤ 0. If it is the case that ||mτ1 + nτ2 || = ||τ1 ||, then by
† we have that
0 =
m2 – 1 ||τ1 ||2 + 2mn hτ1 , τ2 i + n2 ||τ2 ||2 .
Since all terms are positive the only way for this expression to be zero is if the coefficients of each term are
zero. We thus have that m = ±1, andn = 0, so the only solutions are (m, n) ∈ {(±1, 0)}. Now, consider
if instead we have that ||mτ1 + nτ2 || = ||τ2 ||. It follows similarly by † that
0 = m2 ||τ1 ||2 + 2mn hτ1 , τ2 i + n2 – 1 ||τ2 ||2 .
Since all terms are positive the only way for this expression to be zero is if the coefficients of each term are
zero, so n = ±1, and m = 0. The only solutions are (m, n) ∈ {(0, ±1)}
Now we approach the final cases, where we assume m and n are nonzero and are of opposite parity. First,
if we assume that m < 0 and n > 0, we will suppose to the contrary that m is strictly positive, with the
intention of showing that n is identically zero. First, for clarity, it is acceptable for us to reorient the plane
so that τ1 = e1 , the standard basis vector in R2 . As a result, we have that τ2 = (r cos θ, r sin θ) where θ is
the angle between τ1 and τ2 , and r = ||τ2 || > 1. Since, by assumption ||τ1 – τ2 || > ||τ2 ||, we have that
0
<
||τ1 – τ2 ||2 – ||τ2 ||2
=
||(1 – r cos θ, r sin θ)||2 – r2
=
1 – 2r cos θ
1,
2
It follows that r cos θ <
so the first coordinate of τ2 is bounded above by 21 . We know that r cos θ is
positive since
0 < ||τ1 + τ2 || – ||τ1 – τ2 || = (1 + 2r cos θ) – (1 – 2r cos θ) = 4r cos θ.
And since ||τ2 || > 1, we have that
1 < ||τ2 ||2 = r2 cos2 θ + r2 sin2 θ <
√
1
+ r2 sin2 θ
4
Thus, it follows that 23 < r sin θ. Given our vector v = mτ1 + nτ2 , consider the dot product of this vector
with the vector e2 . Since we are assuming that ||v|| = ||τ1 || = 1, it follows that
√
3
1 > v · τ2 = (mτ1 + nτ2 ) · τ2 = n (r sin θ) > n
2
PLANAR EUCLIDEAN CRYSTALLOGRAPHIC GROUPS AND PARABOLIC ORBIFOLDS
17
We confirm that n < √2 , implying that n = 1. Given this, by †, if we assume that ||τ1 ||2 = m2 +
3
2mnr cos θ + n2 r2 . Utilizing the quadratic formula we attain
q
–2r cos θ ± (–2r cos θ)2 – 4(r2 – 1)
m =
2
q
= –r cos θ ± 1 + r2 cos2 θ – 1 .
√ 2
Note that n2 r2 cos2 θ – 1 = –n2 r2 sin2 θ, and so under the radical we have 1– n2 r2 sin2 θ < 1– 23 < 0
and thus there is no possible real m. We have reached a contradiction, and thus conclude that if m < 0
and n > 0, then ||mτ1 + nτ2 || 6= ||τ1 ||. Now, if we instead assume that ||mτ1 + nτ2 || 6= ||τ2 || , then we
have instead that ||τ2 ||2 = m2 + 2mnr cos θ + n2 r2 , so 0 = m2 + 2mr cos θ, giving us the solutions m = 0
or m = 2r cos θ > –1. Since m is negative, we conclude that m = 0, a contradiction to the assumption that
m > 0. Now, consider instead when n < 0 and m > 0. We instead consider taking the dot product with
(–τ2 )
√
3
1 > |v · (τ2 )| = |(mτ1 + nτ2 ) · (τ2 )| = |0 + n (r sin θ)| > |n|
.
2
It follows that |n| is less than √2 , which implies that n = –1. Given this, we have that ||τ1 ||2 = m2 –
3
2mr cos θ + r2 . Utilizing the quadratic formula we attain
q
2r cos θ ± (2r cos θ)2 – 4(r2 – 1)
m =
2
q
= r cos θ ± 1 + r2 cos2 θ – 1 .
Note that r2 cos2 θ – 1 = –r2 sin2 θ, and thus we have yet again under the radical we have 1– n2 r2 sin2 θ <
√ 2
1 – 23 < 0 and thus there is no possible real m. We have reached a contradiction, and thus conclude
that if m < 0 and n > 0, then ||mτ1 + nτ2 || 6= ||τ1 ||.
Finally, we assume that ||mτ1 + nτ2 || = ||τ2 ||, which implies by † that 0 = m2 – 2mnr cos θ. Thus,
m = 2r cos θ < 1, which implies that m = 0, a contradiction to the assumption that m > 0. We conclude
that if m > 0 and n < 0, then ||mτ1 + nτ2 || 6= ||τ2 ||. We conclude if R ∈ O2 preserves an oblique lattice,
then the possible pairs are (R(τ1 ), R(τ2 )) ∈ {(±τ1 , ±τ2 )}.
Acknowledgments
The author thanks Dr. Daryl Cooper of University of California at Santa Barbara for his inspiration,
patience and guidance beyond the scope of this document, Blake Allison from University of California Irvine
for his many revisions and constant support, Dr. Mark Stankus of California Polytechnic State University
at San Luis Obispo for his aid, especially with lattice proofs in the appendix. Finally, the author thanks Dr.
Linda Patton and Dr. Matthew White of California Polytechnic State University at San Luis Obispo, who
initially inspired her to pursue a career in mathematics and have been constantly encouraging her since.
References
[1] M. Armstrong. Groups and Symmetry, pages 136-165. Springer Verlag, 1988.
[2] J. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetry of Things, pages 1-123. A K Peters,
Ltd., 2008.
[3] J. Conway, D. Huson, ’The Orbifold Notation for Two-Dimensional Groups’, Structural Chemistry,
Vol. 13, Numbers 3-4, pages 247-257. DOI: 10.1023/A:1015851621002.
[4] R. Gamelin, T. Greene, Introduction to Topology 2nd Ed., pages 59-104. Dover Publications Inc.,
1999.
[5] B. O’neill, Elementary Differential Geometry, pages 364-375. Elsevier Inc., 2006.
[6] P. Scott, ’The Geometries of 3-Manifolds’, Bull. London Math. Soc., Vol. 5 (1983) pages 401-437.
18
CASEY LYNN KELLEHER
[7] T. Urabe, 2012: ’Seventeen Kinds of Wallpaper Patterns’, Ibaraki University Mathematics Laboratory.
[http://mathinfo.sci.ibaraki.ac.jp/open/mathmuse/urabe/indexE.html.]
Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407
E-mail address: ckellehe@calpoly.edu