George Chemmala

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Wallpaper Groups and Orbifolds

A brief introduction to wallpaper groups and classification using orbifolds

For my Fundamental Problems of Geometry class I did my final paper and presentation on wallpaper groups. Here’s the paper remade for the web using this $\LaTeX$ to markdown/html converter.

Wallpaper groups represent the complete classification of all two-dimensional repetitive patterns based on their symmetries. These 17 distinct groups encode the ways in which shapes can repeat to cover the Euclidean plane, incorporating translations, rotations, reflections, and glide reflections. This paper offers an intuitive exploration of wallpaper groups, from geometric fundamentals to the basic theory of orbifolds.

Introduction #

Wallpaper groups combine elements of geometry, algebra, and art. At their core lies a fascinating question: how many distinct ways can a two-dimensional pattern repeat across the plane? Interestingly, the answer is only 17.

What Are Wallpaper Groups? #

A wallpaper group is a group of isometries of the Euclidean plane that contains two linearly independent translations.

Let’s break this definition into two parts:

Isometries of the Euclidean Plane #

An affine transformation $T$ of the Euclidean plane is called an isometry if it preserves distances; that is, for all pairs of points $X, Y$, $$d(X, Y) = d(T(X), T(Y))$$

The four types of isometries in the Euclidean plane can be defined as follows:

  • Translations, denoted $T_{\vec{v}}$, where $\vec{v} \in \mathbb{R}^2$. These move each point in the plane by the vector $\vec{v}$.

  • Rotations, denoted $R_{c,\theta}$, where $c$ is the center of rotation and $\theta$ is the angle. These rotate the plane around point $c$ by angle $\theta$.

  • Reflections, denoted $F_L$, where $L$ is a line in $\mathbb{R}^2$. These reflect the plane across line $L$, referred to as the mirror or axis of reflection.

  • Glide Reflections, denoted $G_{L,d}$, where $L$ is a line and $d$ is a distance. These combine a reflection across $L$ with a translation along $L$ by $d$.

A wallpaper group will be a group of isometries that can be generated by these four types of transformations.

The set of all isometries in $F \subset \mathbb{R}^2$ is a group under function composition.

Let $G$ be the set of all isometries in $\mathbb{R}^2$ that map a set $F$ onto itself where $T_1, T_2, T_3 \in G$ . Then we have:

  • Associativity: For all isometries $T_1, T_2, T_3$, $(T_1 \circ T_2) \circ T_3 = T_1 \circ (T_2 \circ T_3)$ since composition is associative.

  • Closure: For all isometries $T_1, T_2$, $T_1 \circ T_2$ is also an isometry since composition of isometries preserves distances.

  • Identity: There exists an identity transformation $I$ such that for any isometry $T$, $I \circ T = T$ since $I$ leaves all points unchanged.

  • Inverses: For every isometry $T$, there exists an inverse transformation $T^{-1}$ such that $T^{-1} \circ T = I$. This is because isometries are bijective.

Two Linearly Independent Translations #

This condition ensures the pattern repeats in two non-parallel directions, tiling the entire plane. Here are two non-examples:

  • The Hong Kong flag is not a wallpaper group — it has no translational symmetry since it has only a rotational symmetry around the center.


  • The Greek meander, a frieze pattern, has only one translational direction along the pattern and thus fails the definition.


The group $\ast2222$ features two independent translations, four rotation centers of order 2, and reflection lines. It satisfies both conditions and qualifies as a wallpaper group.

Here, we see that one “tile” can be translated in two independent directions, following the arrows, relected across the lines, and rotated $\frac{\pi}{2}$ over the rotation centers.

There are only 17 distinct wallpaper groups—a complete classification of plane symmetry patterns.

This classification is not simple to derive, but it can be done systematically using group theory and the crystallographic restriction - a condition that limits the possible rotation symmetries in these groups to $n = 2, 3, 4, 6$. Later in this paper, we will discuss an alternate approach to this classification using orbifolds.

Visual Intuition: What Symmetries Look Like #

It’s a difficult to find all the symmetries in a complex wallpaper pattern, so let’s observe some simpler patterns that are easier to analyze. These patterns contain the translations, and one of the other symmetries (rotations, reflections, or glide reflections).

A pattern with only translations:

A pattern with 2-fold symmetry rotational symmetry:

A pattern with reflections:

A pattern with glide reflections:

Historical Context: Wallpaper Groups in the Alhambra #

The Alhambra is a palace in Granada, Spain renowned for its decorative geometric patterns and tilings. It was built in the mid-13th century by the Nasrid Kingdom, the last Muslim dynasty in Spain, as a military zone and later became the royal residence. The palace had been continuously occupied, leading to it being the only surviving example of a palatine city of the Islamic period in the Iberian Peninsula.

Because of its immense size and complexity, the Alhambra is famously thought to contain examples of all 17 wallpaper groups; however, this claim is likely false. Although the Alhambra is not a perfect example of all 17 wallpaper groups, it does contain at least 13 of them and was later the inspiration for M.C. Escher’s work.

Esher had left the Netherlands in 1922 with the intention of visiting the Alhambra, but he was unable to do so until 1936. He was fascinated by the intricate geometric patterns and tilings found in the palace and his brother had encouraged him to study crystal patterns and later tessellations, which led to some of his most famous works. Escher’s art often features repeating patterns, reflections, and symmetries, which are all elements of the wallpaper groups.

How the 17 Groups Arise #

Orbifolds and Fundamental Domains #

An orbifold is a topological space that is locally a finite group quotient of a Euclidean space.

Each wallpaper group corresponds to a distinct orbifold with a characteristic fundamental domain—the smallest part that can tile the plane through the group’s symmetries.

An ant crawling on a pattern with only translations would perceive its world as a torus-shaped space where it sees one edge of the fundamental domain as the edge right behind it i.e. if it continues walking in a line parallel to one of the edges, it will eventually come back to the same point. This reveals how the symmetries of the wallpaper group “fold” the plane into a finite space.

Orbifold Notation #

John Conway’s introduced orbifold notation to succinctly describes the symmetries. In this notation, the orbifold is represented as a string of symbols, each representing a symmetry type.

He also uses a different terminology to describe the symmetries. Translations, reflections, rotations, glide reflections are renamed to wonders, kaleidoscopes, gyrations, and miracles respectively.

A gyration ($n$) is a point in the orbifold where the symmetry group has a rotational symmetry. The notation for a gyration is $n$, where $n$ is the order of the rotation which can be $2, 3, 4,$ or $6$ because of the crystallographic restriction.

In Figure rotational_symmetry, we see a 2-fold rotation across 4 different centers, denoted $2222$.

A kaleidoscope ($\ast$) is a point in the orbifold where the symmetry group has a reflection. The notation for a kaleidoscope is $\ast n$, where $n$ is the number of lines of symmetry intersecting at that point.

In Figure reflections, we see a pattern with reflection lines, denoted $\ast\ast$. The number $1$ can be omitted like in this case, where we have 2 lines of symmetry not intersecting at a point.

A miracle ($\times$) is a point in the orbifold where the symmetry group has a glide reflection.

In Figure glide_reflections, we see a pattern with glide reflections, denoted $\times\times$. Similar to the previous symmetry, the number $1$ can be omitted.

A wonder ($o$) is a point in the orbifold where the symmetry group has only translations. Allthough a wallpaper group must have translations, we use the notation $o$ to denote the presence of translations only.

In Figure only_translations, we see a pattern with only translations, denoted $o$.

“Cost” of an Orbifold #

The Euler Characteristic is given by the formula:

$$ \chi = V - E + F $$

where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces of an obejct in a surface $S$. This formula is a topological invariant, meaning it remains unchanged under continuous deformations of the surface.

Orbifolds can be thought of as a “fractional” version of a manifold. And we can simialrly define the Euler characteristic for an orbifold. Here is a brief outline of how we can think of the orbifold Euler characteristic and how we can calculate it for our wallpaper groups.

Consider the example of a brick divided into $8$ equal bricks.

The total map is made of of $26$ vertices, $48$ edges, and $24$ faces. We can calculate the Euler characteristic of this map using the formula:

$$ V - E + F = 26 - 48 + 24 = 2$$

This is the Euler characteristic of a sphere. Since the symmetry group of the brick has order $8$, its orbifold comes from one eighth of the surface of the celestial sphere, so the calculation of its orbifold Euler characteristic should be one eighth of the Euler characteristic of the sphere.

The brick has $3 = 24 / 8$ faces (red, yellow, and blue). And we should expect $E = 48 / 8 = 6$ edges, but we see $9$ edges (three edges meet at the corner of the brick and six edges run around the boundary of the orbifold).

We can resolve this by considering the $6$ edges on the boundary are really half edges, because each of them is shared equally between two copies of the orbifold.

So we have 3 full edges, and $6$ half-edges, so that:

$$ E = 3 + \frac{6}{2} = 6$$

Similarly, we can consider the vertices. There is one full vertex (lying in only one copy of the orbifold), three half-vertices (shared between two copies), and three quarter-vertices (each lying in four copies of the orbifold), so that:

$$ V = 1 + \frac{3}{2} + \frac{3}{4} = \frac{26}{8}$$

So now we can think of a map on the orbifold as a map on the sphere divided by some symmetry group of order $N$, so our orbifold euler characteristic will be:

$$ \frac{V - E + F}{N} = \frac{2}{N}$$

Therefore, the orbifold Euler characteristic of a spherical pattern is the Euler characteristic of the sphere divided by the order of the symmetry group of the pattern. Now we can apply this idea to our wallpaper group.

Conway’s Magic Theorem #

For a wallpaper group, the orbifold Euler characteristic is zero. Hence, the total cost of the orbifold must equal 2.

Using this theorem and the costs from the previous section, we can find what orbifold corresponds to a wallpaper group by trying all the different combinations of the costs.

Eventually, we get the following table.

Classification of Wallpaper Groups Based on Rotation and Reflection Properties #

Smallest RotationReflections?Orbifold Group
$\tau$/ 6Yes$*632$
No$632$
$\tau$/ 4Yes$*442$
No$4*2$
$442$
$\tau$/ 3Yes, off mirror$3*3$
No, on mirror$*333$
$333$
$\tau$/ 2Yes, perpendicular refl.$*2222$
No, perpendicular refl.$2*22$
Glide reflection$22*$, $22×$, $2222$
noneGlide axis off mirror$*×$
No glide axis$**$
Glide reflection (yes)$××$
Glide reflection (no)$o$

Conclusion #

Wallpaper groups classify all two-dimensional repetitive patterns through their symmetries, with exactly 17 unique groups. These groups, defined by isometries of the Euclidean plane—translations, rotations, reflections, and glide reflections—are elegantly labeled using orbifold notation, where each group corresponds to a distinct orbifold. The fact that the orbifold Euler characteristic for wallpaper groups is zero leads to Conway’s Magic Theorem, which states that the total cost of the orbifold must equal 2 and leads to a systematic classification of the groups.

References #

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