SingSurf Home Wallpaper app Help Maths Orbifolds Groups Code

Wallpaper patterns - the mathematics

Below we describe the four basic transformations, translations, reflection, rotation and glide reflection that all patterns are mode of; the 17 different Wallpaper patterns the frieze groups, cyclic group and dihedral groups. Further sections describe the alternative orbifold notation and how orbifolds can be used to find the Chromatic number of each pattern giving the minimum number of colours needed to colour symmetrical tilings so that no two tiles of the same colour share an edge.

Separate pages give more details on group theory and the mathematics behind the code.

Basic transformations

A translation	A reflection - M	A glide-reflection: a translation follow by a reflection - G
180° rotation - P2	120° rotation - P3	90° rotation - P4	60° rotation - P6

Each wallpaper pattern is based on a pair of translations in different directions. The translations are applied multiple times to give a repeating image. This defined a regular grid of points called a lattice.
The blue shape in the program is called the fundamental domain. The image inside it is translated, rotated and reflected to produce the final image. The yellow parallelogram is one cell in the lattice, the image inside every other cell will be identical to this.
The symmetry lines can be shown. In this pattern the final image is produced by reflecting in the magenta lines. This pattern also has rotational symmetry: each cyan-rectangle is a center of 180° rotation.
Some groups allow some flexibility in the shape of the fundamental domain. For instance the P1 group allows curved edges as used in some of Escher's woodcuts. Here, for simplicity, the fundamental domain is restricted to be a polygon although alternate version of the P1 and P2 rules based on hexagonal cells and the P31M rule with a kite-shaped fundamental domain. Other groups are more restrictive, if the group contains rotation points then the fundamental domain must pass through these point and if the pattern contains a line of reflection then that line will form one edge of the fundamental domain.

The patterns

Lattice type: parallelogram
	P1: the simplest pattern, two translations in different directions.		P2: this pattern has an extra 180° rotation.
Lattice type: diamond
	CM: a reflection along a diagonal		CMM: two reflections along the diagonal
Lattice type: rectangle, the translations are at right angles
	PM: A reflection		PG: A glide reflection
	PMG: A reflection and a glide reflection		PGG: Two glide reflections at right angles		PMM: Two reflections at right angles
Lattice type: square
	P4: a 90° rotation		P4M: a 90° rotation and a reflection		P4G: a 90° rotation and a glide-reflection
Lattice type: hexagon
	P3: a 120° rotation		P3M1: a 120° rotation and a reflection. Line of reflections pass through opposite vertices of the hexagon		P31M: a 120° rotation and a reflection. Line of reflections pass through centers of the edges the hexagon
	P3: a 60° rotation		P3M: a 60° rotation and a reflection

Frieze groups, cyclic groups and dihedral groups

As well as the wallpaper groups their are three other families of symmetries in the plane.

• The seven frieze groups
• An infinite family of cyclic groups: rotations around a point.
• An infinite family of dihedral groups: generated by two lines of reflection which pass through a point.

Whilst the wallpaper groups repeat in two directions the frieze groups only repeat in a single direction

	F1 translation in one direction Symmetry of the infinite sequence of letters pppppp
	F2 a glide-reflection pdpdpd.
	F3 translation and a parallel reflection CCCCCC
	F4 translation and a pair of parallel reflections pqpqpq
	F5 translation and 180° rotation pdpdpd
	F6 translation and a perpendicular reflection and 180° rotation pdbqpdbq.
	F7 translation, perpendicular reflection, parallel reflection XXXXXX.

Frieze groups are often found in decorative borders. The top border of this page is F2 and the left hand side border is F4.

The cyclic and dihedral groups all have rotational symmetry about a single point. Cyclic groups have n-fold rotation symmetry (rotation by 360°/n). Dihedral groups have an n-fold rotational symmetry and two reflections in lines at 180°/n to each other.

		Cyclic groups
C2	C3	C4	C5	C6
		Dihedral groups
D2	D3	D4	D5	D6

The group D1 is technically classed as a dihedral group it just consists of a single reflection.

D1

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SingSurf Home Wallpaper app Help Maths Orbifolds Groups Code

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