# Context help

### The tiling

The tiling is a regular filling of the plane, done with a single figure.

The basic figure is the tile:

The fundamental set of tiles is a group of several tiles which allows to do the tiling in 2 directions of translations:

### Fundamental set of tiles

The fundamental set of tiles is a group of several tiles which allows to do the tiling in 2 directions of translations.

### Tile

The tile is the basic element for a tiling

As a tile is in contact with another, each part of its **contour** appears twice.

### Group of displacement

Each type of tiling is associated to a group of displacement of the plane. This group is made of the displacement allowing to go from one tile to another.

The displacements are:

- The translations
- The rotations
- The central symmetries
- The axial symmetries
- The glide reflections

The groups associated to a different type of tiling are always the results of a finite number of displacements, within which are 2 translations.

### Translations

In a translation, you displace the figure without turning it over or upside down.

The translation is represented by an arrow called **vector**.

The 2 translations associated to each tiling allow to go from one fundamental set of tiles to another.

### Rotations

In a rotation, the figure turns around a point called **center**.

It is also necessary to know the **angle** of rotation.

The rotations associated to a tiling always have their angles being equal to:

- 1/6 of rotation and multiple angles (2/6=1/3, 3/6=1/2, 4/6=2/3 and 5/6)
- 1/4 of rotation and multiple angles (2/4=1/2 and 3/4)
- 1/3 of rotation and multiple angles (2/3)
- 1/2 of rotation (these are central symmetries)

The centers of rotation are always on the **contour** of the tile:

### Central symmetries

A central symmetry is a rotation the angle of which is a 1/2 turn.

### Axial symmetries

In an axial symmetry, the figure is turned around an **axis**.

When a tiling has an axis of symmetry, a part of the **contour** of the tile is along this axis and cannot be distorted.

The tilings with several directions of angles of symmetry, often have a fixed contour:

### Glide reflections

You can get a glide reflection if you compose an axial symmetry with a translation the **vector** of which is parallel to the axis of symmetry.

### Group

A group is a set (let us call it G) with a law of internal composition (if x and y are parts of G , x¤y too) associative (x ¤ y) ¤ z = x ¤ (y ¤ z),We can then write x¤y¤z) as a neutral element exists (e is the neutral element if, for each x, we’ve got x ¤ e = e ¤ x = x) and if each element has an inverse (reciprocal) (for each x , y exists as x ¤ y = y ¤ x = e).

**Examples:** * the integers (positive or negative) form a group for addition (the neutral element is zero and the opposite (additive inverse) of 5 is -5 * however, positive integers , don’t form a group for addition * the displacements of the plane form a group for composition.

### Composition

The composition of two displacements of the plane consists in doing two displacements one after the other.

**Example:**

- The composition of two translations is a translation:
- The composition of a translation and a rotation is a rotation:
- The composition of two axial symmetries is a translation or a rotation:

# The 17 types of tilings

### Type nb 1: p1

### (asymmetric parallelogramic)

The fundamental set of tiles is made of a single tile:

The group of displacement is generated by only two translations.

The **contour** of the tile has got 2 parts:

### Type nb 2: p2

### (symmetric parallelogramic)

The fundamental set of tiles is made of 2 tiles:

The group of displacement is generated by

The **contour** of the tile has got 4 parts:

### Type nb 3: p3

### (hexagonal 3-rotative)

The fundamental set of tiles is made of 3 tiles and both the vectors form an equilateral triangle:

The group of displacement is generated by

- 2 translations
- a rotation (1/3 of a turn)

The **contour** of the tile has got 3 parts:

### Type nb 4: p4

### (square 4-rotative)

The fundamental set of tiles is made of 4 tiles, and both the vectors form a half square:

The group of displacement is generated by

- 2 translations
- a rotation (1/4 of a turn)

The **contour** of the tile has got 3 parts:

### Type nb 5: p6

### (hexagonal 6-rotative)

The fundamental set of tiles is made of 6 tiles, and both the vectors form an equilateral triangle:

The group of displacement is generated by

- 2 translations
- a rotation (1/6 of a turn)

The **contour** of the tile has got 3 parts:

### Type nb 6: pg

### (gliding rectangular)

The fundamental set of tiles is made of 2 tiles, and both the vectors form a half rectangle:

The group of displacement is generated by

The **contour** of the tile has got 3 parts:

### Type nb 7: pgg

### (bi-gliding rectangular)

The fundamental set of tiles is made of 4 tiles, and both the vectors form a half rectangle:

The group of displacement is generated by

The **contour** of the tile has got 4 parts:

### Type nb 8: pm

### (mono-symmetric rectangular)

The fundamental set of tiles is made of 2 tiles, and both the vectors form a half rectangle:

The group of displacement is generated by

The **contour** of the tile has got 2 parts:

### Type nb 9: cm

### (mono-symmetric rhombic)

The fundamental set of tiles is made of 2 tiles, and both the vectors form an isosceles triangle:

The group of displacement is generated by

The **contour** of the tile has got 3 parts:

### Type nb 10: pmg

### (symmetric gliding

rectangular)The fundamental set of tiles is made of 4 tiles, and both the vectors form a half rectangle:

The group of displacement is generated by

The **contour** of the tile has got 3 parts:

### Type nb 11: pmm

### (bi-symmetric rectangular)

The fundamental set of tiles is made of 4 tiles, and both the vectors form a half rectangle:

The group of displacement is generated by

The **contour** of the tile is a rectangle.

### Type nb 12: cmm

### (bi-symmetric rhombic)

The fundamental set of tiles is made of 4 tiles, and both the vectors form an isosceles triangle:

The group of displacement is generated by

The **contour** of the tile has got 3 parts:

### Type nb 13: p4g

### (gliding 4-rotative square)

The fundamental set of tiles is made of 8 tiles, and both the vectors form a half square:

The group of displacement is generated by

- 2 translations
- an axial symmetry
- a rotation (1/4 of a turn)

The **contour** of the tile has got 2 parts:

### Type nb 14: p3m1

### (tri-symmetric hexagonal)

The fundamental set of tiles is made of 6 tiles, and both the vectors form an equilateral triangle:

The group of displacement is generated by

- 2 translations
- an axial symmetry
- a rotation (1/3 of a turn)

The **contour** of the tile is an equilateral triangle.

### Type nb 15: p31m

### (symmetric 3-rotative hexagonal)

The fundamental set of tiles is made of 6 tiles, and both the vectors form an equilateral triangle:

The group of displacement is generated by

- 2 translations
- an axial symmetry
- a rotation (1/3 of a turn)

The **contour** of the tile has got 2 parts:

### Type nb 16: p4m

### (totally symmetric square)

The fundamental set of tiles is made of 8 tiles, and both the vectors form a half square:

The group of displacement is generated by

- 2 translations
- an axial symmetry
- a rotation (1/4 of a turn)

The **contour** of the tile is a right and isosceles triangle (half square).

### Type nb 17: p6m

### (totally symmetric hexagonal)

The fundamental set of tiles is made of 12 tiles, and both the vectors form an equilateral triangle:

The group of displacement is generated by

- 2 translations
- an axial symmetry
- a rotation (1/6 of a turn)

The **contour** of the tile is a right triangle, the angles are 30°, 60° and 90°.