Symmetries: Identity | Rotations | Reflections | Rotoreflections

## Rotational Symmetries 23

Rotational symmetries are transformations involving the rotation of the cube algorithms around its symmetry. A cube has 13 symmetry axes: three mid (tetrad) axes, six edge (diad) axes and four corner (triad) axes.

A cube has three types of axes and their respective rotational symmetries:

- Rotational symmetries around mid axes 9
- Rotational symmetries around edge axes 6
- Rotational symmetries around corner axes 8

A cube has a total of 23 rotational symmetries as described below.

### Rotational Symmetries around mid (tetrad) axes 9

Symmetry group: `C`

_{4}

Rotational Symmetries around mid (tetrad) axes pertain to transformations involving the 4-fold 90 degree (1/4 turn) rotation of a cube algorithm around its three distinct axes: x-axis, y-axis, and z-axis. There are nine such symmetries.

#### Rotational Symmetries around y-axis 3

Symmetry group: `C`

Order: _{4}`2`

and `4`

Rotational Symmetries around y-axis involve a 4-fold rotation of a cube algorithm around the y-axis that passes through the center points of the upper and bottom faces of the cube. There are three such symmetries.

#####
`2.`

y Rotation

Symmetry group: `C`

Order: _{4}`4`

In a `y`

Rotation symmetry, the cube algorithm turns by 90-degree clockwise around the y-axis.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the `y'`

symmetry.

#####
`3.`

y2 Rotation

Symmetry group: `C`

Order: _{4}`2`

In a `y2`

Rotation symmetry, the cube algorithm turns by 180-degree around the y-axis.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply it again.

#####
`4.`

y' Rotation

Symmetry group: `C`

Order: _{4}`4`

In a `y'`

Rotation symmetry, the cube algorithm turns by 90-degree counterclockwise around the y-axis.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the `y`

symmetry.

#### Rotational Symmetries around x-axis 3

Symmetry group: `C`

_{4}

Rotational Symmetries around x-axis involve a 4-fold rotation of a cube algorithm around the x-axis that passes through the center points of the right and left faces of the cube. There are three such symmetries. There are three such symmetries.

#####
`5.`

x Rotation

Symmetry group: `C`

Order: _{4}`4`

In a `x`

Rotation symmetry, the cube algorithm turns by 90-degree clockwise around the x-axis.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the `x'`

symmetry.

#####
`6.`

x2 Rotation

Symmetry group: `C`

Order: _{4}`2`

In a `x2`

Rotation symmetry, the cube algorithm turns by 180-degree around the x-axis.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply it again.

#####
`7.`

x' Rotation

Symmetry group: `C`

Order: _{4}`4`

In a `x'`

Rotation symmetry, the cube algorithm turns by 90-degree counterclockwise around the x-axis.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the `x`

symmetry.

#### Rotational Symmetries around z-axis 3

Symmetry group: `C`

_{4}

Rotational Symmetries around z-axis involve a 4-fold rotation of a cube algorithm around its z-axis that passes through the center point of the front and back faces of the cube. There are three such symmetries.

#####
`8.`

z Rotation

Symmetry group: `C`

Order: _{4}`4`

In a `z`

Rotation symmetry, the cube algorithm turns by 90-degree clockwise around its z-axis.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the `z'`

symmetry.

#####
`9.`

z2 Rotation

Symmetry group: `C`

Order: _{4}`2`

In a `z2`

Rotation symmetry, the cube algorithm turns by 180-degree around its z-axis.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply it again.

#####
`10.`

z' Rotation

Symmetry group: `C`

Order: _{4}`4`

In a `z'`

Rotation symmetry, the cube algorithm turns by 90-degree counterclockwise around its z-axis.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the `z`

symmetry.

### Rotational Symmetries around edge (diad) axes 6

Symmetry group: `C`

Order: _{2}`2`

Rotational symmetries around edge axes involve a 2-fold, 180-degree (1/2 turn) rotation of a cube algorithm around its six distinct diad axes that passes through the midpoints of two opposite edges of the cube. There are six such symmetries.

#####
`11.`

FR Rotation

Symmetry group: `C`

Order: _{2}`2`

In a `FR`

Rotation symmetry, the cube algorithm turns 180 degrees around the axis connecting the midpoints of the Front-Right and Back-Left edges.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply it again.

#####
`12.`

FL Rotation

Symmetry group: `C`

Order: _{2}`2`

In a `FL`

Rotation symmetry, the cube algorithm turns 180 degrees around the axis connecting the
midpoints of the Front-Left and Back-Right edges.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply it again.

#####
`13.`

UR Rotation

Symmetry group: `C`

Order: _{2}`2`

In a `UR`

Rotation symmetry, the cube algorithm turns 180 degrees around the axis connecting the
midpoints of the Upper-Right and Down-Left edges.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply it again.

`14.`

UL Rotation

Symmetry group: `C`

Order: _{2}`2`

In a `UL`

Rotation symmetry, the cube algorithm turns 180 degrees around the axis connecting the
midpoints of the Upper-Left and Down-Right edges.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply it again.

#####
`15.`

UF Rotation

Symmetry group: `C`

Order: _{2}`2`

In a `UF`

Rotation symmetry, the cube algorithm turns 180 degrees around the axis connecting the
midpoints
of the Upper-Front and Down-Back edges.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply it again.

#####
`16.`

UB Rotation

Symmetry group: `C`

Order: _{2}`2`

In a `UB`

Rotation symmetry, the cube algorithm turns 180 degrees around the axis connecting the
midpoints
of the Upper-Back and Down-Front edges.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply it again.

### Rotational Symmetries around triad (corner) axes 8

Symmetry group: `C`

Order: _{3}`3`

Rotational symmetries around triad (corner) axes involve a 3-fold, 120-degree (1/3 turn) rotation of a cube algorithm around its four distinct triad axes that pass through the opposite corners of the cube. There are a total of eight rotational symmetries of this type.

#####
`17.`

FUR Rotation

Symmetry group: `C`

Order: _{3}`3`

In a `FUR`

Rotation symmetry, the cube algorithm turns 120 degrees clockwise around the axis passing
through the Front-Upper-Right corner and its opposite Back-Down-Left corner.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the RUF Rotation symmetry.

#####
`18.`

RUF Rotation

Symmetry group: `C`

Order: _{3}`3`

In a `RUF`

Rotation symmetry, the cube algorithm turns 120 degrees counterclockwise around the axis
passing through the Right-Upper-Front corner and its opposite Left-Down-Back corner.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the FUR Rotation symmetry.

#####
`19.`

BUL Rotation

Symmetry group: `C`

Order: _{3}`3`

In a `BUL`

Rotation symmetry, the cube algorithm turns 120 degrees clockwise around the axis passing
through the Back-Upper-Left corner and its opposite Front-Down-Right corner.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the LUB Rotation symmetry.

#####
`20.`

LUB Rotation

Symmetry group: `C`

Order: _{3}`3`

In a `LUB`

Rotation symmetry, the cube algorithm turns 120 degrees counterclockwise around the axis
passing through the Left-Upper-Back corner and its opposite Right-Down-Front corner.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the BUL Rotation symmetry.

#####
`21.`

LUF Rotation

Symmetry group: `C`

Order: _{3}`3`

In a `LUF`

Rotation symmetry, the cube algorithm turns 120 degrees clockwise around the axis passing
through the Left-Upper-Front corner and its opposite Right-Down-Back corner.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the FUL Rotation symmetry.

#####
`22.`

FUL Rotation

Symmetry group: `C`

Order: _{3}`3`

In a `FUL`

Rotation symmetry, the cube algorithm turns 120 degrees counterclockwise around the axis
passing through the Front-Upper-Left corner and its opposite Back-Down-Right corner.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the LUF Rotation symmetry.

#####
`23.`

RUB Rotation

Symmetry group: `C`

Order: _{3}`3`

In a `RUB`

Rotation symmetry, the cube algorithm turns 120 degrees clockwise around the axis passing
through the Right-Upper-Back corner and its opposite Left-Down-Front corner.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the BUR Rotation symmetry.

#####
`24.`

BUR Rotation

Symmetry group: `C`

Order: _{3}`3`

In a `BUR`

Rotation symmetry, the cube algorithm turns 120 degrees counterclockwise around the axis
passing through the Back-Upper-Right corner and its opposite Front-Down-Left corner.

To apply this symmetry, tap the menu button within the CubePal app and navigate as follows:

To revert this symmetry, apply the RUB Rotation symmetry.

Symmetries: Identity | Rotations | Reflections | Rotoreflections