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: C4
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: C4
Order: 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: C4
Order: 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: C4
Order: 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: C4
Order: 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: C4
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: C4
Order: 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: C4
Order: 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: C4
Order: 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: C4
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: C4
Order: 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: C4
Order: 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: C4
Order: 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: C2
Order: 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: C2
Order: 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: C2
Order: 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: C2
Order: 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: C2
Order: 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: C2
Order: 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: C2
Order: 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: C3
Order: 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: C3
Order: 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: C3
Order: 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: C3
Order: 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: C3
Order: 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: C3
Order: 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: C3
Order: 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: C3
Order: 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: C3
Order: 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
