Symmetries: Identity | Rotations | Reflections | Rotoreflections

## Rotoreflection Symmetries 15

Symmetry group: `m`

and _{f}r4`m`

Order: _{e}r4`2`

, `4`

and `6`

Rotoreflection (rotary reflection) symmetries involve transforming a cube algorithm by mirroring it across a specific symmetry plane and rotating it around its axis. There are two kinds of rotoreflection symmetry:

### Rotoreflectional Symmetries across midplanes 7

Symmetry group: `m`

Order: _{f}r4`2`

and `4`

Rotoreflection Symmetries across midplanes involve transforming a cube algorithm by mirroring it across one of its midplanes (the three coordinate planes that intersect the cube's center and are parallel to its faces) and simultaneously rotating it around a specific mid (tetrad) axis.

#####
`34.`

Mx Rotoreflection

Symmetry group: `m`

Order: _{f}r4`4`

In a `Mx`

Rotoreflection symmetry, the cube algorithm is mirrored across the midplane M (intersecting the
cube's center and parallel to its Right and Left faces) while simultaneously rotating it 90 degrees clockwise around
its x-axis.

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

To revert this symmetry, apply the `Mx'`

symmetry.

#####
`35.`

Mx2 Rotoreflection

Symmetry group: `m`

Order: _{f}r4`2`

In a `Mx2`

Rotoreflection symmetry, the cube algorithm is mirrored across the midplane M (that intersects
the cube's center and is parallel to its Right and Left faces) while simultaneously rotating it 180 degree around
its 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.

#####
`36.`

Mx' Rotoreflection

Symmetry group: `m`

Order: _{f}r4`4`

In a `Mx'`

Rotoreflection symmetry, the cube algorithm is mirrored across the midplane M (intersecting
the cube's center and parallel to its Right and Left faces) while simultaneously rotating it 90 degrees
counterclockwise around its x-axis.

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

To revert this symmetry, apply the `Mx`

symmetry.

#####
`37.`

Ey Rotoreflection

Symmetry group: `m`

Order: _{f}r4`4`

In an `Ey`

Rotoreflection symmetry, the cube algorithm is mirrored across the midplane E (intersecting the
cube's center and parallel to its Upper and Down faces) while simultaneously rotating it 90 degrees clockwise around
its y-axis.

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

To revert this symmetry, apply the `Ey'`

symmetry.

#####
`38.`

Ey' Rotoreflection

Symmetry group: `m`

Order: _{f}r4`4`

In an `Ey'`

Rotoreflection symmetry, the cube algorithm is mirrored across the midplane E (intersecting
the cube's center and parallel to its Upper and Down faces) while simultaneously rotating it 90 degrees
counterclockwise around its y-axis.

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

To revert this symmetry, apply the `Ey`

symmetry.

#####
`39.`

Sz Rotoreflection

Symmetry group: `m`

Order: _{f}r4`4`

In a `Sz`

Rotoreflection symmetry, the cube algorithm is mirrored across the midplane S (intersecting the
cube's center and parallel to its Front and Back faces) while simultaneously rotating it 90 degrees 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 `Sz'`

symmetry.

#####
`40.`

Sz' Rotoreflection

Symmetry group: `m`

Order: _{f}r4`4`

In a `Sz'`

Rotoreflection symmetry, the cube algorithm is mirrored across the midplane S (intersecting
the cube's center and parallel to its Front and Back faces) while simultaneously rotating it 90 degrees
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 `Sz`

symmetry.

### Rotoreflectional Symmetries across diagonal planes 8

Symmetry group: `m`

Order: _{e}r4`6`

Rotoreflection Symmetries across midplanes involve transforming a cube algorithm by mirroring it across one of its vertical diagonal planes (that intersect two opposite vertical edges) and simultaneously rotating it around a specific mid (tetrad) axis.

#####
`41.`

RFx Rotoreflection

Symmetry group: `m`

Order: _{e}r4`6`

In a `RFx`

Rotoreflection symmetry, the cube algorithm is mirrored across its diagonal plane RF
(intersecting its opposite Right-Front and Left-Back vertical edges) while simultaneously rotating it 90 degrees
clockwise around its x-axis.

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

To revert this symmetry, apply the `RFz'`

symmetry.

#####
`42.`

RFx' Rotoreflection

Symmetry group: `m`

Order: _{e}r4`6`

In a `RFx'`

Rotoreflection symmetry, the cube algorithm is mirrored across its diagonal plane RF
(intersecting its opposite Right-Front and Left-Back vertical edges) while simultaneously rotating it 90 degrees
counterclockwise around its x-axis.

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

To revert this symmetry, apply the `RFz`

symmetry.

#####
`43.`

RFz Rotoreflection

Symmetry group: `m`

Order: _{e}r4`6`

In a `RFz`

Rotoreflection symmetry, the cube algorithm is mirrored across its diagonal plane RF
(intersecting its opposite Right-Front and Left-Back vertical edges) while simultaneously rotating it 90 degrees
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 `RFx'`

symmetry.

#####
`44.`

RFz' Rotoreflection

Symmetry group: `m`

Order: _{e}r4`6`

In a `RFz'`

Rotoreflection symmetry, the cube algorithm is mirrored across its diagonal plane RF
(intersecting its opposite Right-Front and Left-Back vertical edges) while simultaneously rotating it 90 degrees
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 `RFx`

symmetry.

#####
`45.`

LFx Rotoreflection

Symmetry group: `m`

Order: _{e}r4`6`

In a `LFx`

Rotoreflection symmetry, the cube algorithm is mirrored across its diagonal plane LF
(intersecting its opposite Left-Front and Right-Back vertical edges) while simultaneously rotating it 90 degrees
clockwise around its x-axis.

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

To revert this symmetry, apply the `LFz'`

symmetry.

#####
`46.`

LFx' Rotoreflection

Symmetry group: `m`

Order: _{e}r4`6`

In a `LFx'`

Rotoreflection symmetry, the cube algorithm is mirrored across its diagonal plane LF
(intersecting its opposite Left-Front and Right-Back vertical edges) while simultaneously rotating it 90 degrees
counterclockwise around its x-axis.

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

To revert this symmetry, apply the `LFz`

symmetry.

#####
`47.`

LFz Rotoreflection

Symmetry group: `m`

Order: _{e}r4`6`

In a `LFz`

Rotoreflection symmetry, the cube algorithm is mirrored across its diagonal plane LF
(intersecting its opposite Left-Front and Right-Back vertical edges) while simultaneously rotating it 90 degrees
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 `LFx'`

symmetry.

#####
`48.`

LFz' Rotoreflection

Symmetry group: `m`

Order: _{e}r4`6`

In a `LFz'`

Rotoreflection symmetry, the cube algorithm is mirrored across its diagonal plane LF
(intersecting its opposite Left-Front and Right-Back vertical edges) while simultaneously rotating it 90 degrees
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 `LFx`

symmetry.

Symmetries: Identity | Rotations | Reflections | Rotoreflections