Rotoreflection Symmetries

Symmetries: Identity | Rotations | Reflections | Rotoreflections

Rotoreflection Symmetries 15

Symmetry group: mfr4 and mer4 Order: 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: mfr4 Order: 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 Mx Rotoreflection

Symmetry group: mfr4 Order: 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:
Transform...Reflect...Rotary (M E S)...Reflect Mx

To revert this symmetry, apply the Mx' symmetry.


35. Mx2 Rotoreflection Mx2 Rotoreflection

Symmetry group: mfr4 Order: 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:
Transform...Reflect...Rotary (M E S)...Reflect Mx2

To revert this symmetry, apply it again.


36. Mx' Rotoreflection Mx' Rotoreflection

Symmetry group: mfr4 Order: 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:
Transform...Reflect...Rotary (M E S)...Reflect Mx'

To revert this symmetry, apply the Mx symmetry.


37. Ey Rotoreflection Ey Rotoreflection

Symmetry group: mfr4 Order: 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:
Transform...Reflect...Rotary (M E S)...Reflect Ey

To revert this symmetry, apply the Ey' symmetry.


38. Ey' Rotoreflection Ey' Rotoreflection

Symmetry group: mfr4 Order: 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:
Transform...Reflect...Rotary (M E S)...Reflect Ey'

To revert this symmetry, apply the Ey symmetry.


39. Sz Rotoreflection Sz Rotoreflection

Symmetry group: mfr4 Order: 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:
Transform...Reflect...Rotary (M E S)...Reflect Sz

To revert this symmetry, apply the Sz' symmetry.


40. Sz' Rotoreflection Sz' Rotoreflection

Symmetry group: mfr4 Order: 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:
Transform...Reflect...Rotary (M E S)...Reflect Sz'

To revert this symmetry, apply the Sz symmetry.


Rotoreflectional Symmetries across diagonal planes 8

Symmetry group: mer4 Order: 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 RFx Rotoreflection

Symmetry group: mer4 Order: 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:
Transform...Reflect...Rotary (RF LF)...Reflect RFx

To revert this symmetry, apply the RFz' symmetry.


42. RFx' Rotoreflection RFx' Rotoreflection

Symmetry group: mer4 Order: 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:
Transform...Reflect...Rotary (RF LF)...Reflect RFx

To revert this symmetry, apply the RFz symmetry.


43. RFz Rotoreflection RFz Rotoreflection

Symmetry group: mer4 Order: 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:
Transform...Reflect...Rotary (RF LF)...Reflect RFz

To revert this symmetry, apply the RFx' symmetry.


44. RFz' Rotoreflection RFz' Rotoreflection

Symmetry group: mer4 Order: 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:
Transform...Reflect...Rotary (RF LF)...Reflect RFz'

To revert this symmetry, apply the RFx symmetry.


45. LFx Rotoreflection LFx Rotoreflection

Symmetry group: mer4 Order: 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:
Transform...Reflect...Rotary (RF LF)...Reflect LFx

To revert this symmetry, apply the LFz' symmetry.


46. LFx' Rotoreflection LFx' Rotoreflection

Symmetry group: mer4 Order: 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:
Transform...Reflect...Rotary (RF LF)...Reflect LFx'

To revert this symmetry, apply the LFz symmetry.


47. LFz Rotoreflection LFz Rotoreflection

Symmetry group: mer4 Order: 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:
Transform...Reflect...Rotary (RF LF)...Reflect LFz

To revert this symmetry, apply the LFx' symmetry.


48. LFz' Rotoreflection LFz' Rotoreflection

Symmetry group: mer4 Order: 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:
Transform...Reflect...Rotary (RF LF)...Reflect LFz'

To revert this symmetry, apply the LFx symmetry.


Symmetries: Identity | Rotations | Reflections | Rotoreflections

 

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