Symmetry of a 3D body
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I'm having a little blackout about a pretty simple concept.
Does a formal symmetry of 3D rigid body have to be achievable/doable only by literally moving the body with the bare hands (e.g. a wooden cube)?
A reflexion with respect to the mass point of the cube would be a symmetry but it cannot be achieved by rotating the cube. Is that right or am I missing something obvious?
geometry
asked Dec 19 '18 at 19:11
21säv 21säv
1068
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add a comment |
$begingroup$
I'm having a little blackout about a pretty simple concept.
Does a formal symmetry of 3D rigid body have to be achievable/doable only by literally moving the body with the bare hands (e.g. a wooden cube)?
A reflexion with respect to the mass point of the cube would be a symmetry but it cannot be achieved by rotating the cube. Is that right or am I missing something obvious?
geometry
asked Dec 19 '18 at 19:11
21säv 21säv
1068
$endgroup$
$begingroup$
Right, you need a mirror.
$endgroup$
– Yves Daoust
Dec 20 '18 at 9:10
add a comment |
$begingroup$
I'm having a little blackout about a pretty simple concept.
Does a formal symmetry of 3D rigid body have to be achievable/doable only by literally moving the body with the bare hands (e.g. a wooden cube)?
A reflexion with respect to the mass point of the cube would be a symmetry but it cannot be achieved by rotating the cube. Is that right or am I missing something obvious?
geometry
asked Dec 19 '18 at 19:11
21säv 21säv
1068
$endgroup$
I'm having a little blackout about a pretty simple concept.
Does a formal symmetry of 3D rigid body have to be achievable/doable only by literally moving the body with the bare hands (e.g. a wooden cube)?
A reflexion with respect to the mass point of the cube would be a symmetry but it cannot be achieved by rotating the cube. Is that right or am I missing something obvious?
geometry
geometry
asked Dec 19 '18 at 19:11
21säv 21säv
1068
asked Dec 19 '18 at 19:11
21säv 21säv
1068
edited Dec 20 '18 at 9:09
21säv
asked Dec 19 '18 at 19:11
21säv 21säv
1068
asked Dec 19 '18 at 19:11
21säv 21säv
1068
asked Dec 19 '18 at 19:11
21säv 21säv
1068
1068
$begingroup$
Right, you need a mirror.
$endgroup$
– Yves Daoust
Dec 20 '18 at 9:10
add a comment |
$begingroup$
Right, you need a mirror.
$endgroup$
– Yves Daoust
Dec 20 '18 at 9:10
$begingroup$
Right, you need a mirror.
$endgroup$
– Yves Daoust
Dec 20 '18 at 9:10
$begingroup$
Right, you need a mirror.
$endgroup$
– Yves Daoust
Dec 20 '18 at 9:10
add a comment |
1 Answer
1
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votes
$begingroup$
A symmetry in 3D can be a rotation, a plane reflection, a line reflection (180 degree rotation), a point reflection (plane reflection together with 180 degree rotation), or a combination of a rotation and a reflection.
If we want to restrict ourselves to moving with the bare hands, we only have rotations (including line reflections) and identity, which are called positive symmetries.
And indeed, the reflection with respect to the mass point (point reflection or inversion), cannot be achieved by rotating the cube.
For a cube there are:
- 6 quarter rotations.
- 8 third rotations.
- 9 half rotations (or line reflections).
- 1 identity.
- 9 plane reflections.
- 15 combinations of a rotation and a plane reflection (including point reflection).
answered Dec 19 '18 at 20:03
I like SerenaI like Serena
4,2571722
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$begingroup$
A symmetry in 3D can be a rotation, a plane reflection, a line reflection (180 degree rotation), a point reflection (plane reflection together with 180 degree rotation), or a combination of a rotation and a reflection.
If we want to restrict ourselves to moving with the bare hands, we only have rotations (including line reflections) and identity, which are called positive symmetries.
And indeed, the reflection with respect to the mass point (point reflection or inversion), cannot be achieved by rotating the cube.
For a cube there are:
- 6 quarter rotations.
- 8 third rotations.
- 9 half rotations (or line reflections).
- 1 identity.
- 9 plane reflections.
- 15 combinations of a rotation and a plane reflection (including point reflection).
answered Dec 19 '18 at 20:03
I like SerenaI like Serena
4,2571722
$endgroup$
add a comment |
$begingroup$
A symmetry in 3D can be a rotation, a plane reflection, a line reflection (180 degree rotation), a point reflection (plane reflection together with 180 degree rotation), or a combination of a rotation and a reflection.
If we want to restrict ourselves to moving with the bare hands, we only have rotations (including line reflections) and identity, which are called positive symmetries.
And indeed, the reflection with respect to the mass point (point reflection or inversion), cannot be achieved by rotating the cube.
For a cube there are:
- 6 quarter rotations.
- 8 third rotations.
- 9 half rotations (or line reflections).
- 1 identity.
- 9 plane reflections.
- 15 combinations of a rotation and a plane reflection (including point reflection).
answered Dec 19 '18 at 20:03
I like SerenaI like Serena
4,2571722
$endgroup$
add a comment |
$begingroup$
A symmetry in 3D can be a rotation, a plane reflection, a line reflection (180 degree rotation), a point reflection (plane reflection together with 180 degree rotation), or a combination of a rotation and a reflection.
If we want to restrict ourselves to moving with the bare hands, we only have rotations (including line reflections) and identity, which are called positive symmetries.
And indeed, the reflection with respect to the mass point (point reflection or inversion), cannot be achieved by rotating the cube.
For a cube there are:
- 6 quarter rotations.
- 8 third rotations.
- 9 half rotations (or line reflections).
- 1 identity.
- 9 plane reflections.
- 15 combinations of a rotation and a plane reflection (including point reflection).
answered Dec 19 '18 at 20:03
I like SerenaI like Serena
4,2571722
$endgroup$
A symmetry in 3D can be a rotation, a plane reflection, a line reflection (180 degree rotation), a point reflection (plane reflection together with 180 degree rotation), or a combination of a rotation and a reflection.
If we want to restrict ourselves to moving with the bare hands, we only have rotations (including line reflections) and identity, which are called positive symmetries.
And indeed, the reflection with respect to the mass point (point reflection or inversion), cannot be achieved by rotating the cube.
For a cube there are:
- 6 quarter rotations.
- 8 third rotations.
- 9 half rotations (or line reflections).
- 1 identity.
- 9 plane reflections.
- 15 combinations of a rotation and a plane reflection (including point reflection).
answered Dec 19 '18 at 20:03
I like SerenaI like Serena
4,2571722
edited Dec 19 '18 at 21:40
answered Dec 19 '18 at 20:03
I like SerenaI like Serena
4,2571722
answered Dec 19 '18 at 20:03
I like SerenaI like Serena
4,2571722
answered Dec 19 '18 at 20:03
I like SerenaI like Serena
4,2571722
4,2571722
add a comment |
add a comment |
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$begingroup$
Right, you need a mirror.
$endgroup$
– Yves Daoust
Dec 20 '18 at 9:10