Exploring Rotations in a Cube: Geometric Insights and Examples

[pivoxa15]

Geometrically, how do rotations in a cube look? i.e rotate through 180 degrees about the line joining midpoints of opposite edges, how does it look?


Are the rotations the different ways of getting from one vertex to the opposite vertex in the cube (where opposite is defined by the line joining opposite edges)? So there are 6 different rotatations in that example. Each of them equally valid. Whereas in a square there are only two way of a rotation by 180 degrees?

 

[mathwonk]

rotations of a cube act on the 8 vertices, and on the 6 faces, and on the 12 edges. The number of rotations leaving a given object fixed, multiplied by the number of different objects it could be taken to, equals the full number of rotations. so since there are three rotations fixing one vertex, and 8 vertices, all potential targets of that one, there are in all 24 rotations.

so rotations can occur abut any axis of symmetry, the ones joining opposite vertices (4 of these) or centers of opposite faces (3 of these), or centers of opposite edges (6 of these).

after this case, try visualizing the 60 rotations of an icosahedron, or equivalently a dodecahedron.

 

[pivoxa15]

I am not sure if I understand rotations in a cube let alone higher order stuff.

Lets go back to rotations in a square which I understand. The only things rotating in a square are the vertices. Does that apply to a cube as well? If not than why don't the edges of a square rotate?

What would rotations about centre of opposite faces, edges, vertices look like?

Should I think about the cube actually moving?

Is there a website that shows this stuff geometrically?

 

[DaveC426913]

Lets go back to rotations in a square which I understand. The only things rotating in a square are the vertices. Does that apply to a cube as well? If not than why don't the edges of a square rotate?

It sounds like you're over-thinking this. The edges of a the square do rotate.

Turn the whole square clockwise 90 degrees. It looks exactly the same.
Turn the whole square counterclockwise 90 degrees.
Turn the whole square 180 degrees.

All these are about a single axis, which passes through the page.

What would rotations about centre of opposite faces, edges, vertices look like?

Should I think about the cube actually moving?

Yes.
Take a blank die (i.e. one 6-sider "dice") How many axes can you rotate it around and still have the same shape?
- one through each of 2 opposing faces = 3
- one through each of 2 opposing vertices = 4
- one through each opposing edges = 6

 

[pivoxa15]

Things are a bit clearer now. The cube should rotate not like a rubik's cube but rotate as a whole. What do you mean by "shape"? Do you mean the same orientation with respect to the sourroundings as when you started? I.e. if the cube had edges lined up horizontally and vertically than after the rotation, it should be lined up that way as well.

Are the angles of the rotations defined as 360=1 full rotation.
So 360/(number of rotations to reach full rotation) = degree per rotation.

What reflections can occur in a cube?
I take it that edges, vertices and faces can all reflect?
So in reflections we specifiy which items are fixed and everything else can reflect (although in some objects like a pentagon, there is a vertex which usually has no partners to reflect to). With the cube, are there 13 reflections?
-one through each of 2 opposing faces = 3
- one through each of 2 opposing vertices = 4
- one through each opposing edges = 6

So generally, reflections<= rotations?

 

[DaveC426913]

What do you mean by "shape"? Do you mean the same orientation with respect to the sourroundings as when you started? I.e. if the cube had edges lined up horizontally and vertically than after the rotation, it should be lined up that way as well.

Sorry, how terribly ambiguous of me. Your interpretation is correct.

How many axes can you turn a cube around so that you cannot tell it has been turned at all?

Clearly, if you turn it 1/4 turn, you will be unable to distinguish the new orientation from the old.

 

[pivoxa15]

How many axes can you turn a cube around so that you cannot tell it has been turned at all?

Clearly, if you turn it 1/4 turn, you will be unable to distinguish the new orientation from the old.

Is this when you rotate through 2 oppposite faces?

If you rotate through 2 opposite vertices than 1/3 turn is adequate (because after 3 turns, you are back to where you started).

Is my comment about reflections correct?

 

[DaveC426913]

Is this when you rotate through 2 oppposite faces?

Yes.

If you rotate through 2 opposite vertices than 1/3 turn is adequate (because after 3 turns, you are back to where you started).

After 4 turns you are back where you started.

Is my comment about reflections correct?

Welllll, reflections are treated pretty much as if you're looking in a mirror at the object.

How many ways can you position the object in front of a mirror such that it and its reflection are indistinguishable.
-or-
How many ways can you replace exactly one half of the object with a mirror to make the combined object/image look indistinguishable from the whole object.

So, rather than establishing 1-dimensional axes as you did in rotations, you are now establishing 2 dimensional planes.

 

[pivoxa15]

After 4 turns you are back where you started.

I have a die in front of me and when I try to rotate it along the axis from vertex to opposite vertex, I can only do 3 rotations before I get back to where I started (with the same number on the face as when I started, facing me).

Welllll, reflections are treated pretty much as if you're looking in a mirror at the object.

How many ways can you position the object in front of a mirror such that it and its reflection are indistinguishable.
-or-
How many ways can you replace exactly one half of the object with a mirror to make the combined object/image look indistinguishable from the whole object.

So, rather than establishing 1-dimensional axes as you did in rotations, you are now establishing 2 dimensional planes.

Reflections across 2D planes seems to make sense with a cube. I can see 13 different planes cutting symmetrically across the cube.
Since there are 12 edges, there are 6 planes cutting across opposite edges.
Since there are 6 faces, there are 3 planes cutting across opposite faces.
Since there are 8 vertices, there are 4 planes cutting across opposite vertices.

13 planes correspond to 13 different reflections? Does doing nothing count as a reflection? If so than there are a total of 14 reflections. Correct?

 

[DaveC426913]

I have a die in front of me and when I try to rotate it along the axis from vertex to opposite vertex, I can only do 3 rotations before I get back to where I started (with the same number on the face as when I started, facing me).

Apologies. You are correct. I thought we were talking about a 2D square.

Reflections across 2D planes seems to make sense with a cube. I can see 13 different planes cutting symmetrically across the cube.
Since there are 12 edges, there are 6 planes cutting across opposite edges.
Since there are 6 faces, there are 3 planes cutting across opposite faces.
Since there are 8 vertices, there are 4 planes cutting across opposite vertices.

13 planes correspond to 13 different reflections? Does doing nothing count as a reflection? If so than there are a total of 14 reflections. Correct?

No to the latter. So, 13.

BTW, I can't categorically say there are only 13. There could be more. They can be tricky to spot.

 

[pivoxa15]

Why is 'doing nothing' not included as a reflection? Whereas it is included as a rotation?

This means that it is not possible to form a group of reflections in a cube or square because there would be no identity. Applying a reflection twice in a square or cube would get one back to where one started from. Ideally, it should equal the identity which would be 'doing nothing' but if it is not included as a reflection than reflections of a square or cube would not count as a group.

 

[DaveC426913]

Why is 'doing nothing' not included as a reflection? Whereas it is included as a rotation?

I do not recall suggesting that doing nothing is a valid rotation.

That being said, the initial position does count as a valid orientation for both square and cube.

 

[DaveC426913]

Let's get some concepts straight.

There are two things to consider:
1] the axis or plane about which the translation takes place
2] how many times it can match (in the case of rotation, that is. Mirror reflections always have two.)

A square has
rotational symmetry
- 1 axis. The axis is the point at the centre of the square.
- 4-fold symmetry. It can be rotated such that there are 4 orientations where the square is the same. (0, 90, 180, 270 - Note these are positions, not movements). I do not believe it matters about clockwise vs. counterclockwise.
mirror symmetry
- 4 mirror lines. There are 4 lines about which it can be flipped and still look the same (x-axis, y axis,y=x, y=-x).

BTW, I've pulled some of my info from here:
http://en.wikipedia.org/wiki/Rotational_symmetry
http://www.mathsisfun.com/geometry/symmetry-rotational.html

 

[pivoxa15]

I do not recall suggesting that doing nothing is a valid rotation.

You mean 'valid reflection'?

That being said, the initial position does count as a valid orientation for both square and cube.

The elements inside the group must be operations. Two reflections about the same axis should be the same as doing one operation which can only be 'doing nothing'. Maybe it depends on the group. For example, in the dihedral group for the square, there is a 0 rotation but no 0 reflection, which is okay as one 0 is enough. This might be where you recalled no 0 reflection. But if we were to form a group of only reflection for the square or cube than we would be forced to add a 0 reflection otherwise a group wouldn't form.

 

[DaveC426913]

You mean 'valid reflection'?.

No, I mean valid rotation. As in: doing nothing is NOT a valid rotation (just as it is not a valid reflection).

Don't worry about the convolutions of our discussion. Doing nothing is not valid either way.

The elements inside the group must be operations. Two reflections about the same axis should be the same as doing one operation which can only be 'doing nothing'. Maybe it depends on the group. For example, in the dihedral group for the square, there is a 0 rotation but no 0 reflection, which is okay as one 0 is enough. This might be where you recalled no 0 reflection. But if we were to form a group of only reflection for the square or cube than we would be forced to add a 0 reflection otherwise a group wouldn't form.

This is getting beyond my ability to talk intelligably about it. I'm but a layperson.


Suffice to say, reflections and rotations are "verbs". Orientations are "nouns".

Three rotations are sufficient to give rise to four orientations in the same way that only three steps are required to ensure you've stepped on four stone tiles.

One reflection is sufficient to give rise to two orientations.

 

[masudr]

Doing nothing is not valid either way.

Although doing nothing is the identity of the corresponding symmetry group.

 

[Gokul43201]

Why is 'doing nothing' not included as a reflection? Whereas it is included as a rotation?

Go over the definitions of rotation and reflection to see why this is true. If you wish, write down the matrices for rotations and reflections (in 2D, about a point and line respectively) . Which of these two families includes the identity?

Note: There is a trivial reflection which performs the identity operation - the reflection of a space about itself.