# Parity, how many dimensions to switch?

Parity is described in Wikipedia as flipping of one dimension, or - in the special case of three dimensional physics - as flipping all of them.

Is there any simple rule that generalises both for any dimension? Like: "Flip an odd number of dimensions."?

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This is described in the introductory paragraphs of the article you linked to. Could you clarify what you didn't understand about them? –  David Z Aug 22 '12 at 8:01
I tried to find a generalisation of: 3d-> flip one or three, 2d -> flip one. (And came up with "flip any odd number" without beeing able to distinguish whether this is interesting or just stupid.) –  Falko Aug 22 '12 at 19:28

If you have two coordinate systems with the same origin, you can represent a (linear) transformation of coordinates from one to another as a matrix. This matrix has either positive or negative determinant. This sign of the determinant is what gives the transformation its parity. (All this applies to any number of dimensions, not just 3.)

If you compose multiple linear transformations, the matrix of the final transformation is the matrix product of their matrices. And the determinant of the result will be positive if and only if an even number (including 0) of the original matrices have a negative determinant.

So, you can categorize linear transformations using the sign of their determinant, using their parity. Some (like rotations, scaling or sheering) preserve parity when composed with another, others (like reflection) flip it.

Knowing this, it's easy to see that flipping $n$ of coordinates (regardless of the number dimensions) produces a matrix with $-1$ appearing $n$-times on the diagonal, so the transformation has odd parity if and only if $n$ is odd.

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I think David is being a bit harsh, because I had to read the Wikipedia article a couple of times to see what they were getting at.

As the article states at the beginning, a parity transformation is the flip of a single spatial co-ordinate. In effect it's like looking in a mirror: when you look in a mirror your height and width co-ordinates are unchanged bu the depth (normal to the mirror) is flipped.

However in 3-D flipping all three co-ordinates is equivalent to a rotation plus a reflection, so it also flips the parity. It isn't the same as flipping a single co-ordinate, but it changes the parity in the same way.

I must admit I'm not sure how this extends to higher dimensions. In 3D two flips is equivalent to a rotation around the axis normal to the two axes being flipped, but in >3D obviously there is more than one axis normal to the two axes being flipped and my grasp of hyper-dimensional geometry isn't good enough to work out what happens. However I think that you are correct and an odd number of flips will always change the parity while an even number leaves it unchanged.

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Hm... well, I thought the article was quite clear about it. But maybe it's just me. –  David Z Aug 22 '12 at 9:01
John, I would like to accept two answers, but can't, sorry. –  Falko Aug 23 '12 at 10:07
Petr's answer is better anyway :-) –  John Rennie Aug 23 '12 at 10:20