Is there a general test for chirality?

[Vorde]

I have been wondering:

Given an arbitrary geometric shape is there a sequence of tests I can perform on individual parts of the shape to determine whether or not it is achiral?

I was hoping for something a little less trivial that simply seeing if the mirror image could be adjusted to match the original, more something along the lines of looking at the composition of the shape itself.

Gratias.

 

[Bacle2]

You can try polynomial invariant, like Jones, or Homfly, but this is not an iff way, i.e., achiral knots may be assigned the same polynomial (usual problem one has with algebraic topology).

EDIT: Sorry, I did not read carefully-enough; I think you may be looking for a different approach.

 

[Vorde]

I am way, way, outside of my knowledge zone here, so I apologize if everything I say is wrong.

I wasn't thinking achiral with knots as much I was with just plain shapes like sticks or hands. I sort of was thinking that it seemed to me that if you took a shape and found an arbitrary center. The shape would be chiral if you could rotate the shape such that it was asymmetrical in two separate dimensions.

I'm pretty sure I've falsified that idea but I was wondering if there was a geometrical approach similar to what I tried to do that works.

 

[Bacle2]

No need to apologize for (the possibility of ) being wrong; main thing is to be curious and open-minded.

Let me see if I get what you're asking:

So you want to see if, say a hand , to be isotopic to their respective mirror images?

And then you want to see if you can find a point p in the shape rotate the shapes along

p, the rotated shape will be symmetric (meaning you can decompose it into two

parts that are mirror images of each other)? If I understood you well, a hand

cannot be rotated into being composed of two disjoint mirror images.

 

[Vorde]

What I meant was to ask is there a simple geometric process from determining if an arbitrary shape is achiral or not.

I don't think that my idea about the rotations works, because I think I found a shape which is achiral yet satisfies my rotations process, but I'll think about that more tomorrow.

I'm just wondering if a process exists, I doubt mine is the correct one.

 

[Shellsunde]

Back up to the definition...

Would someone help with a proper definition of chirality? What I read on Wikipedia, in the dictionary and elsewhere online didn't sink in for me. What I understood these sources to say is chirality is such that a thing and its mirror image can't be superposed, positioned so that, effectively, every point on one of the things can be connected as with a straight line to the corresponding point on the other without any of those connecting lines 'crossing', i,e., intersecting with any other.

Is this correct? Is it comprehensive? When I hold my hand up to a mirror, the image and the hand do map one onto the other so are achiral, but all the examples given of handedness indicate they're chiral. A left and a right hand can't map one to the other without crossing lines only when they're oriented the same way, both palms facing you, for instance. When they're facing one another, palm to palm or back to back, they do map without crossing lines.

So, does chirality depend on orientation, and relative orientation at that? It seems not from what I read, but from how I understand it presently, it seems to.

Perhaps with a better statement or explanation of the definition, we can answer the thread's poster's question, and as for me, I just want to precisely understand it. (Perhaps this should be a new thread in Topology.)

 

[disregardthat]

I will give an informal explanation, but I'm open for corrections here.

You have three general classes of isometries of a 2-dimensional geometric shape: translations, rotations around a point, and reflections over a line.

If your shape X is a subset of $\mathbb{R}^2$, then you may
- Translate it by a translation map $t_a : \mathbb{R}^2 \to \mathbb{R}^2$ where $a$ is a vector in $\mathbb{R}^2$,
-Rotate it by a rotation map $r_{p,\theta} : \mathbb{R}^2 \to \mathbb{R}^2$ where $p$ is a point and $\theta$ is an angle,
- Reflect it by a reflection map $s_l : \mathbb{R}^2 \to \mathbb{R}^2$ where $l$ is a line (defined by some function linear $y = ax+b$).

You may also compose these different maps, like first translate it, then rotate it about a point, then reflect it about a line, and then rotate it about another point again. Any arbitrary series of compositions will again yield an isometry.

The definition of a chiral shape is that for any line $l$ in $\mathbb{R}^2$, you may not end up with the reflection about this line by translations and rotations alone. Note that which line $l$ is does not matter, as any line may be mapped to another line by a translation or a rotation.

Translations and rotations are said to preserve orientation, and reflections are said to reverse orientation. Any isometry preserves or reverses orientation.

This stuff is defined in detail here: http://en.wikipedia.org/wiki/Euclidean_plane_isometry

 

[disregardthat]

Another way of describing chirality is the following:
The isometry maps form a group. The identity map is the identity of the group, and every isometry has an inverse isometry. The symmetry group of a shape is the set of isometries that does not change the shape. This set forms a group.

A shape is chiral if every isometry in the symmetry group preserves orientation. A test for chirality would be to check that the generators of the group are all orientation preserving. In fact, the symmetry group cannot contain translations, and will be generated by rotations and reflections alone. If you can find a generating set of the symmetry group consisting purely of rotations, you know that the shape is chiral.

EDIT: edited out something that was wrong