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Everyone says that our left and right hand are mirror images of each other and are non-superimposable, unlike a sphere whose mirror image is exactly superimposable. But shouldn't our left and right hand be considered achiral? If I move my hands in front of me such that palm of both the hands face towards me and move my say left hand horizontally such that it can be superimposed on the right hand clearly they can't be because the thumbs point in opposite directions.

But if I rotate my left hand by 180 degrees so that now the palm of my left hand faces away from me then both the left and right hand are superimposable. So shouldn't they be achiral? I know there are two more similar questions akin to this but none of the answers answer my simple question.

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Holding your hands in this way merely proves that your hands are mirror images. If you take any object (chiral or not) and hold it up to a mirror, you can always align common features.

Imagine instead placing one of your hands inside the other. You may be able to align the overall thumbs and fingers, but they will be facing opposite directions, thus are not completely superimposable.

If you'd like further confirmation that your hands are chiral, try shaking someone's right hand with your left. Or try putting a left hand glove on your right. The interaction of mirror image forms (enantiomers) of a chiral object (like your hands) is different with the mirror image forms of another chiral object (like a glove or someone else's hand).

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But if i rotate my left hand by 180 degrees ie now palm of my left hand faces away from me then both the left and right hand are superimposable .

Turning your hands this way only makes them superimposable if you make the assumption that they are two-dimensional objects, where the normal vectors coming out of both sides of a given hand are indistinguishable. To note, this sort of assumption is what is typically made when analyzing planar molecules.

From a fully three-dimensional perspective, in this configuration, your fingers and thumbs are indeed aligned but your palms are now facing opposite directions. If the normal vectors from each hand are taken as distinguishable (the "palm side" is different from the "back side"), you cannot just choose to ignore the misalignment of your palms and declare your hands superimposable, and thus achiral.

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    $\begingroup$ So does this mean that a planar molecule can be chiral ? $\endgroup$ – Matt Jan 10 '17 at 16:43
  • $\begingroup$ @RaghavSingal I think not; I'm pretty sure planar molecules are always achiral -- but I'm not completely sure. Planar molecules are treated as something of an idealization, where both normal vectors off of the plane of the system are indistinguishable. Hands are three-dimensional objects, and the palm-side and back-side normal vectors are typically not treated as indistinguishable. $\endgroup$ – hBy2Py Jan 10 '17 at 16:46
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    $\begingroup$ But still my palms are facing opposite directions when I try to superimpose them $\endgroup$ – Matt Jan 10 '17 at 16:52
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    $\begingroup$ Planar molecules possess a mirror plane of symmetry which renders them achiral ($\sigma \equiv S_1$). $\endgroup$ – orthocresol Jan 11 '17 at 6:10
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    $\begingroup$ @AlwaysConfused Good point -- one has to make an even stronger assumption than what I did. $\endgroup$ – hBy2Py Aug 9 '18 at 14:47
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"Superposable" means exactly identical in shape. Superposability is a thought-experiment to know 2 shapes are identical or not.

Our 2 hands are mirror-symmetrical, and of similar size; but no; they are not of identical shape.

Here is a small test. Stand facing the East. Bring your hands at front side of your body. Bring your palms together as shown in FIG2. Seems the shapes are same? (I'm ignoring tissue-details, just look the overall shape) Look closer. Think. Got anything?

For your left hand, now the palm-side is facing south. And its knuckle+nails side is facing north.

For your right hand, now the palm side is facing North. and knuckle+nail side is facing south.

They are so alike but they are so different!

If you now try to bring the both palm facing same direction, as in fig.3; your finger won't match. you may also change the orientation of wrist, and see any difference. No benifit, they wont "superpose". They are so similar but so different. They are sort of "opposite" shapes!

That's chirality.

Yes our hands (left and right) are chiral and enantiomorphic from external morphology.

Chiral and Achiral Objects.
FIG 1

Image source: The Chemistry LibreTexts library. 25.7. Chirality in Organic Chemistry

Our 2 hands are not superposable i.e. not identical, because if you try to merge theme as follows the nails do not match.

palm face-to-face
FIG2
The nails, knuckles, and darker, outer side of your left hand stay towords your left. And the nails, knuckles and the darker side of your right-hand, stay towords your right.

Reference:

  1. This particular example was taught in our college chemistry class.

  2. Solomon & Fryhle's Organic Chemistry, 10th Edition, by T.W. Graham Solomons and Craig B. Fryhle, Wiley Student Edition.

Chapter 5 (stereochemistry)

Here is a google book result of an older-edition; Ed-8

Google Book Result FIG3

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In general, when an object is asymmetrical (reflectional) in n dimensions within n-d euclidean space, it is chiral. Make an object that has a definite 'up' and 'down'. A definite 'forward' and 'back' and a definite 'right' and 'left', and enforce a rule that objects are only permitted to travel through the space in an unbroken line.

Suppose left is 'anticlockwise' to up on one 2d object and we take a copy of this item but swap the position of up and down, now we claim that left is 'clockwise' to up. These two objects are indistinguishable in 3d because we can rotate one 180 degrees to make the other. In 2d they are chiral. But if we introduce an asymmetry in the third dimensions, the act of fixing the chirality in the second dimension will create a different image in the third dimension. A hand is chiral because there is 3d asymmetry of reflection.

It is impossible for the designer of the object to be able to to use some mathematical tool or understanding to 'tell' whether their object has a left or right chirality without having something with an already known chirality to compare with it. The words 'clockwise' and 'anticlockwise' have no meaning without a given chirality in the first place. What I mean is that there is nothing left about a left hand, other than it is simply not a right hand. They can only be told apart from each other.

You may ask why we say that rotated objects are indistinguishable and reflected objects are not. There is really no good mathematical reason for this terminology. From a physical perspective, this only makes sense because of the conservation of angular momentum.

https://en.wikipedia.org/wiki/Noether%27s_theorem#Examples

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