Why is it such a royal pain in the ass to comb your hair using multiple mirrors (for the purpose of seeing better angles), and how come the left-right axis gets switched around when the up-down isn't?

With simple brevity... it's all an illusory effect.

To fully appreciate this, have one individual lie on his/her side and ask what axis appears to be reversed. They will claim it is their left-right axis that differs, when to all other people this axis is the up-down axis. Now you have two totally independant sets of witnesses, the verticals and the horizontals, who have totally contradictory claims.

If a fancy dress party were held, in which all attending members came in harlequin outfits, we would have no problem with this illusion, as the left and right halves of the body would differ in color, thus making things easy. To better illustrate this, place a sign which points to the right between mirrors. No matter how many mirrors reflect the sign, all enantiomorphs of the sign will point right also.

This is because mirrors simply interchange the front-back axis. This illusory "paradox" presents itself only because people incorrectly imagine themselves having turned around, as if they were their enantiomorph; when, in truth, they are not the creature behind the mirror. The mirror world is essentially different from our own.

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