A visual and literate metaphor about a "spectral" world
This post is the last one of a triptych started last saturday that may appear as cryptic as the former ones but it is (explicitly) free from farfetched mathematic(al physic)s at least!
... imagine you are the king of ants living in a two-dimensional flatland. One day your main court astrologist gives you a piece of exciting news that there is a deep sense in the notions of left and right, because nature does not respect parity symmetry and so the absolute meaning of the left, as the side preferred by stars, can be established. You immediately decide to notify your subject ants to what is left – the lucky side. So you send couriers with this mission throughout your kingdom. It may happen however that your world has a non-trivial global structure in the higher dimensional space and constitutes, for example, a Möbius strip. Then after some time one of your couriers can be found in a land, your main astrologist calls the land of shadows. You can not see him but can communicate with him using gravity. Gravitationally you feel as if he were somewhere very close. And really he is just beneath you on the Möbius strip ...
But you are flat, as are all of your subjects, and so have no idea about extra dimensions. You can’t say that your courier ant is turned upside-down, because he is two-dimensional. And his two-dimensional appearance, checked by gravity, looks the same as for all other ants. Simply in his zeal to fulfill your order he traveled too far away. And everybody knows in your kingdom that if you travel long enough way you will return the same place, but will return as an invisible shadow. Your main astrologist says that one can reach the land of shadows after very long journey. But anyway this land of shadows is a part of your kingdom – nobody, even your main astrologist, can tell you where the ordinary land ends and the land of shadows begins. So naturally you want your shadow subjects also to have the correct notion to what is the left side. And here a great surprise is awaiting you. For your main astrologist horror, you shadow courier indicates completely different side as the left side – the side which originally was marked as right by the very same courier before he left the court. Hence in a such Möbius world the absolute difference between left and right has meaning only locally. No such difference can be established globally – the world as the whole is parity invariant! If you do not like worlds to have edges, you can consider, for example, a Klein’s bottle universe instead. In this case you need at least four space dimensions to realize such (two-dimensional) world without self-intersections.
(Submitted on 24 Feb 2000 (v1), last revised 6 Dec 2000 (this version, v2))