On 12/17/2017 3:24 PM, Helmut Raulien wrote:
Now, do you think that there is chirality also in other contexts than molecules, e.g. in signs?
To illustrate that issue, consider the analogs in 2 dimensions and 3 dimensions. For example, any circle on a plane can be made congruent with any other circle by two transformation: movement and size. Given two circles A and B, move A to B so that the center point of A coincides with the center point of B. Then enlarge or contract the radius of A until its circumference coincides with B. But if you put an arrowhead on A that points clockwise and an arrowhead on B that points counterclockwise, there is no way to make A and B congruent by those two transformations: the arrows will always point in opposite directions. However, if you're allowed to move A out of the plane into 3-D space, you can flip it over, put it back on the plane, and make it congruent with both the circle and arrow of B. The same issue holds for chiral pairs in 3-D space: there is no transformation by movement and size that can make your left and right hands coincide. But if you could move out of 3-D space into 4-D space, it would be possible to "flip" your left hand to give yourself two right hands. (But don't do that. It would have bad effects on the rest of your body.) To generalize: In a space of any number of dimensions, the operations of movement and size can be specified by a dyadic relation of A to B. But the operation of "flipping" requires some space (a Third) that cannot be specified within the original space. John
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