Index Key:  PHY058
Author:     murphy
Subject:    String theory
Text:       If string theories require us to have more than 4D space time, and 
these other dimensions are "folded" over each other somehow, how can they meet 
the criterion of being literally perpendicular to all previous dimensions and 
still be topographically curved?  And how does the number of dimensions affect 
the number of proposed electron types (muon taon etc)?

Response #:  1 of 1
Author:      Arthur Smith
Text:        Imagining strange higher-dimensional objects is one of the big 
joys and frustrations of many mathematicians and physicists who do this sort 
of thing.  But it is usually a little difficult to put what they have imagined 
into ordinary language, since these objects usually just cannot exist in our 
space - and nothing like them really can, at least at the level of our normal 
perceptions.  One of the simplest analogies that does not really work is 
simply the circle.  What mathematicians and physicists discovered is that you 
can separate out the "closed-in-on-itself" property of the circle from the 
"curved" property - you kind of imagine a straight line segment with ends 
labeled A and B, and then declare that "A = B"!  Such a circle of course 
cannot exist in the ordinary space we know, but as a mathematical object there 
is nothing wrong with it.  Anyway, those extra dimensions you talked about all 
look sort of like this circle - they are straight line segments closed in on 
themselves without curving at all.  You can also imagine a cylinder as an 
example - the axis of the cylinder is one dimension that extends to infinity 
both ways, while the circumference of the cylinder is a circle that is 
perpendicular to this axis.  However, That is embedding the 2-dimensional 
cylinder into 3- dimensional space, and it turns out you do not actually need 
to even imagine embedding these higher dimensional spaces to work with them, 
although usually it is possible.