On 10/25/2012 4:58 AM, Richard Ruquist wrote:

Stephan,Since yesterday it occurred to me that you may be thinking of the 10 or more dimensions of string theory as being orthogonal because they were so before the big bang. But the dimensions that curled-up/compactified went out of orthogonality during the big bang according to Cumrun Vafa. I'll look up that reference if you are interested. According to Vafa 2 dimensions compactified for every single space dimension that inflated. In over simplified terms, 2 dimensions (actually in strips of some 10,000 Planck lengths) to be compactified lined up say in the east-west space dimension so that space in an orthogonal direction could expand. So some semblance of orthogonality exists in the compactification process, but it is clear that the compactified dimensions become embedded in 3D space for inflation to occur.

`It's implicit in the definition of dimensions of a Riemannian manifold that there are as`

`many orthogonal directions as dimensions. Compactified dimensions are just small; they're`

`small, not infinite, because they have closed topology. That property is completely`

`independent of having orthogonal directions.`

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