Where to begin --
For starters, he only shows nine dimensions. A point is clearly zero dimensions, then he goes through three normal dimensions, numbers them 1, 2, 3, collapses this to a point (which doesn't add a dimension since points are zero-dimensional), numbers his next three dimensions 4, 5, 6, collapses
them to a point, numbers his next three dimensions 7, 8, 9, collapses
them to a point, and then calls that point dimension 10. This is inconsistent with all he had shown before.
It would be hard to describe his higher dimensions as being like either the normal three, or like the string theorists curled up dimensions. They don't even really deserved to be called such.
There is no evidence that the folding and forking he describes is real or possible. There is similarly no evidence for higher spatial dimensions as normally understood, but it is wrong to accept one set without proof but reject the other due to lack of proof.
General relativity seems to imply at least one higher dimension, other than the three of space and one of time. This is the direction in which space is curved. Just as the curved 1D surface of a ring implies a second dimension to curve through, and the curved 2D surface of the Earth implies a third dimension, it seems to me that the curvature of spacetime implies at least one higher dimension to curve through. This curvature dimension seems unrelated to either the string-theorists curled dimensions and folding and forking dimensions. However, no relativity book I have seen talks about this curvature dimension, so it may not be necessary.
One of the relativity books I have read gives an interesting test for whether something is a dimension or not:
Is it possible in principle to rotate an object through it? For instance, you can make a map of the temperature of all points on a plane, but it doesn't mean that this temperature is a dimension. There is no sense in which you can rotate an object through temperature. The whole point of special relativity is that you
can rotate an object through the combination of space and time, and in fact cannot avoid doing so when you speed an object up. Likewise, you can rotate an object through the string theory curled up dimensions, if the object is small enough.
It doesn't seem like the forking and folding dimensions are subject to being rotated through.
In any case, it seems that his dimensions add up only to nine, and there are several other kinds of dimensions he did not consider, which make it hard to accept the proposition that there are and can only be 10 dimensions.
For a good picture of higher dimensions, I recommend
Flatland, available on project Gutenburg.
Dimensions is a good visualization of the mathematical part of
Flatland, plus a visualization (the best we can do) of the 4-dimensional regular solids, and a bunch of other good stuff. Euclidean geometry as currently understood now extends to an unlimited number of discrete straight dimensions.
Brian Greene's
The Elegant Universe (the book, not the Nova show) gives a decent portrayal of the string theory idea of higher curled-up dimensions.
