reply to post by Manjushri Bodhisattva
More from that Article, excerpts from Wiki.
 Number of dimensions
One intriguing feature of string theory is that it involves the prediction of extra dimensions. The number of dimensions is not fixed by any
consistency criterion, but flat spacetime solutions do exist in the so-called "critical dimension." Cosmological solutions exist in a wider variety
of dimensionalities, and these different dimensions—more precisely different values of the "effective central charge," a count of degrees of
freedom which reduces to dimensionality in weakly curved regimes—are related by dynamical transitions.
Nothing in Maxwell's theory of electromagnetism or Einstein's theory of relativity makes this kind of prediction; these theories require physicists
to insert the number of dimensions "by hand," and this number is fixed and independent of potential energy. String theory allows one to relate the
number of dimensions to scalar potential energy. Technically, this happens because a gauge anomaly exists for every separate number of predicted
dimensions, and the gauge anomaly can be counteracted by including nontrivial potential energy into equations to solve motion. Furthermore, the
absence of potential energy in the "critical dimension" explains why flat spacetime solutions are possible.
This can be better understood by noting that a photon included in a consistent theory (technically, a particle carrying a force related to an unbroken
gauge symmetry) must be massless. The mass of the photon which is predicted by string theory depends on the energy of the string mode which represents
the photon. This energy includes a contribution from the Casimir effect, namely from quantum fluctuations in the string. The size of this contribution
depends on the number of dimensions since for a larger number of dimensions, there are more possible fluctuations in the string position. Therefore,
the photon in flat spacetime will be massless—and the theory consistent—only for a particular number of dimensions.
When the calculation is done, the critical dimensionality is not four as one may expect (three axes of space and one of time). Flat space string
theories are 26-dimensional in the bosonic case, while superstring and M-theories turn out to involve 10 or 11 dimensions for flat solutions. In
bosonic string theories, the 26 dimensions come from the Polyakov equation. Starting from any dimension greater than four, it is necessary to
consider how these are reduced to four dimensional space-time.
Calabi-Yau manifold (3D projection)
Calabi-Yau manifold (3D projection)
 Compact dimensions
Two different ways have been proposed to resolve this apparent contradiction. The first is to compactify the extra dimensions; i.e., the 6 or 7 extra
dimensions are so small as to be undetectable in our phenomenal experience. In order to retain the supersymmetric properties of string theory, these
spaces must be very special. The 6-dimensional model's resolution is achieved with Calabi-Yau spaces. In 7 dimensions, they are termed G2 manifolds.
These extra dimensions are compactified by causing them to loop back upon themselves.
A standard analogy for this is to consider multidimensional space as a garden hose. If the hose is viewed from a sufficient distance, it appears to
have only one dimension, its length. Indeed, think of a ball just small enough to enter the hose. Throwing such a ball inside the hose, the ball would
move more or less in one dimension; in any experiment we make by throwing such balls in the hose, the only important movement will be one-dimensional,
that is, along the hose. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant
crawling inside it would move in two dimensions (and a fly flying in it would move in 3D)"