As I understand it, all this thinking about dimensions comes from the problem of trying to figure out what gravity is. Einstein’s general relativity theory does not distinguish between time and space, so our familiar three spatial and one time dimension are, in his theory, all of the same character and they create what he called a four dimensional “space-time.” In this theory, gravity is a result (consequence or effect) of the geometry or local shape of space-time. A quantum theory of gravity is needed to reconcile general relativity with known principles of quantum mechanics. An easy example is reconciling light “waves” with “photons.” Gravity in quantum terms is a force, but it is an effect in general relativity. String theory is an attempt to reconcile relativity and quantum mechanics by substituting one dimensional objects (strings) for point particles of no dimensions (particle physics). Strings vibrate and move through space. One of the vibrational states creates a “graviton” which is a quantum mechanical particle that has gravitational force (analogous to the photon).There are a number of string and super-string theories. The “M” theory attempts to unify all these string theories into a single quantum mechanics theory that describes gravity as a force, not an effect. Edward Witten suggested that all five super-string theories were in fact just different limiting cases of a single theory in eleven space-time dimensions, which he called “M theory”. Witten said that the M could stand for “magic”, “mystery”, or “membrane” because he hadn’t yet worked out the entire theory. The term brane is derived from membrane.
A point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. Higher dimensions are labelled “p.” Thus higher dimensional branes are called p-branes. There is something vaguely satisfying when you say p-brane out loud.
The idea of dimensions that can be compacted is a mathematical convenience to avoid the awkward mathematics of a ten or eleven dimensional space-time. If at least some dimensions are slightly curved, extending the limits of space-time results in a number of dimensions that can be assumed to curl up on themselves to a “point” of essentially no dimensions. Think of a making a garden hose. we could start with a piece of rubber that is long and not very wide; a 2-dimensional plane. By curving the width dimension, we can complete the curve of the width dimension to create an inner space bounded by one dimension – the circumference of the hose we just made. Now if we back away far enough, the curved dimension (the circumference) essentially disappears and we are left with a line – of one dimension. So at a distance the hose looks like a one-dimensional object (a line or string). Mathematically at extremes, the two-dimensional hose can be considered as a compacted single dimension composed one linear and one curved dimension. Mind you, an ant crawling on the circumference of the hose would not be confused.
At this stage, my “pea-brain” is about to collapse under the weight of multi-dimensional p-brane concepts which might turn out to be real strings and membranes, and not just magic or mystery. The notion of living in a universe, only some of which may be within our perception or even our conception, is both intriguing and humbling.