Pardon my terminology, I might be using the words wrong. I watched Susskind string theory course, where he explained the attempt to describe particles using vibrating strings instead of point particles. If you do this you need to have more than 3 spatial dimensions or otherwise there is some problem that I cannot remember (I cannot find the part in the video where he talks about this). Then he explained that string theory also predict all kind of other stuff such as d-branes.
My question is: If instead of attempting to describe elementary particles by vibrating open strings, you attempt to do it by vibrating open 3d volumes (so a particle would have a small volume in our 3D space, and an "open" volume, not one closed into hidden higher dimensions), then, would you still need extra dimensions?