My two cents....In physics terminology, a 'dimension' is a degree of freedom that can be represented by a single numerical value. Most scientists speak of "3 spatial dimensions plus time", rather than '4 dimensions' since time isn't the same sort of quantity as the others. All of these concepts are approximations which don't fully describe the physical world. One way of looking at quantum mechanics is that what is being described is actually less than three dimensional spatially, but more than two dimensional, which makes it seem sort of weird when described in terms of a 3 dimensional metric space. In my experience, when non-physicists speak of higher "dimensions", usually they're not talking about dimensions in a mathematical sense, so its hard to make comparisons against physics theories. I'm always interested in hearing about what people are aware of however, in hopes that I can extend my own awareness in a similar manner.
Like most people here I've had interesting experiences in dreams in this context. As I may have mentioned elsewhere, a few months ago I viewed a scene coherently from several directions at once, which was strange since visual experience is usually projected onto a surface. I've also dreamed of geometries that can't be embedded in 3-dimensional euclidean space. According to a narrator in a couple of these dreams, these reflect something of the everyday experience of other entities. Maybe more people will have these experiences after the seed of the idea is planted.
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