The Structure of the Inner Planes Revealed

Back in Dragon Magazine #8 when Gygax presents the first diagram of the standard planes of D&D (which wasn't yet a "Great Wheel") he assures us the image is "a 2-dimensional diagram of a 4-dimensional concept." Gygax doesn't explain what he means (is the entire conception 4D or only some part>), and so far as I know, no one else seems to have picked up this thread. 

In Dragon #42, for example, Lafoka makes the both suggestive and hard to parse statement about travel to the Elemental Planes from the Prime: 

A figure with ethereal access can freely travel on the Prime Material, go “up” into the Elemental Plane of Air, “down” into the volcanic Elemental Plane of Fire, can go into the Elemental Plane of Water (if a large body of water is nearby), or can go “down” into the Elemental Plane of Earth. 

I think this is mainly saying that areas of the element on the Prime Material are effectively portals in the Ethereal, but it could be more clearly worded if so, and why are up and down in quotes as if they are only so-called? Anyway, unless that scare-quoted up and down are referencing directions other than the usual, this doesn't offer anything.

Next, in Dragon #73, Gygax (inspired likely by Swycaffe's article in Dragon #27, though he doesn't credit it here) proposes a cubic model of the Inner Planes to accommodate the Positive and Negative Material Planes and the various para- and quasi-elemental stuff. Still no indication of dimensions beyond three, though.

I've written posts about the much-maligned inner planes before, I've never addressed this aspect either, so now, in full recognition of what has been written about them by above, I'm going to suggest that the inner planes exist in a 4-dimensional space. So, a better model for them and their relationships would be a hypercube or tesseract (to use the word coined by Charles Hinton to refer to such). Here's a 2D representation of the spatial relationship of the 3D "faces" of the 4D structure:

So this means the elemental planes (with the Prime Material unpictured in the center) are all 3D cells accessible by travel along the 4th axis. Hinton calls these directions kata and ana, and they stand with left and right, forward and backward, and up and down, to define location in a 4D space. This video shows how the above projection is arrived at by "unfolding" the 4D shape in 3 dimensions.

Of course, the Inner Planes don't really form a 4D hypercube any more than they were a cube. It's a model to show their spatial relationships.