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Whenever our discussions have turned to talking about dimensions, it’s pretty easy to derail them over the issue of nomenclature and definitions. This is probably because there’s a rather arbitrary explanation or understanding that has been agreed upon wrt the use of certain words. As most experts already understand, dimensions can be different things (both conceptually and mathematically) depending on the application.
The way that it’s mathematically expressed, Euclidean 3-space actually comprises three sets of 2D planes. It would be more meaningful to call it 2D3 rather than 3D. It’s the reason why Euclidean 3-space has octants… 2X2X2=8.
There also exists a deeper understanding of what a 2D plane actually is, though. It’s most commonly defined as two lengths (or axes) perpendicular to one another, sure, but what does that really mean? The concept of orthogonality brings the concept of relative direction (or orientation) into the picture. This structure of relative orientation, when combined with length, is the defining characteristic of a two-dimensional plane.