Much of the mathematical terminology, as indeed in all science, is set largely with little regard to the real meaning of the word. When the correct meaning of the word is then revisited, one invents a new word. All of this appears to be coming from learned people, and thus assumed to be correct.
One gets then words that have an entirely different meaning in common speech to the supposed 'learned' use. Not in one example, have i found a meaning of edge to mean anything other than a bounding condition. The edge of one's wits, the edge of space, are limits of a linear and three-dimensioal things, so a measure in feet and inches would be hardly appropriate.
Geometry in its usual sense is "plane geometry", which one does with paddocks, and "solid geometry" which one does with boxes and ladders. So both things are relative to our experiences with three dimensional space. The occasional forays into "hyperspace" can then be excused.
It's a whole different story when one comes to deal with something like 4d space as if it were the norm. A paddock is three dimensional, and the space one does with ladders is four dimensional. It gets worse as the dimensions increase. One gets some rather silly items as "over all space" is a line-segment, rather like saying "upstairs" to refer to the second floor, when you are already on the sixth floor.
A good deal of this can be alleviated by realising that words might be tied to solid space, rather than a specific dimension. For example, solid is a referenct to the containing space, rather than a particular dimension.
One starts by using the fabric idiom. Space is made of a fabric, and one cuts patches therefrom. When one has a sufficient set of words for this, there is little need to set the meainig of words like face to mean a 2d thing. Instead, it can resolve back to its original meaning of a dividing entity, which might be pointed at something. The mindnumming idea of a facet having several faces can then be dispenced with.
First imagine that N is all-space. This means that if you're considering the space where a six-dimensional thing occupies all dimensions, then N=6. S is some subspace. Usually, one supposes that S is the result of gravity in N, and the creatures are standing on the surface of a N-sphere. But S could just as much mean time, or even all-space, where no gravitational inpediment holds.
A boundary might be read as S-1. This means, one is free to roam inside a space of S, but when one comes to a boundary, one is no longer inside it. The boundaries of time or wages, make a point. For example, one might talk of a range of numbers with an upper and lower bound. Thhe boundary of a galactic empire is a reagion of 2D. One is free to wander in the 3-space, but there are walls which mark exiting.
An edge is a boundary. One is hard pressed to find any context which supports the meaning of 'space of one dimension', save for the example of "an edge of a cube".
A cell is a bubble in a foam. The foam can be of any dimension. One has for example, the american expression "cell-phone" mobile telephone, where the cells are actually areas on the ground, where when the phone is in that area, it talks to a particular tower. Cellular automata, like Conway's game of life, as well as many role-playing and war games, are played on cells, which are the same two dimensions as the board itself.
When one detangles the "learned" meanings from words, and use a different set of idioms such as my fabric idioms, the text describing large dimensional things becomes quite natural.