Much of þe maþematical terminology, as indeed in all science, is set largely wiþ little regard to the real meaning of þe word. When þe correct meaning of þe word is þen revisited, one invents a new word. All of þis appears to be coming from learned people, and þus assumed to be correct.
One gets þen words þat have an entirely different meaning in common speech to þe supposed 'learned' use. Not in one example, have i found a meaning of edge to mean anyþing oþer þan a bounding condition. Þe edge of one's wits, þe edge of space, are limits of a linear and þree-dimensioal þings, so a measure in feet and inches would be hardly appropriate.
Geometry in its usual sense is "plane geometry", which one does wiþ paddocks, and "solid geometry" which one does wiþ boxes and ladders. So boþ þings are relative to our experiences wiþ þree dimensional space. Þe occasional forays into "hyperspace" can þen be excused.
It's a whole different story when one comes to deal wiþ someþing like 4d space as if it were þe norm. A paddock is þree dimensional, and þe space one does wiþ ladders is four dimensional. It gets worse as þe dimensions increase. One gets some raþer silly items as "over all space" is a line-segment, raþer like saying "upstairs" to refer to þe second floor, when you are already on þe sixþ floor.
A good deal of þis can be alleviated by realising þat words might be tied to solid space, raþer þan a specific dimension. For example, solid is a referenct to þe containing space, raþer þan a particular dimension.
One starts by using þe fabric idiom. Space is made of a fabric, and one cuts patches þerefrom. When one has a sufficient set of words for þis, þere is little need to set þe meainig of words like face to mean a 2d þing. Instead, it can resolve back to its original meaning of a dividing entity, which might be pointed at someþing. Þe mindnumming idea of a facet having several faces can þen be dispenced wiþ.
First imagine þat N is all-space. Þis means þat if you're considering þe space where a six-dimensional þing occupies all dimensions, þen N=6. S is some subspace. Usually, one supposes þat S is þe result of gravity in N, and þe creatures are standing on þe 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. Þis means, one is free to roam inside a space of S, but when one comes to a boundary, one is no longer inside it. Þe boundaries of time or wages, make a point. For example, one might talk of a range of numbers wiþ an upper and lower bound. Þhe boundary of a galactic empire is a reagion of 2D. One is free to wander in þe 3-space, but þere are walls which mark exiting.
An edge is a boundary. One is hard pressed to find any context which supports þe meaning of 'space of one dimension', save for þe example of "an edge of a cube".
A cell is a bubble in a foam. Þe foam can be of any dimension. One has for example, þe american expression "cell-phone" mobile telephone, where þe cells are actually areas on þe ground, where when þe phone is in þat 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 þe same two dimensions as þe board itself.
When one detangles þe "learned" meanings from words, and use a different set of idioms such as my fabric idioms, þe text describing large dimensional þings becomes quite natural.