http://www.os2fan2.com/glossn/updown.htmlA word acquires many meanings, and some of these meanings enter popular use. My doctor related to me, that 'genetic disorders', has nothing to do with 'genes', but derive from some earlier meaning of the word (The root means 'born'). The same is true in many of the sciences one encounters, the science uses these as a variable, and the student is supposed to go from room to room, supposing that all arts of science are agreeing and describing the truth. Neither are true.
Polytopic in biology, has nothing to do with polytopes, but is truer to the roots than the geometric one. Poly = many, and tope = place. In biology, polytopic means a feature or gene that has come in the same form in many different places. For example, the same mutation might be observed into isolated populations. Topic is derived from tope, refers to along the lines between local gossip to information on the local transport.
One might note that we live in a reality loosely based on E2H. That is, we live in a euclidean 2d space, with height. This is the flat-earth theory. We can refine this theory, to S2H, &c, but it serves no end to this discussion. So we walk at the bottom of an ocean of air, on a fairly extensive plane. When we use words, they are fixed to this model, but E2H is essentially E3, the words still have some meanings. Euclid's plane geometry deals with the ground, and solid geometry deals with buildings.
When we go randomly into 4D, then that is space over 3D, and 'hyperspace' is largely correct. It's like 'outside' if you never been there. Or 'upstairs'. But a good portion of the art is given to spending a good deal outside, and 'hyperspace' and 'planes' no longer are befitting of their 3D meanings.
To make the student's life less miserable, one ought tease apart the most significant parts of the word from their perception, and try to keep that meaning in higher dimensions.
A thing has both consist and context. An edge of a blade cuts in three dimensions, and we might suppose it consists of 1D element is the cause. But the same blade in four dimensions, would require an edge of two dimensions to cut. In essence, a blade divides space on motion, ie is solid space ÷ cutting ÷ sweeping, or 2d less than solid.
A plane, in the sense of the ground, is represented by h=0, as such, is 1D in 2d space, 2D in 3d space, and 3D in 4d space.
We can now create symbols representing consist (ie number of included dimensions), and context (relation to solid space), along with a special mark for something at the bottom of the ocean of air. The dimensions we mark +, the context (by equal signs like h=0), by =, and the ground itself is ~.
A line is 1D, and this is preserved in shipping line, bus line, bee line. But a line in the sand, to 'toe the line', a ship of the line, dead-line, are actually an equals on the ground, ie =~. Here the line is not to get to get you from A to B (as + infers), but to prevent you from crossing it. (ie =~ a division on the ground). The edge of a blade is not + (1D), but == (a cut by sweeping).
When these are fleshed to four dimensions or higher, one adds as many of the + or = as not mentioned in the definition. So a plane as = becomes +++= (a three dimensional thing), and a blade ( ==) comes to be ++== . The bee-line + is +==~ something that requires two equations as well as h=0, to define.
Hyper simply means 'over', and we add an extra & to the front of the equation, to say that we normally restrict ourselves to 4D, but we need to go over our needs with 5D.
Space is (=) comes +++. Hyperspace is then &+++, but in 4D, it's &++++. A hyperplane is simply &=, gives &++= in 3D, and &+++= in 4D. In other words all space is a hyperplane! 3d space is hyper-space to 2d space.
Space is made of fabric, and we have words to denote fabric (infinite or indefinite boundary), against patches (a definite boundary). Patches are sewn together to make things.
A poly+hedr+on is many 2d patches sewn together to make a closed bag. So poly means many, hedr is taken as 2d, and on as patch. We can simply create huge numbers of words by fixing hedr and on to other things. Putting -ix, -ices, will make a fabric out of the patches. Hedra are cut from a hedrix or hedrid sheet. -id comes from solid, here it means solid in 2d space. -ous means roughly, like like calling a snake a latrous thing (1d-like thing).
Latr-, hedr-, chor- and ter- are the 1d, 2d, 3d, 4d roots. You make fabric by -ix, and patches by -on. From there you add prefixes to say what you mean. Poly means a closed many. Apeiro- means without any boundary in the space they're in (ie a tiling). An apeirochoron is a tiling of 3d patches, literally 'without a fence, 3d patches'.