Understanding the higher dimensions is pretty difficult when the words give you mixed messages.

Cell, in usual parlance, is a solid element of something, the notion being a tiling or foam. For example, people who play board games call the squares or hexagons 'cells'. This is because the cells and the whole of playing space (the plane here), are two-dimensional. In essence, one is playing on a regular 2d foam. Games for zero players, like Conway's "Game of Life" are called 'cellular automata'.

When one looks at quickfur's renders, one sees lots of bubbly things which might be thought of as a foam of cells. They are in effect projections of the surface, and the cells cease to be cells in the same manner that cells on a square lattice cease to be cells on the surface of a cube.

Grasping four dimensions, is more than three dimensions. With the right terminology, things are much easier to hold onto.

Any view of four dimensions, is essentially a picture, a plan or a projection. In a picture, there's a top and bottom, so things would fall to the bottom. A plan is a view of the ground from above, so things don't fall in it. A projection is a distorted image of something, usually a plan, such that one can see the innards.

But a projection that looses one dimension is going to be something solid. You have to imagine that you're looking at the solid and its innards, without surface or crossing any bit of it. This is how you see a three-dimensional thing captured to two dimensions. For the greatest part, we can't see the innards of three dimensions like we can see the innards of two dimensions. So people make the bits translucent (so you can see through it), or give an exploded diagram (so you can see part A goes onto part B etc). When you look at quickfur's renders, you have to imagine that the solid thing you build up, is only a picture of something, and you then have to imagine that it has depth perpendicular to the three-dimensional space.

You next have to understand that some ideas are not hooked to specific dimensions, but are related to solid space. One can hide behind a 2d panel in three dimensions, but in four dimensions, hiding behind a 2d panel is like hiding behind a 1d pole in 3d. You need a three-dimensional panel to hide behind. The words that suggest division of space (plane, wall, face, facing) are thus hooked to N-1 space, while the words that suggest 2d (hedron, hedrix) are hooked to a fixed dimension.

Because we're inventing a lot of new words for a lot of new ideas, we can use linguistic concepts to make life easier. Back in the days at school, one might be taught ladin and greek roots, and construct words like tele.phone = far+i see, and tele.vision = far + image etc. Then there is the idea of regular endings, like a verb /to own/ gives a noun to make the object (ownings), and a noun for the subject (owner). There are regular endings in the PG, for various ideas like this.

One has more room to distinguish between ar.round and sur.round /Sur/ means 'on', so a surface is a face (a dividing thing), that is on (ie contains) something. A surface is divided into surtopes: vertices, edges, surhedra, surchora, or counting down from N, faces, margins, ... The surface is measured in the space where the thing is solid. So the 'surface' of a hexagon is the bit of the hexagon that separates it from the hedrix it falls in.

The around-space is the space that is not part of the plane, but is said to 'ring' it. One sees for example, that one dances around the maypole, and the earth spins around its axis. The distinction is faint in 3d, but is useful to persue higher.

A poly.hedr.on is read as many.2d.patches. The patch image is merged into an unbounded cloth hedr.ix. Stems in /a/ and /o/ refer to patches, while those in /i/ refer to unbounded cloths. It then comes a matter of inventing words for 1d, (latr), 3d (chor), 4d (ter) &c. There are other ideas that have specific dimensions like /solid/ (you can have a solid area or a solid part of a line), gives rise to sol.id, and then -id becomes a solid or definite thing in that dimension. A hexagon is always a hedr.id because it is always a 2d solid. The opposit of solid is neblu.ous or cloudy. Something that is approximately 2d, but might have thickness in other spaces, is "hedr.ous".

You can add prefixes to give these things certian shape. A glom(o).id is a globe-shaped thing, a circle, a sphere, etc. A glomo.hedr.on is a globe-shaped thing bounded by a 2d shape, ie a 3d sphere. A glomo.hedr.ix is the 2d cloth that's bent into glomid shape - a sphere-surface.

While there is a cartesian product, there are many different things one can do to modify or multiply shapes. Truncation is about cutting off the corners, usually vertices. When this is used of higher dimensions, it's described as a bevel. A bevelled cube has long hexagon sides where the edges were.

Spheration is about giving any non-solid element a glomic arrounding (eg circular or spheric section). The

Atomium is a spheration of a body-centred cube. The eight vertices and the centre become spheres (since 3d is 'around' 0d) and the edges are turned into thinner cylinders (because the arounding of an edge gives a 2d space, which holds a circle). The sizes of the added aroundings do not have to match.

When we look at eg the 'tiger', we see it's a 'spherated bi-glomohedrix prism'. The glomo.latr.ix is a sphere-shaped 1d cloth, ie something that might cover a 2d sphere (circle). A prism is a cartesian product. So this corresponds to a 2d fabric (hedrix) that lives in 4d. It's the 2d element in a bi-cylinder. Spherating it makes it solid (ie a 4d thing here). It's hard to imagine directly, i fess, but not so hard as something like a bi-curcular tegum.

In any case, start with a circle in 3d. When you spherate its perimeter, you get a torus, anchor ring, or doughnut. You replace the line by a bent cylinder. To get something that works like the 'tiger' you now have to spherate the surface of the 3d torus in four dimensions. This means that you have to imagine spaces on either side of 3d space, and replace each point of the surface with a circle, that is orthogonal to the hedrix. You get in essence, a circle, standing on its edge, so that all you see is a 1d section. The centre is the old 3d torus, the inner and outer points are the new surface in 4d.