# -: Walls - Dimensional Roles :-

Hyperspace:Home Intro Time Width Walls Bridges Rounds

Walls divide. Specifically, walls divide solid space. The names under here are more about roles rather than fabric. Whatever the dimensionality of the wall, the role is constant: it divides space. Bridges deal with the dimensionality of the fabric: a bridge carries a line, which is a one-dimensional fabric.

While the notion of solid space is used, one refers to higher and lower dimensions accordingly.

## Solid Space

For our purposes, we make a solid any figure that contains every point of some n-sphere as part of its body. A hexagon is solid in 2D, because it can contain a circle, while it is not solid in 3D, because it does not contain all of a sphere. None the same, the circle is solid in the plane, because it contains all of a disk.

Cells are to be thought of as a fragment of foam, being solid in the space they fill. Cells (like rooms) are divided by each other by Walls. Walls meet at Sills. Note that it is legitimate to talk of cells in different dimensions to all-space. In three dimensions, one might speak of hexagons in a tiling as cells, bounded by linear walls.

## Planes

In the polygloss, a plane is a dividing space. If one wants a non-dividing space, one mught use something like a planifold (flat manifold). A plata or plate is something cut from a plane, a polytope solid in the plane.

A face is a fragment of the plane where we can draw our smiley. That is, a face is a kind of plata where we stick eyes, mouth, nose, and any other orifices we might want our poly-being to want. The facing side is the side one can see.

The facing side of a spherical thing (like a planet) is a disk or sphere solid in the plane.

In the first instance, plane (like plain) is something that keeps our way out of gravity. That is, it limits the region of fall. In practice, it is usually safe to imagine the surface of the planet as a sphere or glomoplane. Plat is a form of flat but has the meaning of being solid in a dividing space.

## Spheres and Disks

One might distinguish between the surface of a sphere, and the disk of a sphere.

The surface of a 3d sphere is a glomohedrix. This is a glome-shaped 2-dimensional cloth. A solid sphere is a glomohedron.

A Sphere is taken to be then a solid space, not further than the radius from its centre. A Disk is properly the solid space of an N-1 dimensional sphere.

## Margins and Marginix

A margin divides the surface of a solid into plata or faces.

Margins are made out of marginix, or cutting cloth. Anything made out of marginix, will, by a sweep through time and space, divide two things, and marginix is the only cloth that can be woven.

One demarks a thing, not delineates it. Demarking is the drawing of margins to divide the surface into plata.

## Area and Volume

Areas and Volumes are taken to mean extent of surface and solid space. However, the terms can confuse some, so it's best to use terms like surface content and solid content.
The extent of a given dimension is given in its root stem + age, eg 2-space = hedrage, 3-space = chorage. The extent of 2-space in four dimensions is about as meaningful as total edge-length of polyhedra.
Content can be measured in prismic, tegmic, and spheric units, as well as a few others. If one specifically wants to define eg cubage one should say prismic chorage.

## Hyperspace

Hyperspace is taken to be space over all-space: ie over-solid space. The all-space lives in a hyper-plane, ie a dividing surface in hyperspace.

The significance of Abbot's story is that 3d is hyperspace to 2d, and that we can understand hyperspace by looking at the projections and outcomes in our own space.

## Tilings

Tilings are to be read as the separated surface of a hyper-space polytope. For example, a 2d tiling of squares {4,4} can also be taken to be the surface of a very large polyhedron, {4,4}, with interior as half of all-space.

The surface of polyhedron becomes the surcell of the aperihedron or tiling.

For tilings, it is customary to give the surcell or "surface" dimension, rather than the content or solid dimension of the underlying hyperspace polytope.

Hyperspace:Home Intro Time Width Walls Bridges Rounds