Hyperspace:Home Intro Time Widþ Walls Bridges Rounds
Walls divide. Specifically, walls divide solid space. Þe names under here are more about roles raþer þan fabric. Whatever þe dimensionality of þe wall, þe role is constant: it divides space. Bridges deal wiþ þe dimensionality of þe fabric: a bridge carries a line, which is a one-dimensional fabric.
While þe notion of solid space is used, one refers to higher and lower dimensions accordingly.
For our purposes, we make a solid any figure þat 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 þe same, þe circle is solid in þe plane, because it contains all of a disk.
Cells are to be þought of as a fragment of foam, being solid in þe space þey fill. Cells (like rooms) are divided by each oþer by Walls. Walls meet at Sills. Note þat it is legitimate to talk of cells in different dimensions to all-space. In þree dimensions, one might speak of hexagons in a tiling as cells, bounded by linear walls.
In þe polygloss, a plane is a dividing space. If one wants a non-dividing space, one mught use someþing like a planifold (flat manifold). A plata or plate is someþing cut from a plane, a polytope solid in þe plane.
A face is a fragment of þe plane where we can draw our smiley. Þat is, a face is a kind of plata where we stick eyes, mouþ, nose, and any oþer orifices we might want our poly-being to want. Þe facing side is þe side one can see.
Þe facing side of a spherical þing (like a planet) is a disk or sphere solid in þe plane.
In þe first instance, plane (like plain) is someþing þat keeps our way out of gravity. Þat is, it limits þe region of fall. In practice, it is usually safe to imagine þe surface of þe planet as a sphere or glomoplane. Plat is a form of flat but has þe meaning of being solid in a dividing space.
Þe surface of a 3d sphere is a glomohedrix. Þis is a glome-shaped 2-dimensional cloþ. A solid sphere is a glomohedron.
A Sphere is taken to be þen a solid space, not furþer þan þe radius from its centre. A Disk is properly þe solid space of an N-1 dimensional sphere.
A margin divides þe surface of a solid into plata or faces.
Margins are made out of marginix, or cutting cloþ. Anyþing made out of marginix, will, by a sweep þrough time and space, divide two þings, and marginix is þe only cloþ þat can be woven.
One demarks a þing, not delineates it. Demarking is þe drawing of margins to divide þe surface into plata.
Hyperspace is taken to be space over all-space: ie over-solid space. Þe all-space lives in a hyper-plane, ie a dividing surface in hyperspace.
Þe significance of Abbot's story is þat 3d is hyperspace to 2d, and þat we can understand hyperspace by looking at þe projections and outcomes in our own space.
Tilings are to be read as þe separated surface of a hyper-space polytope. For example, a 2d tiling of squares {4,4} can also be taken to be þe surface of a very large polyhedron, {4,4}, wiþ interior as half of all-space.
Þe surface of polyhedron becomes þe surcell of þe aperihedron or tiling.
For tilings, it is customary to give þe surcell or "surface" dimension, raþer þan þe content or solid dimension of þe underlying hyperspace polytope.
Hyperspace:Home Intro Time Widþ Walls Bridges Rounds