# -: Bridges - Dimensional Fabric :-

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Bridges unite. Specifically, in any given dimension, a bridge carries a line. Correspondingly, we group here terms that do not change their dimensionality. The section on walls deals with role-names.

Because what we're naming does not change as dimension changes, it is essentially the same fabric to everyone. Just as a bridge carries a line, we have names for fabrics (manifolds) of every kind.

## The Fabric of Space

The nature of space is envisaged to be fabric of various dimensions. One then works this fabric into various shapes, accordingly to different results. The technical term for this fabric is a manifold

The fabric does not have to lie straight. We can render it into spheres or whatever. This gives us names for all of the assorted polytopes etc.

Hedrix is the name for two-dimensional cloth. By itself, it can be taken to mean planohedrix, but this usage is best avoided unless one is heavily involved in flat surfaces.

• planohedrix is flat hedrix, having the same curvature as all-space.
• horohedrix is zero-curvature hedrix (eg a horosphere)
• glomohedrix is positive-curvature hedrix (ie sphere-surface)
• bollohedrix is negative-curvature hedrix, eg hyperbolic space
• isohedrix is any uniform curvature figure: ie any of the above.

Note that we can make cloth into any shape, but it retains its essential dimensionality. The manifold of the cloth can give rise to things like sleeves, tunnels, and bridges, without ever loosing its essential hedrix nature at any point.

## Patches: Mounted Polytopes

We cut from hedrix, 'solid' shapes which we sew together to make other things. A mounted polytope is thus something like a patch. Mounting is in the sense of what one does to a picture or a trophy on the wall.

The rules of mounting require that the polytopes share one or more full polytopes. That is, if the interior of something is common to both surfaces, then all of the interior and all of the shared polytope surface is common to the mounted polytope.

One might mount polygons by making them share an edge, or a vertex. But if an edge is shared, then every vertex on the edge is also shared.

One then makes from mounted polygons, many different things.

• polyhedra or polygons mounted with closure.
• multihedra or polygons mounted without implied closure. (it can close).
• infinitohedra or an infinite number of polygons with closure.
• aperihedra or mounted polygons without a final perimeter: a tiling.
• planohedra or mounted polygons in a plane (and an interior).
• surhedron or a surface mounted polygon.
• angulohedron or a surhedron of a surtope, or incident on a surtope. For example, a vertex on a cube has three cornering polygons, or angulohedra. A square has four cornering vertices, and so has four angulotela.

## Cloth names

The names of the cloth and derived patches are as follows.

• teelix and teelon, 0D stuff, like buttons and knots.
• latrix and latron, 1D stuff, like threads, cord, and tape.
• hedrix and hedron, 2D stuff, like cloth, sheeting, and paper
• chorix and choron, 3D stuff, like clay, wood, and solid things.
• terix and teron, or 4D stuff
• petix and peton, or 5D stuff
• ectix and ecton, or 6D stuff formerly exix and exon
• zettix and zetton or 7D stuff
• yottix and yotton or 8D stuff
• solix and solon or general nD stuff

A polytope in 8D is thought of as being covered in 7D patches, and hence is a polyzetton. A polytope in 3D is thought of being covered in 2D patches, and hence a polyhedron.

## Hedrid vs Hedrous

A hedrid thing is something that is solid in 2d, and thin elsewhere. For example, a hexagon is hedrid. The term covers any general shape, and is not related to polytopes etc.

A hedrous thing is some sort of lumpy hedrid thing, that may be tossed and turned into shape. One might describe a rolled out peice of modeling-clay, complete with lumps etc, as a hedrous mass. This term allows us to describe general outlines of shapes in higher dimensions. A ridge in 3d is essentially latrous (line-shaped), although it may have some width, etc. It would not be describe as hedrid.

## Hedrobours and Chorobours

The residents of a general dimension are described as bours (be-ers).

In the Abbot-quote, the Sphere is a chorobour (resident in a chorix or 3-space) while the square is a hedrobour (2d resident).

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