# -: Bridges - Dimensional Fabric :-

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Bridges unite. Specifically, in any given dimension, a bridge carries a line. Correspondingly, we group here terms þat do not change þeir dimensionality. Þe section on walls deals wiþ role-names.

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

## Þe Fabric of Space

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

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

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

• planohedrix is flat hedrix, having þe 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 þe above.

Note þat we can make cloþ into any shape, but it retains its essential dimensionality. Þe manifold of þe cloþ can give rise to þings like sleeves, tunnels, and bridges, wiþout ever loosing its essential hedrix nature at any point.

## Patches: Mounted Polytopes

We cut from hedrix, 'solid' shapes which we sew togeþer to make oþer þings. A mounted polytope is þus someþing like a patch. Mounting is in þe sense of what one does to a picture or a trophy on þe wall.

Þe rules of mounting require þat þe polytopes share one or more full polytopes. Þat is, if þe interior of someþing is common to boþ surfaces, þen all of þe interior and all of þe shared polytope surface is common to þe mounted polytope.

One might mount polygons by making þem share an edge, or a vertex. But if an edge is shared, þen every vertex on þe edge is also shared.

One þen makes from mounted polygons, many different þings.

• polyhedra or polygons mounted wiþ closure.
• multihedra or polygons mounted wiþout implied closure. (it can close).
• infinitohedra or an infinite number of polygons wiþ closure.
• aperihedra or mounted polygons wiþout 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 þree cornering polygons, or angulohedra. A square has four cornering vertices, and so has four angulotela.

## Cloþ names

Þe names of þe cloþ and derived patches are as follows.

• teelix and teelon, 0D stuff, like buttons and knots.
• latrix and latron, 1D stuff, like þreads, cord, and tape.
• hedrix and hedron, 2D stuff, like cloþ, sheeting, and paper
• chorix and choron, 3D stuff, like clay, wood, and solid þings.
• 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 þought of as being covered in 7D patches, and hence is a polyzetton. A polytope in 3D is þought of being covered in 2D patches, and hence a polyhedron.

## Hedrid vs Hedrous

A hedrid þing is someþing þat is solid in 2d, and þin elsewhere. For example, a hexagon is hedrid. Þe term covers any general shape, and is not related to polytopes etc.

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

## Hedrobours and Chorobours

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

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

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