by wendy » Mon Nov 17, 2008 7:00 am
The comb product is a "repetition of surface".
It is known in two forms.
1. horotope (euclidean tilings, etc).
2. solotope (eg solid polytopes)
HOROTOPE
One can regard a euclidean tiling as a surface of an infinite polytope. For example, the apeirogon is an infinite polygon, and the tiling of squares as a polyhedron.
The comb product of such tilings (eg polygons) give rise to a polyhedron (eg tiling of squares). In hyperbolic space, the same is true, except that figures written in euclidean surfaces are not tilings but real (all be it infinite) polytopes. The comb product of two polygons is a polyhedron.
SOLOTOPE.
A solotope is a solid polytope. This is usually implemented as a non-crossing surface and a connected interior.
The comb product is implemented by radially replacing the surface of one polytope by the body of another. The dimension-loss comes from sharing the radius in both figures.
For example, two polygons multiply to give a polyhedron. The result is a torus, where the original surface of say, a decagon, runs around the axis of the tube. The second figure (say a hexagon), is then repeated so to preserve an axis pointing at the centre of the decagon, and its centre at the surface of the decagon.
What you get is a torus, which repeats the surface of the decagon (for each element going on the circle through the hub), which runs aling the rim, and a hexagon for each element of the rim, that goes around the tyre (hub-outside).
The effect of both products is a solid, whose surface is effectively a cartesian product of the surfaces.
Since this tends to produce tunnels [latin, comb], or 'honeycombs', the product becomes the "comb" product.