Comb examples?

Higher-dimensional geometry (previously "Polyshapes").

Comb examples?

Postby Keiji » Fri Nov 14, 2008 2:30 pm

I've heard wendy mention a "comb" product many times by now, but I have no idea what it is. Could anyone give some examples?
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Re: Comb examples?

Postby 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.
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Re: Comb examples?

Postby Keiji » Mon Nov 17, 2008 10:28 am

Then, isn't this exactly the same as spheration? :\
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Re: Comb examples?

Postby wendy » Tue Nov 18, 2008 4:02 am

Spheration is something ye do to something. A classic example would be to replace points and lines with balls and sticks. You often see atoms represented in this way. It's more a way of beefing up lines etc. It has a definition in moving a point around the thing.

Comb product is a 'repeatition of surfaces', is used mainly for generating new polytopes. After having listened to the discussion on the tiger (spherate bi-glomolatrid comb), it is possible to take the product of surfaces directly.

They are different things.
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