Torinder

Discussion of shapes with curves and holes in various dimensions.

Torinder

Postby Keiji » Tue Sep 18, 2007 4:18 pm

Hmm... What genus does the torinder (prism of a torus) have? I think it should be zero, but I'm not sure...
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Postby wendy » Wed Sep 19, 2007 7:25 am

Genus in 4 dimensions does not work like in 3d. The numbers carry two indexes, one for each kind of hole: So it's 1h1c. The actual surface of the figure, carries a single genus of 1hc.
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Postby Keiji » Wed Sep 19, 2007 2:55 pm

1h1c?

The number of types of holes in n dimensions is n-1: in 2D, you can only have what I call "pockets" (a hole that a 2D being can't see from outside the shape), in 3D, you can have a pocket or a torus-like hole, etc...

So, I imagine that the hole in the torinder would be the 2nd of the three types of hole? And a ditorus (RNS "((21)1)") would have both the second and third types?
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Postby wendy » Thu Sep 20, 2007 7:41 am

A hole exists in a space or region, if there exists a "sphere-surface" which can not vanish without crossing the surface. Vanish here means go to zero or to infinity.

In a 3d torus, there are two holes, but we normally only count one. One can imagine a circle that goes around the tyre, like a bike-chain does. Such circles can not vanish because they would need to cross the surface. This hole would become closed by placing a plane like an axis hub.

A second hole is occupied by the tube of the tyre. This is inside the figure, and is likewise can not vanish. In order to prevent this circle from forming, one needs a second hedral (2d) wall inside the tyre.

So the actual genus of a 3d torus is 2h. We usually treat outside holes, which gives G, but this is where the 2G comes from. If you do a surtope count on a set of eight cubes forning a ring (like a torus), you will only get a difference of one, not two.

In four dimensions, there is also a single kind of hole, but it is different from the inside to outside. It exists as 'bridge' and 'tunnel' forms, these refer to the track that would run inside it.

Outside the tunnel, one can go over the hill, and through the tunnel, to form a circle that can not vanish. This would be spanned by a hedron, which would open up the tunnel. So this hole is h.

From underground, it is possible that the tunnel passes through the centre of a sphere, wholy underground, and entirely perpendicular to the tunnel. Such a sphere would be constrained by the tunnel to stop vanishing, and thus represents a hole inside the ground. This hole is a c-type hole (since it is spanned by a 3d patch - a choron).

So in 4d, all holes are either hc (tunnel) or ch (bridge). The surface of the ground can topologically exist in higher space, but it can't tell whether the hc discontinuity is a tunnel or bridge, so it is topologically the same.

So for chorix (3-fabric), the topology has a genus that increases as the presence of hc clusters: in 5d and higher, it can't tell if the thing is a bridge or a tunnel. It's just another sleeve.
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Postby papernuke » Wed Nov 28, 2007 5:55 am

What is a "genus"? On the Wikipedia it has to do with biology..
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Postby wendy » Wed Nov 28, 2007 7:22 am

Genus is simply the lædin word for 'kin, kind'.

In maths hedrices (2d fabric) have the same genus, G, if a map of regions, each spherous ('convex'), produces V-E+F =2-2G.

In ordinary 3d, it corresponds to the surface of a figure with G holes through it.
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