4-point, 4-line, 4-plane, 4-volume,...

Higher-dimensional geometry (previously "Polyshapes").

4-point, 4-line, 4-plane, 4-volume,...

Postby thigle » Sat Mar 11, 2006 10:13 am

it seems to me sometimes reasonable to distinct 0(d)-point from 3-point or 4-point, in other words, to embed information about embedding space of the geometrical object in question into its name.

so for exemple, a 2-point (a point in E2) is different from a point in E3 - a 3-point. one allows a 2-star of rays, a 2d-star of directions to pass through it, other allows/is open to an axis of planar stars - a 3-star of rays.

so some lines can be more "open" than others, according to where they dwell.

but then, considering the non-orientable 2-point within mobius band, it seems inadequate to catch the nuances of different points. or maybe ratios should be used for non-orientable ones ?

anyway, my question is other but close:
in 4-space, flat this time, i have a patch of 2-plane ( which is actually 4-plane, according to the above)
extending it infinitely, in E4, what do we get ? just ordinary E2 ?

now more important to me (bo would say essential, and he's right this time :wink: ) is how this would behave in S4, or elliptic 4-space, and especially with curvature equal to 1.
would the patch close on itself ? if so then what manifold would we get ?
for 3-space, extending a 3-plane (2-plane in S3), is it projective plane that we get for any initially locally flat patch, when we get to including infinity ?

i wonder if the elliptic 4-plane gives a klein bottle or RP3. ?
isn't every totality of 4-plane(=2-plane spread infinitely in 4-space, infinity excluded, a mobius band ? (i think mobius band is a projective plane minus a point (=with a hole) ?
then the answer to the quaestion at the beginning of this paragraph would be RP3, rather than klein's bottle, it seems to me. :?
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Postby wendy » Sun Mar 12, 2006 9:11 am

The polygloss works on the exact opposite: that a point is a point, regardless of the embedding space. None the same, one can talk of what's around the point (or its arounds vs surrounds).

One notes also that a 4d solid that is line-like (eg a tube or chord), is described as a latrous or latrid, depending on how line-like it is.
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Postby thigle » Sun Mar 12, 2006 3:49 pm

but still. what about the 2-patch expanding in elliptic 4-space, and in E4 ? what do we get ? a projective plane and mobius band, or klein bottle instead of projective plane ?
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Postby thigle » Sun Apr 02, 2006 11:50 pm

wendy, do you mean by arounds looking at something from its inside outwards and surrounds is looking at thing fromoutside ?

i just found on the very last page of hilbert's geometry&imagination, that in e4, the only closed surfaces isometric locally with e2, and not ruled, are torus & klein-bottle. in elliptic geometry cannot be realised on other surfaces than sphere & projective plane.

so the answer to my last question would be torus&klein-bottle for flat & projective plane & sphere for elliptic case.

is that so ?

also, wendy states:
The polygloss works on the exact opposite: that a point is a point, regardless of the embedding space. None the same, one can talk of what's around the point (or its arounds vs surrounds).


in geometry& imagination, page 297, hilbert states:
let us now consider a surface - with or without boundaries - that extends to infinity. the topological structure of such a surface will depend on whether we imagine it situated in metric or projective space.

then he proceeds to specify the differences.
does that mean that polygloss dwells in metric space ?
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Postby wendy » Wed Apr 05, 2006 10:16 am

The way that the polygloss operates is that in/out/surface etc apply only in the context of a given space, and that these do not extend to the embedding space.

So the interior of a hexagon is a 2d surface in the spame hedrix, regardless of whether all-space is 2d, 3d, 4d, ...

The terminology for the relation for points not in the embedding space is arrounding, ortho, etc.

It is possible to have, for example, a non-crossing line in 3d, that makes a knot, and a non-crossing hedrid in 4d that has a different topology to the euclidean plane. For example, you can connect the six edges of a hexagon to make opposite sides connect: this makes a torus.

In 4d, you can have some strange knotted surfaces, such as a cross-cup (a klein bottle), etc.

Hilbert seems confused with metric vs projective space. One can indeed apply a metric to projective space. The polygloss does make the distinction between these, in fact, a several-way distinction, according to which of the folloing one chooses to reject

1. lines cross at one point
2. space is orientable
3. space is complete

The common metric space of Euclid is a fragment, and so is rejects proposition 3. This means that the fragment is never large enough to see the effects of antipodes. This is rule F

The mono- forms keep 1 and 3, and thus reject 2. This is the same geometry that arises from projective geometry, since this projects antipodes onto the same line. This is rule M

Keeping 2 and 3 rejects 1, which leads that lines cross twice. The space of crossing corresponds to a straight diteelon, eg antipoles of spheric geometry. This is rule O

The general model of nature is that any measurable space is likely to be rule F, except of the sphere. If one keeps antipodes separate, then rule O applies. If one maps antipodes to each other than rule M applies.

The further thing is that the projective model of the sphere is the gnomic one, which places the viewer at the centre of the sphere. Lines through antipodes pass through the centre, and map as a single point on plane. This makes rule M apply to the projective model.

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