4th dimensionally perpendicular earths

[PWrong]

Hmm. I was thinking that maybe space curves so much that it intersects with itself in some way, allowing the two planets to somehow orbit around the same sun. I'm just visualising the 2D case, of course. Most kinds of curve don't seem to work, but I've tried to imagine wrapping the 2D universe around a klein bottle. It's a bit far-fetched, but I think it actually works.

By the way, a klein bottle is one of these
http://mathworld.wolfram.com/KleinBottle.html

In my Klein bottled 2D solar system, the sun is at the centre of the circle where space intersects with itself. Both intersecting planets rotate around it, and they both orbit the same sun at a different radius.

[PWrong]

I'm not sure about any of this; I've never formally studied topology, but this is how I think it would work. I'm just going into more detail this time.

Apart from the intersection, if 2D space was wrapped around a Klein bottle, the effect would be similar to being wrapped around a torus. This means if you move in straight line from any point, you will eventually get back to your starting point. Actually, on the klein bottle, it would be slightly different

The only difference is the part where space intersects. This is the circle where the "tube" part of the bottle meets the flatter "side".

Now, after wrapping 2D space around the Klein bottle, we place a small circular sun on the "side" at the centre of this circle. A planet happens to orbit this sun exactly over the intersection line. At right angles to this planet is another planet, which exists on the "tube" part. The two planets are anchored together. The second planet doesn't appear to be in orbit around the same sun, but it is. Take a line from the sun to the first planet at any given time, then trace the path of the light (or the gravity), by continuing this line around the bottle, and you will eventually reach the second planet.

[pat]

The only difference is the part where space intersects. This is the circle where the "tube" part of the bottle meets the flatter "side".

Technically, the Klein bottle doesn't intersect at all. It has to intersect to be projected down into three-dimensional space. But, in four-dimensional space, it sits perfectly fine without intersection.

Here's a topological way to think about spheres and toruses and Klein bottles.

Consider a square. Label the edges with arrows. Fold the thing up so that the arrows align. For example, a sphere would just be:

Code: Select all

+---<------+
|          |
^          |
^          v
|          |
+--->->----+

Because, if you glue that so that the top edge is stuck to the right edge and bottom edge is glued to the left edge, you'll end up with something topologically equivalent to a sphere.

Perhaps easier to see is a torus:

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+----->----+
|          |
^          ^
^          ^
|          |
+----->----+

This is the same sort of geometry that the video game Asteroids has. When something goes off the top of the screen, it comes back on the bottom the same distance from each vertical edge. When something goes off the left, it comes back on the right keeping the same distance from the horizontal edges.

Now, a Klein bottle is sort of the Moebius-strip version of the torus. It's got a half twist in one direction.

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+----->----+
|          |
^          v
^          v
|          |
+----->----+

So, something that goes off the left side still comes back on the right side, but if it was 50% closer to the top edge when it left, it will be 50% closer to the bottom edge when it comes back. But, if something goes off of the top edge, it will come back at the bottom edge, keeping the same distance from each vertical edge.

If you twist in both directions, you come up with what is called a projective sphere:

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+----->----+
|          |
^          v
^          v
|          |
+----<-----+

When projected down to three-dimensions, this is called a cross-cap because it ends up taking on a sort of fortune-cookie shape where it gets all pinched together.

I'll read the rest of the thread more closely later. But, I hope this helps you get some firmer footing on the Klein bottle.

p.s. ubb really ticks me off sometimes as does anything that tries to format my text for me.

[PWrong]

Technically, the Klein bottle doesn't intersect at all. It has to intersect to be projected down into three-dimensional space. But, in four-dimensional space, it sits perfectly fine without intersection.

I meant to say that actually, but it doesn't matter. In this case, we want the surface to intersect.

When projected down to three-dimensions, this is called a cross-cap because it ends up taking on a sort of fortune-cookie shape where it gets all pinched together.

Hooray, lets wrap space around a fortune cookie instead

I'll read the rest of the thread more closely later. But, I hope this helps you get some firmer footing on the Klein bottle.

Thanks, that helped a lot. I'm pretty sure my klein bottled solar system will still work (if orbits work in 2D at all). The second planet might appear to be in the opposite direction to the one intended though.:? I'd like to add another dimension, but maybe we should stick to the 2D analogy for now.

[Keiji]

If you twist in both directions, you come up with what is called a projective sphere:

Code: Select all

+----->----+
|          |
^          v
^          v
|          |
+----<-----+

When projected down to three-dimensions, this is called a cross-cap because it ends up taking on a sort of fortune-cookie shape where it gets all pinched together.

Could you give me an image of that, I can't visualise how that would be folded...

[pat]

My terminology was bad... it's actually called the 'projective plane' rather than the 'projective sphere'. Anyhow, here's some pictures of a cross-cap (brought down to three dimensions).

[Keiji]

Yes, I remember that now.