4d Wormhole Picture?

[Hugh]

Here is a 2d representation of a wormhole, showing 2d space bent in 3d, and the shortcut that the wormhole could give:



Has anyone ever attempted to draw a 2d representation of a wormhole, showing 3d space bent in 4d, and the shortcut that the wormhole could give?

[Nick]

No, I don't think so, but I would love to see it if it was

[Hugh]

No, I don't think so, but I would love to see it if it was

Me too.

I've been wondering if there is a logical way to approach this. Let's look at what happens in 2d and see if we can do the same thing with our 3d picture.

Okay, for the picture of the 2d plane wormhole through the 3d dimension, this is the process:
- Take a 2d plane, put it in 3d space.
- Curve the 2d plane in 3d space around so that it brings two distant parts of the 2d plane closer in the 3rd dimension.
- Connect the distant plane parts together with a wormhole, through the closest parts in the 3rd dimension.

So now, we should be able to figure out what to do with our 3d cube wormhole through the 4th dimension picture:
- Take a 3d cube, put it in 4d space.
- Curve the 3d cube in 4d space around so that it brings two distant parts of the 3d cube closer in the 4th dimension.
- Connect the distant cube parts together with a wormhole, through the closest parts in the 4th dimension.

So here's a 3d cube, let's say it's the middle cube, in 4d space.



Any takers as to what we draw next?

[papernuke]

Well, I haven't tried to draw one, but I saw one just like yours (except the color). It was in my Kingfishers Encyclopedia of Sci.

[Hugh]

Well, I haven't tried to draw one, but I saw one just like yours (except the color). It was in my Kingfishers Encyclopedia of Sci.

Wormhole pictures are cool to look at.

I've only ever seen a 2d plane one in 3d space in a picture though Icon. I'm hoping that someone can illustrate how a 3d one looks in 4d space in a 2d picture here.

[papernuke]

It'll probably look like a four dimensional cylinder on its side, with two foruth dimensional peices of paper on the sides. But no matter what, it'll look weird (to us at least).

[tbriggs]

Page 5 of "Exploring Hyperspace with the Geometric Product" (posted in Tetraspace and elsewhere) has a wormhole figure which shows a neglected projection that could be very helpful.

[Hugh]

Page 5 of "Exploring Hyperspace with the Geometric Product" (posted in Tetraspace and elsewhere) has a wormhole figure which shows a neglected projection that could be very helpful.

Hi Tom, I checked out your thread and document, very interesting.

Is the wormhole figure you are talking about Figure 6, the matching segments of projections of hemispherical hypercylinders, which when joined at the edges, gives the interval of spheres orthogonal to 3 space?

[tbriggs]

Thank you for reading my "hyperspace" post. Based on Figure 5, which diagrams a wormhole in "Flatland", Figure 6 shows a projection of a small slice of a wormhole embedded in 4-space that is cut along the axis to produce hemispherical hypercylinders. This longitudinal cut corresponds to the longitudinal slice used in Figure 5 but cannot be readily visualized in our 3-space.

[Hugh]

So what's the next step in the attempt to draw a more complete and fuller overall picture?

[tbriggs]

The next step would be to combine the two halves in Figure 6 as was done in Figure 5. However, the resulting column of difficult to discern overlapping spheres would not add much to an understanding of the wormhole beyond what is implied by Figure 5.

[Hugh]

Well, we'd have to try to understand how that would work when we'd see the final picture and how it would fit in there. The 3d wormhole through the curved 2d plane picture is so easy to understand. The problem is how to show a curved 3d space and the shortcut 4d wormhole, all in a 2d picture.

Is there any way to show it using one of the cubes of a hypercube curving somehow, then the wormhole of overlapping spheres joining two far away parts? What's the best representation we could acheive here? What about using a stereogram image to make it 3d for our brain, then have the image rotate so that we could see it from different ways to get a better understanding of it?

[houserichichi]

In the first picture posted you form a 2D grid and warp it to form that connecting wormhole bridge. If you wanted, you could easily draw little arrows in each square (more than one if you liked) representing the magnitude and direction of the gravitational "pull" or the "intensity" of the gravitational field at that point in each of the little grid squares. Call these little arrows "vectors". Remove the yellow and red arrows as they're not particularly difficult to understand.

Do you see what I mean?

Now since we can't draw a little arrow (physically) at every single point in that picture (because there would be so many arrows that they would all be touching and the picture would just look like we coloured it in) we'll say that we have drawn ourselves an approximate vector field on 2D plane representation of a 3D wormhole.

So what if to expand further we switch from a 2D grid to a 3D one? Instead of having little squares that look like graph paper, why don't we turn each little square into a little cube and do the same thing as last time? At places inside each cube why don't we draw little arrows limited to lying in each cube and not outside representing the direction and intensity of the gravitational field at that point. So each cube will have many more little arrows, but were we to zoom in and keep the cube size proportional to our zoom then we would eventually be able to make out what the field intensity is along any curve in our messy diagram.

To re-iterate, take the green grid diagram in the first post of this thread (the 'standard' picture of a wormhole) and wherever you see a little square you fill in arrows indicating the intensity of the gravitational "pull" at that point. Now, comb through the grid again and wherever you see a little square with arrows you now "raise" that square up and form a cube. So at the base of the cube you have the original arrows and now it's your job to fill in arrows in the remaining space of each of the cubes one by one.

If that didn't make sense then I'll try and draw a diagram tomorrow as I have the day off work. It's actually difficult to explain in words, sorry...Only a thought, anyway. I know it's not as interesting as a proper 4D picture mashed down to 2D but it would at least give us an idea of what kind of beast we're dealing with? I suppose rather than arrows one could use colours. Have red for more "curved" arrows and purple for less, filling in the spectrum in between? At any rate, I digress...

[Hugh]

I'll try and draw a diagram tomorrow as I have the day off work... I know it's not as interesting as a proper 4D picture mashed down to 2D but it would at least give us an idea of what kind of beast we're dealing with?

It is quite a beast we are dealing with here.

I'd like to see your idea in picture form house, would you mind drawing it?