Space bubbles

Imagine that our 3D space is just a 3D surface in four spatial dimenions. Now imagine that this surface is not just a simple flat or curved surface, but that it branches.

Let's say that the 3D surfaces are minimal surfaces, they minimise volume. We can make a lot of complex minimal surfaces, but one of the simplest is bubbles. 

We can start out with a flat 3-surface (our normal world) and add 4D bubbles onto it, which enclose pockets of 4D volume.  

A 3D analogy of this is a soap bubble on a soap film:

Our usual space is then the flatter surface (larger bubble in image). Notice that the middle bubble intersects the usual space at 120 degree angles, obeying Plateau's laws. This bubble is where our usual space branches into two spaces.

So, what would it look like one dimension up, where we are in the 3D surface observing through a 4D bubble?

I think it would look like this:

(Shadertoy)

To simulate it you trace the light rays through each pixel over all possible branches. Those that hit the bubble split in two and follow the curved path through each of the bubble 3D surfaces. At the far end they also split, they can exit the bubble and will get the pixel colour of the background that they are pointing at, or they can enter the other half of the bubble and return to the near side. They will then split again, and each split will consume half the pixel colour intensity, as though half of the energy of incoming light is distributed in each branch. 

The resulting view sees the space bubble a lot like a solid glass sphere, with both forward and reverse images visible, and the strongest image being an inverted forwards image.

I suspect that the light physics could be closely replicated if the glass sphere could be made to increase in refractive index towards the centre, such that entered light follows a circular path through it.

Unlike a glass sphere though, the above bubble holds two separate volumes of space, so one volume could have a feather in it and the other a balloon, and you would see a superposition of both. 

A bubble pair is next.

Above is a real pair of bubbles in the air, but the 4D case, like with the single bubble, splits at 120 degree angles at its surface. One of the consequences of it being a bubble pair in 4D is that the dihedral angle between the two bubbles in 3D is not 120 degrees but about 109 degrees (acos(-1/3)). That's because you aren't actually looking at a threeway 2D surface but at a 6-way 3D surface, which branches at the angle between vertices (and the centre) of a tetrahedron. 

The difference with the bubble pair is that it contains not just two pairs of volumes but a fifth volume equivalent to the central surface in the bubble pair photo above. This surface is indirectly connected to the outside space. Here I simulate a red ball inside this concealed volume:

Shadertoy

The ball can be seen, but access to it is indirect. If you pushed a pole from one end through the bubble pair and out the other, then both main paths through the bubbles would not hit the ball. Only a longer path through will hit the ball, making the pole appear shorter when it exits the far bubble.

This idea can be shown in a 2D version:

Only the bottom-right path touches the ball here. In 4D this bendy path does not mean the pole is bending, it is taking a locally straight ahead route through the 4D bubbles, which is a geodesic.

One useful take-away from this is that we aren't forced to only view 4D things by projecting them down into fewer dimensions, if they are made of 3D-surfaces then we can view them from within by simulating the light travelling through these surfaces.

I should add that these are classical-only light simulations. In quantum physics my understanding is that the light branches would decohere and so an observer would only see the light pass through one of the branches. However, that is if the bubble volumes are identical. If there is separate air in each branch, so they aren't entangled, then the light physics may be more classical, like in the animations above.