Topology is sometimes described as what does not change when you stretch a rubber surface without gluing or cutting it. Geometry is the opposite. So when we glue the edges of two rubber hexagons, we change the topology (to spherical), but not the geometry. When we blow up the balloon obtained, we change its geometry (also to roughly spherical), but not its topology.
0:00 blowing up a balloon
This is a simulation of blowing up a balloon. We start with these two hexagons, and the balloon becomes closer and closer to a sphere as we blow it up.
This simulation acts as follows. Two forces act on every point of the balloon: the tension of the rubber (if two adjacent points are in distance d, it attracts them with force (d-1)) and the air pressure (on every three points defining a triangle, there is a force orthogonal to that triangle, proportional to (amount of air in the balloon / current volume of the manifold).
However, it is the balloon surface which has non-Euclidean geometry. The world is still Euclidean, and light follows Euclidean straight lines.
0:30 net of the round balloon
So let's see the non-Euclidean view! Let's see what happens when the light rays curve together with the surface (balloon × ℝ geometry). Note that the raytracer still actually uses Euclidean geometry here -- because the balloon obtained from the simulation is still a polyhedron. Here is the net of the inflated one.
0:40 inflated balloon x R
When flying over one of the stars, the other star will look like an annulus -- because, on a sphere, if we go half the circumference in any direction, we reach the other pole.
0:50 flat balloon x R
When the balloon is still flat (not yet inflated), the stars still look like stars. The perspective looks Euclidean in general. In the flat case the "cone lines" (corresponding to the 6 vertices of the polyhedron) are clearly visible, because there is only 240 degrees around them. In the inflated case we see only minor glitches and a good approximation of S2xR geometry.
1:00 intermediate balloon x R
And here is what happens when the balloon is only partially inflated.
1:10 intermediate balloon in E3
The intermediate balloon looks like this. It would be interesting to simulate the inner view in a three-dimensional balloon being blown up in four-dimensional space (our Universe could be something like this). Maybe later if there is interest.
Now, some side results... what do we get when the topology of the balloon is not spherical?
1:15 toroidal balloon
Here we have a toroidal balloon. A surprising thing happens at 1:27. The balloon blowing algorithm attempts to find an "optimal" immersion of the given manifold into the three-dimensional Euclidean space.
(This is an "immersion" because the algorithm does not make sure that the balloon does not self-intersect. Even when it does self-intersect, the manifold has a well-defined "volume", so the algorithm is still happy.)
The shape depends on the amount of air pumped in and out; as we give more and more air, it sometimes happens that the manifold changes to a less symmetrical shape. We show one example for each manifold (except for the torus).
1:45 Zebra quotient
Here is the Zebra Quotient (a somewhat symmetric hyperbolic manifold of genus 2, used in HyperRogue as the basis of the land Zebra).
2:00 Klein Quartic
And here is the Klein quartic, a highly symmetric hyperbolic manifold of genus 3. (The balloon has sevenfold symmetry, with more air, we would get threefold symmetry.)
2:15 Macbeath surface
And here is the Macbeath surface, a highly symmetric hyperbolic manifold of genus 7.
2:30 Bolza surface
It generally tries to make it spherical (as the sphere has the smallest surface for the given volume). It kind of succeeds with the Bolza surface -- the result looks like a sphere, but it is actually a double branched cover (there are two sheets, and going 360° around a vertex brings you to the other sheet).
2:45 Bolza surface for reference
Here is the Bolza surface for reference. Viewed as the universal cover (all the locations where the Princess is seen are the same location). The red vertices have 8 triangles next to them, in the last animation we see 4 because it is a double cover.