I've been thinking about geometry and space. Specifically I've been thinking about different spaces in terms of resource gathering and resource delivery. When gathering resources, the most important thing is that a small search radius encompasses a very large volume. One of the great tragedies of living in euclidian space is that volume only grows polynomially with respect to radius. For example, on a flat plane, the area of your search circle is pi*r^2. Emphasis on the r^2.
Hyperbolic space by contrast has far better properties in this regard. For example you might see something like 2^r instead. This could be the difference between having 400 resources in a 20 unit radius and having 1,000,000 resources in a 20 unit radius.
However, hyperbolic space is far worse for resource delivery. If you are transporting a resource from a source to a destnation then it's important to be able to navigate around obstacles. For example, the fat butt of the another guy delivering between the same source and destination.
In hyperbolic space, the circumference of an obstacle with radius r is on the order of 2^r. Exponential again. Meanwhile in euclidian space, the circumference of an obstacle with radius r is linear: 2*pi*r. This means that if 20 people are performing a delivery in euclician space, they might have to travel about 20 units in order to get around eachother. Whereas in hyperbolic space, some poor delivery guy might have to travel a path of 1,000,000 units just to avoid intersecting with the paths of the other 19 delivery guys.
The ideal space has both of these properties, but clearly messing with the curvature of the space isn't enough to accomplish that. Cranking up the number of dimensions far beyond our ordinary three can help a lot, but unlike messing with the curveature, (like for hyperbolic space) the effects of high dimensionality don't depend on scale.
That means that even just existing in such a world would be very difficult. With 100 dimensions you'd need 100 legs just to avoid falling over, and you would need like 100 unit^99s of skin to cover one unit^100 of flesh or your organs would fall out. Not great!
The only way I can think of to avoid this issue is if the space is some sort of very complicated manifold where distant locations are connected to eachother as if by a portal. But that isn't super nice either. For example, in regular euclidian and hyperbolic space you can always make a path slightly shorter by just taking a small shortcut. That's not always the case in more complicated manifolds. There are some cycles you can take where there isn't any small shortcut and there isn't any way to shorten a looping path without going a completely different way.
Additionally, such a manifold would no longer be as symmetric as euclidian and hyperbolic space. Even if there were no stationary objects at all to act as reference points in the space, you may still be able to tell different locations apart just based purely on how the space loops back on itself.
I'll have to keep thinking about this.