#hyperbolic space

A hyperbolic plane is 'the geometric opposite' of a sphere, explains Henderson in an interview with arts and culture magazine Cabinet. 'On a sphere, the surface curves in on itself and is closed. A hyperbolic plane is a surface in which the space curves away from itself at every point.' It exists in nature in ruffled lettuce leaves, in coral leaf, in sea slugs, in cancer cells. Hyperbolic geometry is used by statisticians when they work with multidimensional data, by Pixar animators when they want to simulate realistic cloth, by auto-industry engineers to design aerodynamic cars, by acoustic engineers to design concert halls. It's the foundation of the theory of relativity, and thus the closest thing we have to an understanding of the shape of the universe. In short, hyperbolic space is a pretty big deal.

But for thousands of years, hyperbolic space didn't exist. At least it didn't according to mathematicians, who believed that there were only two types of space: Euclidean, or flat space, like a table, and spherical space, like a ball. In the nineteenth century, hyperbolic space was discovered - but only in principle. And although mathematicians tried for over a century to find a way to successfully represent this space physically, no one managed it - until Taimina attended that workshop at Cornell. Because as well as being a professor of mathematics, Taimina also liked to crochet.

Taimina learnt to crochet as a schoolgirl. Growing up in Latvia, part of the former Soviet Union, 'you fix your own car, you fix your own faucet - anything', she explains. 'When I was growing up, knitting or any other handiwork meant you could make a dress or a sweater different from everybody else's.' But while she had always seen patterns and algorithms in knitting and crochet, Taimina had never connected this traditional, domestic, feminine skill with her professional work in maths. Until that workshop in 1997. When she saw the battered paper approximation Henderson was using to explain hyperbolic space, she realised: I can make this out of crochet.

And so that's what she did. She spent her summer 'crocheting a classroom set of hyperbolic forms' by the swimming pool. 'People walked by, and they asked me, "What are you doing?" And I answered, "Oh, I'm crocheting the hyperbolic plane."' She has now created hundreds of models and explains that in the process of making them 'you get a very concrete sense of the space expanding exponentially. The first rows take no time but the later rows can take literally hours, they have so many stitches. You get a visceral sense of what "hyperbolic" really means.' Just looking at her models did the same for others: in an interview with the New York Times Taimina recalled a professor who had taught hyperbolic space for years seeing one and saying, 'Oh, so that's how they look.' Now her creations are the standard model for explaining hyperbolic space.

-Caroline Criado Perez, Invisible Women

The world was created in "layers," refining the background noise of the cosmos step by step, but with each reduction in perceived chaos, there's an increase in complexity, ultimately resulting in the outermost layer being almost as entropic as the innermost, creating a cyclic structure, like a matryoshka doll whose innermost doll is just the first doll again.

The world isn't a globe or disc or even a bubble with hazy edges, it's a hyperbolic 4D space that folds in on itself, making navigation incredibly difficult. The "layers" all occupy adjacent space, but not in any way that maps onto one another cleanly, and shifting between the layers requires hyperbolic motion along a 4th spatial axis.

Obviously the world has to be infinite in order for this to work, but beyond that, if there's a limit to how fast resources can be transferred, Euclidean space won't do. The amount of space and resources within a certain distance must increase exponentially with distance, which is what you get with hyperbolic space, so let's go with the world being a hyperbolic plane.

Each region has one upline and multiple downlines. It transfers useful resources to its upline, and waste to its downlines. The amount of resources contained in each region increases because it has more downlines than uplines, and this can continue indefinitely because the world is infinite.

If everyone cooperated, this would be enough for it to work, but such a situation isn't particularly stable. Which direction counts as upline or downline is basically arbitrary, and there's nothing to stop any one region from just refusing to provide resources to its upline. I have two different ways around this issue.

#1, each one of the blocks in the grid shown starts out identical, so the infinite world is fully populated with people from the start. Since each block is identical, they know that at least initially, their downlines will behave the same way as them, so they superrationally choose to cooperate. The initial symmetry gets broken somewhat over time, as each block can treat its left and right downlines differently, then they may treat their four downlines differently in response, and so on. I find this a little unsatisfying since it's still pretty unstable and the world has to have some finite start time with people already in it, but it gets points for being interestingly weird.

#2, something about the world inherently breaks the symmetry, and the upline and downline directions are actually different in such a way that it is significantly easier to defeat someone who is downline of your position. Civilizations threaten their downlines into providing tribute, and predators use this advantage to kill the creatures in their downlines and eat them.

I think this version allows the world to develop more naturally. If it starts out with a uniform distribution of resources, and life emerges somewhere, then it starts to spread in all directions, initially consuming the primordial resources, but shifting as the frontier moves on to consuming other living things instead. Because of the natural advantage to attacking in a downline direction, the developing organisms do this preferentially, concentrating resources even while they're used up, so they never actually run out. A lot of the time the organisms left behind by the frontier likely wouldn't develop these behaviours quickly enough to avoid the regions they're in becoming desolate, but so long as it develops somewhere, the pyramid-scheme lifestyle could still come to cover arbitrarily large regions over time. Because of the exponential size of hyperbolic space, the majority of life at any time would be near the frontier, but from the perspective of something (or later, someone) living far away from it, the frontier may as well not exist.

There are probably loads more implications I haven't thought of, but the economy in such a world would be very odd at least. No resources can really be rare, except by being difficult to transport, because it's always possible to increase their concentration by moving them upline. Since the world is infinite and innovations, both biological and technological, can spread exponentially, the overall rate of innovation would be extremely fast. With no plants, there wouldn't be much in the way of farming, except perhaps things like mushrooms to turn rotten things into something edible (since if every bit of some kind of food rots within a few weeks, no amount of moving it around is going to stop you running out). The usual arguments against mushrooms as a major food source don't apply, because of the pyramid scheme thing. Similarly, there could be predators as major livestock. Just get your downline to deliver their livestock to you so you have twice as many, then feed one half to the other half, and keep doing this every time they get hungry. This situation is of course terrible for infectious disease, but having evolved in this world they probably have robust immune systems. There could be sewers with no outlets, just delivering waste forever to the downline to effectively reduce the amount. Anything manufactured, stable and transportable would only need an extremely tiny fraction of people to actually ever make it, maybe even to the point where it's only the person who invented the thing making a few copies that are then passed around forever. Plus of course money wouldn't really work, because you could get an unlimited amount of it very easily. Maybe it would have to be labelled with the height or something, so money from downline of your location is worth less than local money, and money from upline is worth more. If you successfully set up the exchange rates right, it might even be possible for all this to work out as a stable capitalist economy, without having to forcibly extract tribute from anyone. I don't know enough about exchange rates to tell whether that would actually work though.

This is a Model of hyperbolic space. It was designed by Daina Taimina and made by Daina Taimina. Its medium is wool and its technique is crocheted.

The distinctly crenulated, irregular, and wavy features exhibited in this textile are reminiscent of forms seen in living corals, which are examples of a mathematical structure known as hyperbolic space. The discovery that this type of geometric structure could be modeled in crochet was first made in 1997 by Dr. Daina Taimina, a mathematician working at Cornell University. Her inspiration was based on a suggestion put forward in the 1970s by an American geometer, William Thurston. Thurston’s model, however, was made of paper and taped together, making it inherently fragile and hard to handle. Taimina found that the essence of this construction could be created with crochet by increasing the number of stitches in each row. With each increase, the surface naturally begins to ruffle. The result is a model with a cohesive surface and dynamic form that exhibits many of the intrinsic properties of hyperbolic space. Textiles are often used as tools for learning. This work, for example, enhances the understanding of hyperbolic space as a particular mathematical structure in a tangible form that can be experienced and carefully studied. In this way, it is not unlike the museum’s exceptional collection of embroidered samplers, which were stitched “practice sheets” for young girls in the 18th and 19th centuries and, ultimately, became a record of their expertise. These crocheted models of hyperbolic space not only exemplify textiles as “tools,” but also highlight textiles associated with women and the craft tradition, contributing a new meaning and significance to the historical continuum.