In previous posts I talked about creating models of hyperbolic space with crochet or with a differential-growth type algorithm. Here I’ll describe a very different approach.
To get started, let’s first talk about the Euclidean plane. Most people are familiar with the fact that you can tile the plane with hexagons; you can find ceramic floor and wall tile in the shape of hexagons in any tile shop. However, you can’t see an infinite tiling of the plane all at once. We can remedy this by looking through a fisheye lens. (For the mathematicians, I’ll use a stereographic projection.) The result looks something like this:
In theory, an image like this can depict all of the infinitely many hexagons at once (assuming infinite resolution), at an obvious cost: although the hexagons are all identical, the lens creates distortion so they don’t look identical. As a result, it becomes very hard to see the hexagons at the edge of the image.
You can’t tile the Euclidean plane with seven-sided polygons (proving this is a great exercise!), but you can in hyperbolic space. Such a tiling has a similar problem, though: there’s no way to “see” all the heptagons at once, without introducing some distortion. This is the idea behind the Poincaré disk model of hyperbolic space, shown here:
Like the fisheye view of the Euclidean plane, an infinite resolution image of the Poincaré disk would show infinitely many heptagons, at the cost of distortion: although heptagons toward the edge look smaller, in hyperbolic space they are all identical.
Of course, to construct a physical model of anything, it has to be finite. If we’re not trying to depict an infinitely large space, then constructing undistorted models becomes more possible: we can, of course, tile a room with identical hexagonal tiles, and thus create an undistorted model of a part of the tiled Euclidean plane. Similarly, we can assemble finitely many identical heptagons to create a physical model of a part of hyperbolic space, as shown here.
To see more of the space, we can just assemble more heptagons:
However, here we run into a problem: since we’re trying to create larger and larger patches of hyperbolic space inside our Euclidean 3-dimensional world, they will necessarily start to intersect each other. If you look closely enough at the rotating image above, you can see some of these self-intersections.
With this limitation aside, we’ve come to our method to create a physical model of hyperbolic space: choose some physical material to make heptagons out of, and assemble. Many years ago I did this with grade-school children, using paper heptagons which they stapled together. Here is a model I made recently, where the heptagons were assembled virtually, and turned into physical reality with a 3D printer:
This was made in the CAD program Rhinoceros 3D, using the Grasshopper scripting platform and a variety of plug-ins. A central part of the design process used the Kangaroo plug-in for Grasshopper, a physics based modeling package written by Daniel Piker. The idea was to start with the heptagonal tiling of the Poincaré disk depicted above, and create a simulation in which each edge acts as a spring with equal rest length. In addition, a repulsive “charge” was added at each vertex, to avoid self-intersections as much as possible. The resulting model then springs into shape, as shown here:
Many others have made models of hyperbolic space using the same basic idea of attaching polygons. Here’s one by Henry Segerman and Geoffrey Irving, where groups of seven triangles are attached to make several heptagons, and those are attached to make a model of hyperbolic space. In this design hinges were used to attached the triangles, giving the model a floppy, fabric-like feel, reminiscent of the crochet models I discussed in a previous post.
Here is another model I made, using essentially the same Grasshopper code as the colorful design above. These were 3D printed in PA12, a relatively inexpensive and versatile plastic-like material.
This particular piece will be on display as part of Margaret and Christine Wertheim’s Austrian Satellite Reef exhibition at the Linz Palace Museum (Schlossmuseum Linz), October 5th 2023 to April 4th, 2024.
Very cool, Edmund!