## Materializing Hyperbolic Spaces with Gradient-Index Optics and One-Way Mirrors

Burning Man is one week away, so I figured I would share a neat idea I’ve been hoarding that could lead to a kick-ass Burning Man-style psychedelic art installation. If I have the time and resources to do so, I may even try to manifest this idea in real life at some point.

Around the time I was writing The Hyperbolic Geometry of DMT Experiences (cf. Eli5) I began asking myself how to help people develop a feel for what it is like to inhabit non-Euclidean phenomenal spaces. I later found out that Henry Segerman developed an immersive VR experience in which you can explore 3D hyperbolic spaces. That is fantastic, and a great step in the right direction. But I wanted to see if there was any way for us to experience 3D hyperbolic geometry in a material way without the aid of computers. Something that you could hold in your hand, like a sort of mystical amulet that works as a reminder of the vastness of the state-space of consciousness.

What I had in mind was along the lines of how we can, in a sense, visualize infinite (Euclidean) space using two parallel mirrors. I thought that maybe there could be a way to do the same but in a way that visualizes a hyperbolic space.

### One-Way Mirrors and 3D Space-Filling Shapes

Right now you can use one-way mirrors on the sides of a polyhedra whose edges are embedded with LEDs to create a fascinating “infinite space effect”:

This works perfectly for cubes in particular, given that cubes are symmetrical space-filling polyhedra. But as you can see in the video above, the effect is not quite perfect when we use dodecahedra (or any other Platonic solid). The corners simply do not align properly. This is because the solid angles of non-cube Platonic solids cannot be used to cover perfectly 4π steradians (i.e. what 8 cubes do when meeting at a corner):

This is not the case in hyperbolic space, though; arbitrary regular polyhedra can tesselate 3D hyperbolic spaces. For instance, one can use dodecahedra by choosing their size appropriately in such a way that they all have 90 degree angle corners (cf. Not Knot):

### Gradient-Index Optics

Perhaps, I thought to myself, there is a way to physically realize hyperbolic curvature and enable us to see what it is like to live in a place in which dodecahedra tesselate space. I kept thinking about this problem, and one day while riding the BART and introspecting on the geometry of sound, I realized that one could use gradient-index optics to create a solid in which light-paths behave as if the space was hyperbolic.

Gradient-index optics is the subfield of optics that specializes in the use of materials that have a smooth non-constant refractive index. One way to achieve this is to blend two transparent materials (e.g. two kinds of plastic) in such a way that the concentration of each type varies smoothly from one region to the next. As a consequence, light travels in unusual and bendy ways, like this:

### Materializing Hyperbolic Spaces

By carefully selecting various transparent plastics with different indices of refraction and blending them in a 3D printer in precisely the right proportions, one can in principle build solids in which the gradient-index properties of the end product instantiate a hyperbolic metric. If one were to place the material with the lowest refraction index at the very center in a dodecahedron and add materials of increasingly larger refractive indices all the way up to the corners, then the final effect could be one in which the dodecahedron has an interior in which light moves *as if* it were in a hyperbolic space. One can then place LED strips along the edges and seal the sides with one-way window film. Lo-and-behold, one would then quite literally be able to “hold infinity in the palm of your hand”:

I think that this sort of gadget would allow us to develop better intuitions for what the far-out (experiential) spaces people “visit” on psychedelics look like. One can then, in addition, generalize this to make space behave as if its 3D curvature was non-constant. One might even, perhaps, be able to visualize a black-hole by emulating its event-horizon using a region with extremely large refractive index.

### Challenges

I would like to conclude by considering some of the challenges that we would face trying to construct this. For instance, finding the right materials may be difficult because they would need to have a wide range of refractive indices, all be similarly transparent, able to smoothly blend with each other, and have low melting points. I am not a material scientist, but my gut feeling is that this is not currently impossible. Modern gradient-index optics already has a rather impressive level of precision.

Another challenge comes from the resolution of the 3D printer. Modern 3D printers have layers with a thickness between .2 to 0.025mm. It’s possible that this is simply not small enough to avoid visible discontinuities in the light-paths. At least in principle this could be surmounted by melting the last layer placed such that the new layer smoothly diffuses and partially blends with it in accordance with the desired hyperbolic metric.

An important caveat is that the medium in which we live (i.e. air at atmospheric pressure) is not very dense to begin with. In the example of the dodecahedra, this may represent a problem considering that the corners need to form 90 degree angles from the point of view of an outside observer. This would imply that the surrounding medium needs to have a higher refraction index than that of the transparent medium at the corners. This could be fixed by immersing the object in water or some other dense media (and designing it under the assumption of being surrounded by such a medium). Alternatively, one can simply fix the problem by using appropriately curved sides in lieu of straight planes. This may not be as aesthetically appealing, though, so it may pay off to brainstorm other clever approaches to deal with this that I haven’t thought of.

Above all, perhaps the most difficult challenge would be that of dealing with the inevitable presence of chromatic aberrations:

Since the degree to which a light-path bends in a medium depends on its frequency, how bendy light looks like with gradient-index optics is variable. If the LEDs placed at the edges of the polyhedra are white, we could expect very visible distortions and crazy rainbow patterns to emerge. This would perhaps be for the better when taken for its aesthetic value. But since the desired effect is one of actually materializing faithfully the behavior of light in hyperbolic space, this would be undesirable. The easiest way to deal with this problem would be to show the gadget in a darkened room and have only monochrome LEDs on the edges of the polyhedra whose frequency is tuned to the refractive gradient for which the metric is hyperbolic. More fancifully, it might be possible to overcome chromatic aberrations with the use of metamaterials (cf. “Metasurfaces enable improved optical lens performance“). Alas, my bedtime is approaching so I shall leave the nuts and bolts of this engineering challenge as an exercise for the reader…