What is Love? Neural Annealing in the Presence of an Intentional Object

Excerpt from: The Neuroscience of Meditation: Four Models by Michael E. Johnson


Neural annealing: Annealing involves heating a metal above its recrystallization temperature, keeping it there for long enough for the microstructure of the metal to reach equilibrium, then slowly cooling it down, letting new patterns crystallize. This releases the internal stresses of the material, and is often used to restore ductility (plasticity and toughness) on metals that have been ‘cold-worked’ and have become very hard and brittle— in a sense, annealing is a ‘reset switch’ which allows metals to go back to a more pristine, natural state after being bent or stressed. I suspect this is a useful metaphor for brains, in that they can become hard and brittle over time with a build-up of internal stresses, and these stresses can be released by periodically entering high-energy states where a more natural neural microstructure can reemerge.

Furthermore, from what I gather from experienced meditators, successfully entering meditative flow may be one of the most reliable ways to reach these high-energy brain states. I.e., it’s very common for meditation to produce feelings of high intensity, at least in people able to actually enter meditative flow.* Meditation also produces more ‘pure’ or ‘neutral’ high-energy states, ones that are free of the intentional content usually associated with intense experiences which may distort or limit the scope of the annealing process. So we can think of intermediate-to-advanced (‘successful flow-state’) meditation as a reheating process, whereby the brain enters a more plastic and neutral state, releases pent-up structural stresses, and recrystallizes into a more balanced, neutral configuration as it cools. Iterated many times, this will drive an evolutionary process and will produce a very different brain, one which is more unified & anti-fragile, less distorted toward intentionality, and in general structurally optimized against stress.

An open question is how or why meditation produces high-energy brain states. There isn’t any consensus on this, but I’d offer with a nod to the predictive coding framework that bottom-up sense-data is generally excitatory, adding energy to the system, whereas top-down predictive Bayesian models are generally inhibitory, functioning as ‘energy sinks’. And so by ‘noting and knowing’ our sensations before our top-down models activate, in a sense we’re diverting the ‘energy’ of our sensations away from its usual counterbalancing force. If we do this long enough and skillfully enough, this energy can build up and lead to ‘entropic disintegration’, the prerequisite for annealing. (Thanks to Andrés for discussion here)

If this model is true, it feels very important for optimizing a meditation practice. E.g., we should try to figure out some rules of thumb for:

  • How to identify a high-energy brain state, in yourself and others, and how best to create them;
  • Things to do, and things not to do, during an annealing process (‘how to anneal the right things’);
  • Identifying tradeoffs in ‘cooling’ the brain quickly vs slowly.

Off the top of my head, I’d suggest that one of the worst things you could do after entering a high-energy brain state would be to fill your environment with distractions (e.g., watching TV, inane smalltalk, or other ‘low-quality patterns’). Likewise, it seems crucial to avoid socially toxic or otherwise highly stressful conditions. Most likely, going to sleep as soon as possible without breaking flow would be a good strategy to get the most out of a high-energy state. Avoiding strong negative emotions during such states seems important, as does managing your associations (psychedelics are another way to reach these high-energy states, and people have noticed there’s an ‘imprinting’ process where the things you think about and feel while high can leave durable imprints on how you feel after the trip). Finally, perhaps taking certain nootropics could help strengthen (or weaken) the magnitude of this annealing process.

Finally, to speculate a little about one of the deep mysteries of life, perhaps we can describe love as the result of a strong annealing process while under the influence of some pattern. I.e., evolution has primed us such that certain intentional objects (e.g. romantic partners) can trigger high-energy states where the brain smooths out its discontinuities/dissonances, such that given the presence of that pattern our brains are in harmony.[3] This is obviously a two-edged sword: on one hand it heals and renews our ‘cold-worked’ brain circuits and unifies our minds, but also makes us dependent: the felt-sense of this intentional object becomes the key which unlocks this state. (I believe we can also anneal to archetypes instead of specific people.)

Annealing can produce durable patterns, but isn’t permanent; over time, discontinuities creep back in as the system gets ‘cold-worked’. To stay in love over the long-term, a couple will need to re-anneal in the felt-presence of each other on a regular basis.[4] From my experience, some people have a natural psychological drive toward reflexive stability here: they see their partner as the source of goodness in their lives, so naturally they work hard to keep their mind aligned on valuing them. (It’s circular, but it works.) Whereas others are more self-reliant, exploratory, and restless, less prone toward these self-stable loops or annealing around external intentional objects in general. Whether or not, and within which precise contexts, someone’s annealing habits fall into this ‘reflexive stability attractor’ might explain much about e.g. attachment style, hedonic strategy, and aesthetic trajectory.

Links: Annealing (metallurgy)The entropic brain

[3] Anecdotally, the phenomenology of love-annealing is the object ‘feels beautiful from all angles’. This may imply that things (ideas, patterns, people) which are more internally coherent & invariant across contexts can produce stronger annealing effects — i.e. these things are more easy to fall deeply in love with given the same ‘annealing budget’, and this love is more durable.

[4] It’s important to note that both intense positive and intense negative experiences can push the brain into high-energy states; repeated annealing to negative emotions may serve many of the same functions as ‘positive annealing’, but also predispose brains to ‘sing in a minor key’ (see ‘kindling’).


Related Work: Algorithmic Reduction of Psychedelic States, Principia Qualia: Part II – Valence, and Ecstasy and Honesty


Image credit: Fabián Jiménez

Psychedelic Turk: A Platform for People on Altered States of Consciousness

An interesting variable is how much external noise is optimal for peak processing. Some, like Kafka, insisted that “I need solitude for my writing; not ‘like a hermit’ – that wouldn’t be enough – but like a dead man.” Others, like von Neumann, insisted on noisy settings: von Neumann would usually work with the TV on in the background, and when his wife moved his office to a secluded room on the third floor, he reportedly stormed downstairs and demanded “What are you trying to do, keep me away from what’s going on?” Apparently, some brains can function with (and even require!) high amounts of sensory entropy, whereas others need essentially zero. One might look for different metastable thresholds and/or convergent cybernetic targets in this case.

– Mike Johnson, A future for neuroscience

My drunk or high Tweets are my best work.

– Joe Rogan, Vlog#18

Introduction

Mechanical Turk is a service that makes outsourcing simple tasks to a large number of people extremely easy. The only constraint is that the tasks outsourced ought to be the sort of thing that can be explained and performed within a browser in less than 10 minutes, which in practice is not a strong constraint for most tasks you would outsource anyway. This service is in fact a remarkably effective way to accelerate the testing of digital prototypes at a reasonable price.

I think the core idea has incredible potential in the field of interest we explore in this blog. Namely, consciousness research and the creation of consciousness technologies. Mechanical Turk is already widely used in psychology, but its usefulness could be improved further. Here is an example: Imagine an extension to Mechanical Turk in which one could choose to have the tasks completed (or attempted) by people in non-ordinary states of consciousness.

Demographic Breakdown

With Mechanical Turk you can already ask for people who belong to specific demographic categories to do your task. For example, some academics are interested in the livelihoods of people within certain ages, NLP researchers might need native speakers of a particular language, and people who want to proof-read a text may request users who have completed an undergraduate degree. The demographic categories are helpful but also coarse. In practice they tend to be used as noisy proxies for more subtle attributes. If we could multiply the categories, which ones would give the highest bang for the buck? I suspect there is a lot of interesting information to be gained from adding categories like personality, cognitive organization, and emotional temperament. What else?

States of Consciousness as Points of View

One thing to consider is that the value of a service like Mechanical Turk comes in part from the range of “points of view” that the participants bring. After all, ensemble models that incorporate diverse types of modeling approaches and datasets usually dominate in real-world machine learning competitions (e.g. Kaggle). Analogously, for a number of applications, getting feedback from someone who thinks differently than everyone already consulted is much more valuable than consulting hundreds of people similar to those already queried. Human minds, insofar as they are prediction machines, can be used as diverse models. A wide range of points of view expands the perspectives used to draw inferences, and in many real-world conditions this will be beneficial for the accuracy of an aggregated prediction. So what would a radical approach to multiplying such “points of view” entail? Arguably a very efficient way of doing so would involve people who inhabit extraordinarily different states of consciousness outside the “typical everyday” mode of being.

Jokingly, I’d very much like to see the “wisdom of the crowds enhanced with psychedelic points of view” expressed in mainstream media. I can imagine an anchorwoman on CNN saying: “according to recent polls 30% of people agree that X, now let’s break this down by state of consciousness… let’s see what the people on acid have to say… ” I would personally be very curious to hear how “the people on acid” are thinking about certain issues relative to e.g. a breakdown of points of view by political affiliation. Leaving jokes aside, why would this be a good idea? Why would anyone actually build this?

I posit that a “Mechanical Turk for People on Psychedelics” would benefit the requesters, the workers, and outsiders. Let’s start with the top three benefits for requesters: better art and marketing, enhanced problem solving, and accelerating the science of consciousness. For workers, the top reason would be making work more interesting, stimulating, and enjoyable. And from the point of view of outsiders, we could anticipate some positive externalities such as improved foundational science, accelerated commercial technology development, and better prediction markets. Let’s dive in:

Benefits to Requesters

Art and Marketing

A reason why a service like this might succeed commercially comes from the importance of understanding one’s audience in art and marketing. For example, if one is developing a product targeted to people who have a hangover (e.g. “hangover remedies”), one’s best bet would be to see how people who actually are hungover resonate with the message. Asking people who are drunk, high on weed, on empathogenic states, on psychedelics, specific psychiatric medications, etc. could certainly find its use in marketing research for sports, comedy, music shows, etc.

Basically, when the product is consumed in the sort of events in which people frequently avoid being sober for the occasion, doing market research on the same people sober might produce misleading results. What percent of concert-goers are sober the entire night? Or people watching the World Cup final? Clearly, a Mechanical Turk service with diverse states of consciousness has the potential to improve marketing epistemology.

On the art side, people who might want to be the next Alex Grey or Android Jones would benefit from prototyping new visual styles on crowds of people who are on psychedelics (i.e. the main consumers of such artistic styles).

As an aside, I would like to point out that in my opinion, artists who create audio or images that are expected to be consumed by people in altered states of consciousness have some degree of responsibility in ensuring that they are not particularly upsetting to people in such states. Indeed, some relatively innocent sounds and images might cause a lot of anxiety or trigger negative states in people on psychedelics due to the way they are processed in such states. With a Mechanical Turk for psychedelics, artists could reduce the risk of upsetting festival/concert goers who partake in psychedelic perception by screening out offending stimuli.

Problem Solving

On a more exciting note, there are a number of indications that states of consciousness as alien as those induced by major psychedelics are at times computationally suited to solve information processing tasks in competitive ways. Here are two concrete examples: First, in the sixties there was some amount of research performed on psychedelics for problem solving. A notorious example would be the 1966 study conducted by Willis Harman & James Fadiman in which mescaline was used to aid scientists, engineers, and designers in solving concrete technical problems with very positive outcomes. And second, in How to Secretly Communicate with People on LSD we delved into ways that messages could be encoded in audio-visual stimuli in such a way that only people high on psychedelics could decode them. We called this type of information concealment Psychedelic Cryptography:

These examples are just proofs of concept that there probably are a multitude of tasks for which minds under various degrees of psychedelic alteration outperform those minds in sober states. In turn, it may end up being profitable to recruit people on such states to complete your tasks when they are genuinely better at them than the sober competition. How to know when to use which state of consciousness? The system could include an algorithm that samples people from various states of consciousness to identify the most promising states to solve your particular problem and then assign the bulk of the task to them.

All of this said, the application I find the most exciting is…

Accelerating the Science of Consciousness

The psychedelic renaissance is finally getting into the territory of performance enhancement in altered states. For example, there is an ongoing study that evaluates how microdosing impacts how one plays Go, and another one that uses a self-blinding protocol to assess how microdosing influences cognitive abilities and general wellbeing.

A whole lot of information about psychedelic states can be gained by doing browser experiments with people high on them. From sensory-focused studies such as visual psychophysics and auditory hedonics to experiments involving higher-order cognition and creativity, internet-based studies of people on altered states can shed a lot of light on how the mind works. I, for one, would love to estimate the base-rate of various wallpaper symmetry groups in psychedelic visuals (cf. Algorithmic Reduction of Psychedelic States), and to study the way psychedelic states influence the pleasantness of sound. There may be no need to spend hundreds of thousands of dollars in experiments that study those questions when the cost of asking people who are on psychedelics to do tasks can be amortized by having them participate in hundreds of studies on e.g. a single LSD session.

Quantifying Bliss (36)

17 wallpaper symmetry groups

This kind of research platform would also shed light on how experiences of mental illness compare with altered states of consciousness and allow us to place the effects of common psychiatric medications on a common “map of mental states”. Let me explain. While recreational materials tend to produce the largest changes to people’s conscious experience, it should go without saying that a whole lot of psychiatric medications have unusual effects on one’s state of consciousness. For example: Most people have a hard time pin-pointing the effect of beta blockers on their experience, but it is undeniable that such compounds influence brain activity and there are suggestions that they may have long-term mood effects. Many people do report specific changes to their experience related to beta blockers, and experienced psychonauts can often compare their effects to other drugs that they may use as benchmarks. By conducting psychophysical experiments on people who are taking various major psychoactives, one would get an objective benchmark for how the mind is altered along a wide range of dimensions by each of these substances. In turn, this generalized Mechanical Turk would enable us to pin-point where much more subtle drugs fall along on this space (cf. State-Space of Drug Effects).

In other words, this platform may be revolutionary when it comes to data collection and bench-marking for psychiatric drugs in general. That said, since these compounds are more often than not used daily for several months rather than briefly or as needed, it would be hard to see how the same individual performs a certain task while on and off the medicine. This could be addressed by implementing a system allowing requesters to ask users for follow up experiments if/when the user changes his or her drug regimen.

Benefit to Users

As claimed earlier on, we believe that this type of platform would make work more enjoyable, stimulating, and interesting for many users. Indeed, there does seem to be a general trend of people wanting to contribute to science and culture by sharing their experiences in non-ordinary states of consciousness. For instance, the wonderful artists at r/replications try to make accurate depiction of various unusual states of consciousness for free. There is even an initiative to document the subjective effects of various compounds by grounding trip reports on a subjective effects index. The point being that if people are willing to share their experience and time on psychedelic states of consciousness for free, chances are that they will not complain if they can also earn money with this unusual hobby.

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LSD replication (source: r/replications)

We also know from many artists and scientists that normal everyday states of consciousness are not always the best for particular tasks. By expanding the range of states of consciousness with economic advantages, we would be allowing people to perform at their best. You may not be allowed to conduct your job while high at your workplace even if you perform it better that way. But with this kind of platform, you would have the freedom to choose the state of consciousness that optimizes your performance and be paid in kind.

Possible Downsides

It is worth mentioning that there would be challenges and negative aspects too. In general, we can probably all agree that it would suck to have to endure advertisement targeted to your particular state of consciousness. If there is a way to prevent this from happening I would love to hear it. Unfortunately, I assume that marketing will sooner or later catch on to this modus operandi, and that a Mechanical Turk for people on altered states would be used for advertisement before anything else. Making better targeted ads, it turns out, is a commercially viable way of bootstrapping all sorts of novel systems. But better advertisement indeed puts us at higher risk of being taken over by pure replicators in the broader scope, so it is worth being cautious with this application.

In the worst case scenario, we discover that very negative states of consciousness dominate other states in the arena of computational efficiency. In this scenario, the abilities useful to survive in the mental economy of the future happen to be those that employ suffering in one way or another. In that case, the evolutionary incentive gradients would lead to terrible places. For example, future minds might end up employing massive amounts of suffering to “run our servers”, so to speak. Plus, these minds would have no choice because if they don’t then they would be taken over by other minds that do, i.e. this is a race to the bottom. Scenarios like this have been considered before (1, 2, 3), and we should not ignore their warning signs.

Of course this can only happen if there are indeed computational benefits to using consciousness for information processing tasks to begin with. At Qualia Computing we generally assume that the unity of consciousness confers unique computational benefits. Hence, I would expect any outright computational use of states of consciousness is likely to involve a lot of phenomenal binding. Hence, at the evolutionary limit, conscious super-computers would probably be super-sentient. That said, the optimal hedonic tone of the minds with the highest computational efficiency is less certain. This complex matter will be dealt with elsewhere.

Concluding Discussion

Reverse Engineering Systems

A lot of people would probably agree that a video of Elon Musk high on THC may have substantially higher value than many videos of him sober. A lot of this value comes from the information gained about him by having a completely new point of view (or projection) of his mind. Reverse-engineering systems involves doing things to them to change the way they operate in order to try to reconstruct how they are put together. The same is true for the mind and the computational benefits of consciousness more broadly.

The Cost of a State of Consciousness

Another important consideration would be cost assignment for different states of consciousness. I imagine that the going rates for participants on various states would highly depend on the kind of application and profitability of these states. The price would reach a stable point that balances the usability of a state of consciousness for various tasks (demand) and its overall supply.

For problem solving in some specialized applications, for example, I could imagine “mathematician on DMT” to be a high-end sort of state of consciousness priced very highly. For example, foundational consciousness research and phenomenological studies might find such participants to be extremely valuable, as they might be helpful analyzing novel mathematical ideas and using their mathematical expertise to describe the structure of such experiences (cf. Hyperbolic Geometry of DMT Experiences).

Unfortunately, if the demand for high-end rational psychonauts never truly picks up, one might expect that people who could become professional rational psychonauts will instead work for Google or Facebook or some other high-paying company. More so, due to Lemon Markets people who do insist on hiring rational psychonauts will most likely be disappointed. Sasha Shulgin and his successors will probably only participate in such markets if the rewards are high enough to justify using their precious time on novel alien states of consciousness to do your experiment rather than theirs.

In the ideal case this type of platform might function as a spring-board to generate a critical mass of active rational psychonauts who could do each other’s experiments and replicate the results of underground researchers.

Quality Metrics

Accurately matching the task with the state of consciousness would be critical. For example, you might not necessarily want someone who is high on a large dose of acid to take a look at your tax returns*. Perhaps for mundane tasks one would want people who are on states of optimal arousal (e.g. modafinil). As mentioned earlier, a system that identifies the most promising states of consciousness for your task would be a key feature of the platform.

If we draw inspiration from the original service, we could try to make an analogous system to “Mechanical Turk Masters“. Here the service charges a higher price for requesting people who have been vetted as workers who produce high quality output. To be a Master one needs to have a high task-approval rating and have completed an absurd number of them. Perhaps top score boards and public requester prices for best work would go a long way in keeping the quality of psychedelic workers at a high level.

In practice, given the population base of people who would use this service, I would predict that to a large extent the most successful tasks in terms of engagement from the user-base will be those that have nerd-sniping qualities.** That is, make tasks that are especially fun to complete on psychedelics (and other altered states) and you would most likely get a lot of high quality work. In turn, this platform would generate the best outcomes when the tasks submitted are both fun and useful (hence benefiting both workers and requesters alike).

Keeping Consciousness Useful

Finally, we think that this kind of platform would have a lot of long-term positive externalities. In particular, making a wider range of states of consciousness economically useful goes in the general direction of keeping consciousness relevant in the future. In the absence of selection pressures that make consciousness economically useful (and hence useful to stay alive and reproduce), we can anticipate a possible drift from consciousness being somewhat in control (for now) to a point where only pure replicators matter.


Bonus content

If you are concerned with social power in a post-apocalyptic landscape, it is important that you figure out a way to induce psychedelic experiences in such a way that they cannot easily be used as weapons. E.g. it would be key to only have physiologically safe (e.g. not MDMA) and low-potency (e.g. not LSD) materials in a Mad Max scenario. For the love of God, please avoid stockpiling compounds that are both potent and physiologically dangerous (e.g. NBOMes) in your nuclear bunker! Perhaps high-potency materials could still work out if they are blended in hard-to-separate ways with fillers, but why risk it? I assume that becoming a cult leader would not be very hard if one were the only person who can procure reliable mystical experiences for people living in most post-apocalyptic scenarios. For best results make sure that the cause of the post-apocalyptic state of the world is a mystery to its inhabitants, such as in the documentary Gurren Lagann, and the historical monographs written by Philip K. Dick.


*With notable exceptions. For example, some regular cannabis users do seem to concentrate better while on manageable amounts of THC, and if the best tax attorney in your vicinity willing to do your taxes is in this predicament, I’d suggest you don’t worry too much about her highness.

**If I were a philosopher of science I would try to contribute a theory for scientific development based on nerd-sniping. Basically, how science develops is by the dynamic way in which scientists at all points are following the nerd-sniping gradient. Scientists are typically people who have their curiosity lever all the way to the top. It’s not so much that they choose their topics strategically or at random. It’s not so much a decision as it is a compulsion. Hence, the sociological implementation of science involves a collective gradient ascent towards whatever is nerd-sniping given the current knowledge. In turn, the generated knowledge from the intense focus on some area modifies what is known and changes the nerd-sniping landscape, and science moves on to other topics.

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):

n-light-objects-header

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”:

dodecahedra_hyperbolic

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.

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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…