Generated by Midjourney

How Many Dimensions Can You Handle?

Or how to improve your simulation by constraining your generation

By MARTIN REZNY

So, yeah, I’ve spent a lot of time playing Minecraft lately. While listening to wonderful world and AI news. Which led me to the obvious question:

Why is playing Minecraft so damn satisfying?

Seriously, I’ve been playing computer games my whole life. Tens of thousands of hours by now. Nothing touches Minecraft. Not Doom, not Diablo, not Civilization, not Elder Scrolls. I specifically tried to avoid Minecraft, for years, until I was forced to play it with my friends.

You think I’d be kidding, but when I showed the slightest hint of resistance, a friend of mine, well in their thirties, genuinely upset, scolded me by almost yelling “Stop being so childish and play Minecraft with us!”

The game sliced right through my cube rejection syndrome in minutes, as well as through my general distaste for non-story based multiplayer games. In the following years, I ended up playing Minecraft more than all of my friends combined. So, why is arguably the best game ever cubes?

I think I might have figured it out, architecturally speaking. And it could be a useful insight for the purposes of better AI design in other areas. The key engineering term to consider here is dimensionality reduction.

The “D” Stands for Disappointment

Whenever I used to hear physicists and engineers throw the D word around, I felt like there’s something profound being discussed. That it must be some sort of mind-bending concept, one that mere mortals are ill-equipped to even begin to understand. It also felt very clear, definitive.

Turns out it means “more things”. Vaguely.

In terms of physical dimensions, you start with a point, then you strech points into a line, then lines into planes, then planes into volumes, and then it starts getting difficult to picture. Still, it’s just the last thing you had, but stretched out, which forms a new thing. Let’s go beyond space.

Because that’s what 3D is — space. You could try to think of 4D as space, but more. A hyperspace, if you will. Cube to hypercube, sphere to hypersphere, that sort of thing. But that’s not usually what the fourth dimension is reserved for. For legacy reasons, the fourth dimension is usually time. You take 3D spaces and you stretch them through different moments in time.

You’ve got a spacetime, or what pop culture tends to call timeline.

If you continue adding more timeline-like things, instead of creating progressively more weird shapes folded unto themselves, you get 5D as all of the possible timelines at once, like lines on a flat page. Then you can think of 6D as all of the pages in a book. Then 7D as all of the books in a library. Then 8D as all of the libraries, of a particular type.

What would a page correspond to in reality? Maybe a universe. The book a multiverse. The library a particular family of multiverses, like MCU, DCEU, and all other specifically comic book-based multiverses. Then you could have 8D as some sort of multiplex of all kinds of multiverses. It goes on.

Or you could just stick to making more and more tangled shapes within shapes within shapes. Thinking about extra dimensions of spacetime with emphasis on time feels more useful to me, but maybe that’s just my preference. What I’m trying to say is, hyperdimensionality isn’t profound, despite how technically complicated it can get to calculate or draw.

Anyone who’s ever replayed multiple different computer games has had no issues understanding and navigating at least a 6D spacetime. A single game with any player choices at all is 5D. Playing a bunch of games thus means navigating a 6D phase space. You may not be able to diagram the shape of this type of hyperobject, but you can handle navigating through it.

Admittedly, some of those games might have been 2D spatially, but that brings me back to the vagueness or flexibility of the general concept of dimension. If you think about it, technically, Minecraft is a 3D game. But you don’t get all the 3D shapes, do you. In games, 2-and-a-half-D exists.

I think that both the nuance and practical banality of “dimensions” becomes more clear when you start thinking about them as an engineer. In an engineering sense, an extra dimension just becomes an extra thing to worry about. One more variable. One more column in a spreadsheet. How many columns can you add? As many as you wish, until your PC crashes.

Mathematically or geometrically speaking, the one more column in a spreadsheet can turn into a literal additional axis in a graph, as if it was a dimension of space. You can represent the interaction of different variables with those weird shapes within shapes within shapes that are difficult to graph. LLMs do this, for any variables from “height” to “hate”.

The Effing of the Ineffable

Returning back to the question of how 3D Minecraft really is, I think it’s clearly not all of 3D, shape-wise. How many dimensions is 3D, but not all the shapes? If you think about it further, Minecraft doesn’t have all of the colors, sounds, or quirks of physics of the real 3D world either. It doesn’t even have all of the 3D that computer games are capable of.

By design. That’s where the dimensionality reduction thing comes in. No matter the context, not all the variables you can identify or imagine are useful, or, to use a technical term, “salient”. Some dimensions are just noise, and you want the signal. They’re not helpful to know or factor into your calculations or choices. In science, this is called “parsimony”.

That’s probably why our physical models of the universe don’t factor in “dimensions” like “good/evil” or “happy/sad”. Or any subjective qualia or phenomena whatsoever. Those can’t be usefully calculated. One could argue phenomena feel pretty real, and leaving them out makes our models of the universe woefully reductive and incomplete, but that’s a side issue.

For a particular purpose, like calculating an orbital trajectory of a spaceship, you probably genuinely don’t need to factor in how hangry the astronauts are. But if you’re engineering software, for users, you probably do need to care about how the users may or may not be feeling.

This is where LLMs changed the whole game recently. Using semantic vector mapping, all kinds of extra “dimensions” can now be usefully calculated. Calling LLMs “word calculators” is about right. This includes fuzzy dimensions with no objective definition. No matter their number.

Which is a problem. The number of dimensions that general LLMs are factoring in tends to be way too high. Too high to “parse”, which is a distinct kind of not parsimonious. Dimensionality reduction for AI is about weeding out those “dimensions” or variables that are redundant, meaning that if you ignore them, the useful result of the calculation doesn’t change.

It’s also about making the process of AI reasoning or generating anything more manageable for the programmers and the users. The fewer dimensions you have to calculate, the simpler and faster the calculation becomes. The fewer dimensions the user has to parse, the better user experience they’re going to have. That’s why Minecraft is so satisfying.

In a perfect game, you’d like to be able to have as many choices as possible, while having as few “dimensions” to worry about as possible. Ideally, you’d only like to have to deal with salient dimensions of choice and experience. Only the signal, no noise. In this sense, less becomes literally more.

In Minecraft, you can make infinite things, but only from blocks, plus half-blocks and flat things. Having the ability to make infinite things in infinite shapes from infinite shapes wouldn’t make the game better. It would make most players not know what to make most of the time, or fail to understand how to make anything. The same goes for all of the other game mechanics.

You can have pets and breed animals, but not every single variety of animal, and each animal has simplified behavior. You can make logical circuits and smart machines using redstone, but you only get several specific components to work with. There are different biomes, but each is distinctly different from the others and has consistent rules of generation.

In this way, open world games like Skyrim are similar in terms of their broad variety of constrained features. But the ultimate advantage of Minecraft is that it applies the same approach to the procedural generation of the world as a whole. The reduction of the space into blocks allows for a grammar of world generation that doesn’t repeat, but always makes sense.

The Case for the Purple Pill

Go through the same exact map or story line again, over and over, and eventually, you’ll get bored. Be thrown into a whole new world where nothing is familiar and where too many things are going on for you to be able to make sense of them, and you’ll get lost. There have to be new things to do all the time, but you need to be able to keep up with your choices.

Which is where this becomes enlightening about the nature of all simulations. Arguably, aiming for optimal dimensionality is neither staying in the Matrix, nor leaving it for the “desert of the real”. It’s the purple pill option — living in a Matrix, but one that keeps up with your dynamic level of awakening from the illusion. Turns out “reality” is a gradient.

If every variable in life you could conceivably perceive, understand, or care about is a dimension of reality, then the more things you perceive, understand, or care about, the more dimensions your life has, and the more real your life is. Alas, no one’s ability to handle dimensionality is infinite. Maybe afterlife is too much for you. Or Earth being round.

In this sense, living with a particular worldview that reduces the dimensionality of reality makes one live in a layer of simulation. Living with the program of left-right politics in your head, for example, is like exploring an environment with only the angular shapes. Reality of politics isn’t binary, but such dimensional reduction could be useful or enjoyable.

Or terrible. It all depends on your dimensional processing capacity. A todler on a rollercoaster probably isn’t going to have a great time. But neither is an adult in jail, or a teenager in a retirement home. In the simulation metaphor, each new dimension added is basically a new game mechanic. Which variables and gameplay modes are absent from your life?

Personally, I don’t want to drive, or have to deal with any of the related variables. For most people on the planet, this is a core mechanic, a crucial set of dimensions of real life. On the other end of the spectrum, I dabble in multiple classical philosophies and arts. Most people have to deal with little to none of that their whole lives. Which life is more “real”? Hard to say.

Specifically, it’s hard to say because math doesn’t tell you which dimensions are more objectively “salient” than others, or how they compare exactly. Mathematicians and physicists prefer to model dimensions of space and time, but that’s a convention, a choice. Marketers and psychologists can and do compute dimensions of space and time, but find them far from foundational. Useful salience depends on use case.

A person’s physical volume or age can correlate with salient life experiences or outcomes, but as far as I can tell, there’s no way in science to convert quanta or coordinates of spacetime into why a poem makes you sad. Or is there a way? Let’s say there may be a way.

Making Your Constraints More Satisfying

It’s not found in science, not exactly, but there has been one attempt at converting a literal spacetime coordinate into all of personality and life experience. Yes, I’m talking about astrology again. But in a different way than before, I promise. Let’s look at astrology as a symbolic grammar designed for putting constraints on the dimensionality of simulations.

What simulations? Maybe simulated conversations with an AI. Or simulated images or videos generated by an AI. Or simulated pieces of music. Or simulated NPCs taking simulated actions in simulated events that are part of simulated histories of simulated societies in simulated worlds. Or layers of reduced dimensionality overlaying the real world.

But first, let’s bring out some relevant constraints from Minecraft’s rules for the generation of environmental biomes, which are put on top of perlin noise algorithm that creates smooth, natural-looking, fractal-like terrain:

• There are only about 60–90 distinct biomes.
• They are distributed following temperature and humidity gradients (combinations of hot/cold and dry/wet spots).
• Each biome’s terrain can only be generated using specific types of blocks at different ranges of elevation, and populated with specific objects or groupings of objects in locations that meet relative criteria.

There’s a lot more to it, I imagine, but this will suffice as an example. Thanks to constraints like these, the randomly generated terrain might always be unique, but the player is never lost.

By looking around, they can identify the biome they’re in, as there aren’t too many biomes to remember and they’re distinct enough. The player always has a general idea what they might expect to find, at what heights or distances, or beyond the biome they’re in. And yet, surprise is a constant.

Astrology is also a constrained generative grammar, but for different types of “terrain” and different classes of “objects”. The main reason this might not be obvious to modern engineers is that until recently, we could only run it using human brains. Which are, engineering-wise, neural networks. Turns out astrology can be run on artificial neural networks as well.

Imagine you want to generate an NPC for your game. You ask an LLM to generate one. How would you put constraints on the generation process? You could try to freestyle it, but then the results will vary and not scale. I don’t think it’s controversial to say you should use some kind of constraint system. At present, you could either go scientific, or old school gamer.

If you go scientific, you could use any modern personality typology. You could say, for example, that you’d like the character to be INTJ, or Introverted Intuitive Thinking Judging-type person from Myers-Briggs Type Indicator. Or you could set scalar values for each of the Big 5 traits. That would narrow down the outputs, and the AI will understand you because there’s quite a bit of relevant documentation in its training data on this.

But how much more specific would that make your character, really? How close exactly to any specific idea, or functional story need, you might have? Not a lot. There are only 16 MBTI types, or something like 4 dimensions, and Big 5 is literally just five dimensions to a person. I think it’s safe to say there are more than just 4 or 5 dimensions to all that could be depicted character-wise in all of myth, literature, cinema, and gaming.

What you want aren’t few overly broad (barely constrained) types. You want the opposite — near infinity of highly constrained types. Let’s look at how the old school gamer approach would fare. Yes, I’m talking about DnD, and any variations and spinoffs thereof. Following this methodology, you could specify constraints on character generation such as:

• Character race and gender as background (let’s say 20 combinations)
• Character alignment as ethics (Lawful/Neutral/Chaotic combined with Good/Neutral/Evil)
• Character class as an archetype (around a dozen of classes from artificer to warlock)
• Character level as a developmental stage (about 4–6 stages from novice to master)

Now we’re somewhere in the thousands of combinations. That’s much more useful, particularly because unlike modern scientific typologies of personality, DnD doesn’t pretend that ethics don’t exist, or character growth. The danger here is different than uselessness. It’s the product of the generation being too formulaic — it’s not enough dimensionality.

At full specification, there’s not actually that much meaningful difference between a chaotic good elven rogue novice and a lawful evil dwarven bard master. It could be thought of as the same hero at different points in the same life. Minus the race or gender thing, mostly. You could add subclasses to the classes, which helps, but that’s about where this system tops out.

And I’m not hating on DnD, just like I’m not actually hating on theoretical physics or clinical psychology. But DnD as a system or grammar, the state of the art for character creation, was designed to be run by a single human mind in the conditions of real-time improvisation. Astrology was also designed to be run by human minds for a similar purpose, but as many human minds as you could spare, taking all the time in the world. It scales.

For example, let’s apply astrology to refine insufficient DnD constraints. When you say good, neutral, or evil, what do you mean? In the astrological model, you can stay archetypal and virtue-philosophical, but add as much situational and developmental nuance as you’d like, systematically:

• There are subsets of virtues and vices (aries good is different from scorpio good, as libra bad is different from taurus bad, etc.).
• Each type of vice is in fact a specific debilitated or misapplied type of virtue.
• Misapplication of virtue or vice types is determined by timing or domain context rules (in libra times or libra places or libra relationships, aries good may be situationally bad, etc.).

I could go on, but I’d like you to notice something important — I’m putting more and more refined constraints on the concepts of good and bad, but I’m not really invoking new symbols to do it. In astrology, the twelve signs are universal semantic constraints that can be applied in any context. They can also be subdivided or stringed together using only a few rules.

Taken all together, you have 12 signs (essential semantic vectors), 10 planets (components of identity), 12 houses (local contexts), and about 6 aspects (relationships). You could add some extra things, or fine tune the grammar for any particular purpose, but that’s it. And yet, there are enough operations you can run in this grammar to generate billions upon billions of unique “types” of anything. And you’ll never get lost doing it. AI won’t.

You start from however many dimensions you can handle, and then put on just enough layers of dimensional constraints on your problem. Do you want to have 12 distinct characters in your story or game? Signs, or planets, will be enough. You could pick if you want a complete set, or 12 variations on the same theme, or just the introverted half of the spectrum twice.

Or maybe you’d like a matrix of hundreds, or thousands, or quintillions. Makes no difference. Just like you can decide to generate particular blocks only above or below a particular height in Minecraft, you can astrologically constrain character generation so that you create any particular balance or imbalance of any possible personal traits in the generated population. All you need is a few symbols, a few rules, and a seed set of coordinates.

Disagree with any particularities of traditional astrological semantic geometry or mapping? No problem. Feel free to tinker with the primitives or relational rules, or run your own data-driven re-mapping project. Astrology is in fact just one possible application of a higher class of systems, numerology. The Wizards of the Coast had made their own unique color wheel system following similar (ancient) principles around 5-ness.

Imagine how great of a game you could get if you had Minecraft-level generation capability for not just terrain, but for character and narrative. Infinite variation, with optimally manageable dimensionality. Manageable fully for both developers and players. Better yet, LLMs already understand this language, or this whole class of languages. Human engineers, less so.

Alternatively, instead of imagining it, you could help me engineer this. This is just the highest level overview. I’m already about five or six steps beyond this initial realization. If you can also see the potential, feel free to DM me.