Abstraction Design

The Introduction to a talk I gave recently...


Creating an abstraction means to take a phenomenon or concept and reduce it to a simpler version by ignoring certain factors. We can then use an abstraction to predict things about the more complex phenomenon or concept.
For example, we can predict the motions of entire planets just using a few simple equations, despite the fact that planets are extremely complicated, by reducing planets to points with mass.

When building abstraction with the goal of predicting something the goal is to simplify as much as possible, but not too much so that we don't lose the ability to predict what we want to predict. An abstractions that is more detailed (ignores less effects) can potentially predict more about the world, but at the cost of being more complicated.
In some sense that is what we have evolved to do as humans, is to learn to ignore the parts of the chaos of our universe that are not necessary for our survival.


If we take a model/abstraction and look at a particular case, we get a prediction.
If we are doing a thought experiment we can imagine what the future state might be.
Or if we have an equation that models some part of the world we can plug in some numbers into it and it will output predictions of what the state of the object will be in the future, for example.
But it recently got very interesting because computers have made this step much easier to do.

For example to produce images of a 3D scene a computer uses abstractions of how light bounces around in the world before it reaches our eyes to produce the images we see, and computes this 30 or 60 times per second.
In some way, by instancing abstractions computers turn them back into “realities” because they can feel surprising, interactive, etc...


There is another thing we can do an abstraction. We can take any abstraction and generalize it without concern for whether it has a physical representation in our universe. You could say this is what pure mathematics is about.
For example, we can add a fourth number to space (x,y,z,w) and see what that would be like.

Any game does this and in a sense games are a natural part of a process to understand the universe, a universe in a more general sense including its generalizations.
For example, Sim City is based on a model (i.e. is an instance of an abstraction) of a city created by Jay Wright Forrester. Adding Godzilla to Sim City is a generalization.

Designing Abstractions

When you look at the world to design an abstraction, ideally if you are a physicists the abstraction might contain surprising results because it might encode more about the world than what you based in on when you designed it.

For example one of the things that made Einstein famous is that his theory of relativity predicted that light’s path would get bent by heavy objects, and in 1919 during a solar eclipse, it was confirmed by looking at starlight as stars passed behind the sun.

When you are making a Video Game, you have to decide what the abstraction level for each component is going to be.

The highest abstraction (i.e. the less detailed) might be “if statements,” for example: if (main character does this) then { react like this } else { react like that }.

For more detail we could use an equation for example to compute the path of a projectile. The equation might become more detailed: we could consider air friction, etc...

An equation might return something that the designer of the equation didn't expect. That's interesting because it means that systems that are at a more detailed level of abstraction are more interesting to explore since the designers themselves more often do not know everything that could happen with them. This effect is what I care about a lot as a designer.

If statements are essentially step functions, which also create discontinuities that are often a source of problems in models. It is possible to combine lots of step functions to make a smooth-looking curve, but it takes a lot more effort.

By the way, I am not saying that all if statements are bad, just that as a designer we chose to focus on a region to explore, and where we explore it is better to use more detailed abstractions. For example, in Miegakure there are characters that walk around using simple state machines. I choose to not explore their internal state very much, as just their change in position is interesting enough with respect to the core of the game.

Almost done with puzzles! Counting Levels of each type/main mechanic

I am nearly done with making puzzles for the game! I might still add a couple puzzles here and there but the bulk of the work is done.

I was curious about how many levels of each type (i.e. main mechanic) there were so I made this (approximate) chart:

The colors indicate main mechanics and the subcolors indicate relative difficulty within a mechanic. (Some levels use more than one mechanic, however...). The surprising thing for me is the large "blue" section, which is one fourth of the game, and which I added almost as an afterthought and didn't anticipate would be so rich, while some other sections didn't end up as relatively interesting as I thought they would be.

I am very happy with the result as every single one of these levels contains a unique idea. Luckily the 4D mechanic is extremely rich so I didn't have to look much higher than the low-hanging fruits.

It seems like I almost doubled the number of puzzles in the game these past seven months? The initial prototype had about 30 puzzles, which defined the basic rules and most of the mechanics. Then I fleshed out the tutorial section and added two main mechanics: two years later I had about twice that amount. Then I mostly stopped making puzzles (even though all the mechanics were set) to focus on improving the graphics (which in turn improved the gameplay), but I added a few puzzles here and there, and I had about 80. This final push increased that to around 140: I added puzzles to all the mechanics to fill-in blind spots and make the difficulty ramps smoother, then made most of the puzzles for two main mechanics and replaced a small mechanic that had been bothering me for a long time with one that is much cooler.

Now I will do a last pass improving various graphical things, while we work on making each of these puzzles and the rest of the game look and sound as good as possible.

Finally, here's a recent screenshot I didn't post on the blog yet:

Miegakure is coming to Steam.


Deep within the Ancient's Grove one can find broken-down stone columns, erected to worship old gods.

I could have officially announced this much sooner, but I wanted to wait until the game was further along. Miegakure will release on Steam when it is done. It will be out for Windows/Mac/Linux.

I have been working on finishing up most of the puzzles for the game. There are about 130 right now, and not that many left to do! I am not shooting for a specific number, but rather exploring all the mechanics in the game.

I am also having fun making levels that are less puzzle-y and more about showing off cool 4D things, like the 120-cell level from the trailer, and the above Spherinder grove. My design philosophy has been that each puzzle in the game should be about a cool consequence of what you can do because you can move in 4D but I realized this extends naturally to more visual consequences as well.

Understanding Miegakure, and the 4D as Parallel Universes.

When looking across worlds the windmill appeared strange, its swift blades moving in and out of sight. I could hear their faint echo ripple through the dry desert air.

There's something very mysterious about a fourth spatial dimension. We can't directly see or touch it. We don't know if it exists, and if so in what form. It is difficult imagine, because our sensory system is built for three dimensions. Furthermore, most people have never tried imagining it at all.

And yet Miegakure can be understood and played by any random puzzle-game player, at PAX for example. How come?

[I messed] about with this strange toy until I quickly understood most of the problems that I faced. - Jim Rossignol (Rock Paper Shotgun)

When I did finally get it, I realized how fantastic Miegakure could be. -Tyler Wilde (PC Gamer)

It's amazing how fluid the transitions between dimensions are, and how much sense it makes once you play. -Chloi Rad (IGN)

For me the main reason might be the fact that Miegakure starts by purposefully framing the fourth dimension from a particular point of view, one that we are very familiar with, that of parallel universes.

Think of a stack of paper, each piece of paper is a 2D square, but together they form a 3D cube. Each piece of paper is literally parallel to the other pieces of paper; they don't intersect. The same thing happens in lower dimensions: we can build a 2D square out of parallel (1D) line segments, or a line out of (0D) dots. This pattern works in any number of dimensions: we can think of a 4D cube as being a stack of parallel 3D cubes. They are stacked along the fourth dimension.

So a 3D world can be seen as a stack of 2D spaces, as is shown in the trailer:

Similarly, the fourth dimension can be seen as literally parallel universes (A 4D world can be seen as a stack of 3D spaces). The fourth dimension is a way to mathematically define parallel universes in a rigorous way.

Zelda A Link to the Past overworld

As a culture, we have been thinking about parallel worlds for a long time. Here's a long list of Parallel Universes in Fiction on Wikipedia, going back to Through the Looking-Glass and The Lion, the Witch and the Wardrobe.

Some parallel universes are completely separate from each other, but some are connected in some way. In games there's the Dark World and the Light World from Zelda A Link to the Past and A Link between worlds. There are also parallel worlds that are the same world but at different times, like Back to the Future, and Chrono Trigger.

Miegakure happily leverages all this experience we have thinking about parallel universes, but extends the concept as contained in the concept of 4D space.

I find it a bit similar to skeuomorph interfaces used previously on the iPhone, where for example the calculator looked like an actual old calculator. “it makes it easier for those familiar with the original device to use the digital emulation by making certain affordances stronger.” [Wikipedia] We can recreate something that people are familiar with, but also extend it, freed from the physical limitations.

In Miegakure, especially at the beginning of the game, to help players understand the game we texture the ground differently at intervals, to group parallel worlds together and visually differentiate them. So the first main thing that needs to be figured out when playing Miegakure is how do the literally parallel worlds (that are a natural consequence of a 4D world) manifest themselves in the game. How do you move between them? How do they relate to each other? Which point in one world is closest to this other point in another?

While any interaction with a video-game is very instinctive (especially at first, and since I chose to make the game teach non-verbally), at a basic level these questions do have relatively simple answers that can be expressed in words, in part because of the vocabulary we have built for parallel universes.

2D/3D version of the game

Of course, A 4D world is more than an stack of independent 3D worlds, just like a 3D world is more than an stack of independent 2D worlds.. Something deeper is going on, something that takes longer to grasp. Something that players tend to feel but can't express in words.

For example, while these worlds are parallel, they are not necessarily independent. So while each piece of paper in our stack can contain its own 2D world, independent of all the other worlds, this stack is different from a cube, which is a single continuous object. If we still insist on seeing the 3D object from a multiple-2D-worlds perspective, we can say that the worlds can somehow be connected/attached to each other. They can also rotate together by rotating the whole thing, etc...

Furthermore, a true 3D object might look very complicated and confusing if we only saw it through 2D slices. And so similarly if you look at the shape at the beginning of the 2nd trailer or the end of the first, you can see that it is not made out of layers (parallel worlds). It is a 4D shape called the 120-cell.

My design goal in creating Miegakure is to use the very familiar concept of parallel worlds as a strong foundation for understanding, acknowledging it as a part of the concept of a fourth dimension, but to not limit the game to it. Since the game is properly programmed in 4D if players wish to dig deeper there are plenty of things to discover and try to understand, things that I sometimes don't even fully understand myself.

Miegakure on PS4. New Screenshot. Interviews. Collision Detection.

Windmill Rotated

The windmill appeared strange from this perspective, its swift blades moving in and out of sight.

A few things.

  1. Miegakure is coming to PS4. I wrote a blog post introducing the game over at the Playstation Blog. Interestingly I had never written an introductory post like that. We are showing the game at the Playstation Experience in Las Vegas on December 6th and 7th, 2014. Of course, it will also be on PC/Mac/Linux.
  2. A couple interviews I had not posted on the blog: Chris Suellentrop wrote a piece about me for Wired's December 2014 Issue, which was guest directed by Christopher Nolan. In the same issue there is also a great "XKCD guide to higher dimensions." I also did this interview with Cathlin Sentz on N4G a while ago and this interview on Kill Screen turned out well: Videogame architecture allows us to visualize the impossible:

    To a fourth dimensional person, it’s like, ‘you forgot two walls to your house,’”

  3. Now that the tech has solidified a bit we have been working on lots of new art, which I am exited to show soon. On the code side I have been finishing up smaller things like collision detection with objects that are not the tesseract tiles. It used to be possible to walk inside trees and lanterns, etc...
    You know the game is strange when you can finally add collision detection many years in! For the longest time it was not clear how to embed 3D objects in the 4D space and how to display them, or what a 4D mesh even is, etc... So collision detection came as an afterthought. Note that you can almost never stand stand on these objects, so them having no collision never affected gameplay. Anyway I literally just added that and I am excited to test how it feels at the show, ahah.

Nature as Designer (There was only one way to design Miegakure)

Go board

There are some games that are so simple, so pure, so fundamental that they feel like they were discovered, not invented. Go is a perfect example. Probably Tetris.

If I were to get really good at a game, I feel like such games would be more worthy of my time. It's part of my game design philosophy to try and make games that are discovered, not invented.

When I started working on Miegakure, I had experimented a little bit with making higher-dimensional games and so I knew that the player would be looking along three vectors out of four (these vectors could be oriented any which way in 4D). Why did I chose to let the players see only along three dimensions (taking a 3D slice), and not project the entire four dimensions down to three, then to two for the screen?

First of all, I wanted the 4D world to feel like an extension of the 3D world we live in. What if our world actually had four dimension, but we didn't know it? I love this idea of a mysterious fourth dimension, rumored, but never seen (that's true in the real world too!). And as a player you are the only person you know that is capable of reaching it.

Second, if you use a projection a lot of objects are going to overlap on the screen, objects that you can't actually touch because they are too far away. You could try to solve this problem by coloring objects differently based on where they are along the fourth dimension, but this is unnecessarily difficult to visually parse.

In retrospect, one thing that makes Miegakure special as compared to the few other 4D visualizations that exist is that it lets you touch the 4D objects as if they were real objects, and create entire 4D generalization of our world. This is something that is much harder to do using projections, which is the usual way of representing 4D objects, such as this more commonly seen, confusing-looking, projection of a Tesseract (the 4D equivalent of a cube).

Then there was the question, how should the player be allowed to move along the fourth, perpendicular vector? The obvious way would be to have the player press another couple of buttons to move up or down the 4D.

It quickly became clear that you don't want the player to move blindly along the fourth direction; you want vision and movement to be coupled: if you could move without being able to see where you are going, you would bump into invisible objects, and the whole world would change at each step (In the 2D/3D version of the game shown on the right and in this trailer, if you could side-step the world you see would change at each step).

So the idea of swapping a dimension for the fourth one came about. Inspired by Ikaruga (which is a beautiful shooter where all the complexity is derived only from the ability to switch colors by pressing one button), the simplest thing you can do is to have one button that swaps a dimension for the fourth one back and forth.

I loved the idea that the game plays like a regular platformer, except for this one special button that you press once in a while. Braid is also this way.

If you are only allowed to move along three dimensions, but you can pick which ones they are, then you can move anywhere in 4D. The following question remains: which direction will be swapped for the fourth one? If you name the three dimensions X,Y,Z,W, and decide that gravity will point down Z, then you don't want to swap Z out, because it would look very confusing, and pressing the jump button should probably always move you in Z. So you're left with swapping either X or Y. If doesn't really matter which one we pick, in my case it's Y. For simplicity X is left untouched. It would not be interesting enough to let players swap in the X direction to justify adding that ability. It also means that levels can be made harder or easier by simply rotating them 90 degrees in the XY plane (i.e. swapping X and Y)!
It turns out that a swap can be implemented as a 90 degree rotation, which can be interpolated smoothly (this is what is happening when the world looks like it is deforming).

As you can see, from first principles there was only one way to design Miegakure, and even though I was especially lucky in this case, that was very much something that I was trying to do. The rest was just exploration of this rule set.

But the next problem was: how do you make it so that the interactions are meaningful? 4D space is exponentially harder to fill with meaningful stuff. It takes 102=100 data points to fill a 10x10 grid, 103=1000 data points to fill a 10x10x10 grid, and 104=10000 data points to fill a 10x10x10x10 grid! This means that even if we take a small region of 4D space (10 units in each direction), we need a huge number of things to fill it with.

We want the number of objects to keep track of to be small to help the player hold them in their head. This essentially means that we want our "base" (the number that is raised to the power 4) to be small. This is how building the game out of 4D tiles, some of them pushable, with small levels (4x4x7x4 for example) came about. (Note that other, more detailed objects can always be placed on the tiles).

I think I may have been partly inspired by this puzzle from Braid (probably my favorite in the game!). The entire puzzle can fit on the screen; it's just two doors and a key. It is extremely compressed. Everything extraneous to the puzzle itself has been removed. But it is still interesting and difficult to solve. All the difficulty is in the understanding of the systems at play, such that when you understand them properly the puzzle becomes trivial. (A video of it, spoilers!).

Because the number of objects to keep track of is small, it's possible for the player to hold an entire level in their head. This is very important to me, because that means they are truly thinking in 4D, as opposed to looking at a bunch of 3D spaces one at a time.

At this point I could vaguely picture how to walk through walls using the fourth dimension in my head. I knew that since two dimensions are always visible an entire plane would stay the same after rotating, and therefore objects on this plane would be reference points, and that they would help the players orient themselves.

So I didn’t really know what the game would look like! I especially could not picture how the transition between the two states would look like. But I programmed it and found out and I was the first person to discover how to play the game.

[Part 1] [Part 2] [Part 3] (Part 4) [Part 5] [Part 6]

Miegakure at PAX Prime

We are back from PAX Prime! We were showing as part of the Indie Megabooth. I think the show went very well. People seem really excited about the game. I am glad that the new graphics are making a ton of people curious about what the game is (Not sure why I didn't notice the difference as much at PAX East). They were a lot of work but clearly worth it.


I tried to remember what the most frequently asked questions were, and I plan to write blog posts/ make videos about them. Questions were: is the fourth dimension time? (Nope!) How did you come up with the concept? How is this game even implemented?
If you have more questions that you would like me to answer please let me know (Excluding: when will the game be out?)

We had 4 stations initially but because the wait to play was getting too long, I added a fifth station saturday afternoon. Like last PAX, the main trailer was playing on a big screen, and the explanation trailer was playing on a smaller screen below that.

My favorite moment was witnessing a 9 year old girl play the game better than her father, giving him tips on how to solve the puzzles.

PC Gamer did an awesome preview of the game (they totally got it!).

For the first six or so levels, I did not get Miegakure at all. I was completing the early puzzles, but I had no idea how I was doing it.

I gathered some statistics about most of the playthroughs. It seems the game was played over 400 times, which gives an approximate average play time of around 23 minutes. That's about how long it takes to play the first 13 "intro" levels. Many people played for one or two hours, but I have no trivial way of counting how many.




New Trailer: How to walk through walls using the 4th Dimension

An explanation of how walking through walls would actually look like if you could move in 4D.

This doubles as an explanation of how Miegakure works (finally! Also SPOILER ALERT) and what the fourth dimension is.

Ever since people discovered the concept of a fourth dimension of space around a century and a half ago, they have tried to come up with what would be possible if space was actually four-dimensional. Walking "though" walls would be one of the simplest consequences of being able to move in 4D space. But what would it actually look like? It's not often that watching a video-game trailer actually teaches you about real math.

Of course, if you have watched the video, you know that it is not "through," but rather "around" walls, despite what this movie poster or this comic book character, for example, would lead you to believe.

Additional Notes:

  • From the perspective of a regular 3D observer standing next to the wall the player character would suddenly disappear, and a few moments later reappear on the other side of the wall (assuming the player character is very thin along the fourth dimension).
  • There are infinitely many 3D worlds stacked on top of each other, even if in the Wall level the ground texture makes it seem like there are only two. A more complex example is the shape at the beginning of the trailer, which is a true 4D shape called the 120 Cell, as explained here.
  • Yup, the 2D/3D section is part of the game. There are more 2D levels. Not going to spoil how you get access to them.
  • The 2D/3D section has been part of the game since I first showed it at the Experimental Gameplay Workshop 2009 a couple months after making the prototype. At first I made it in order to explain the game, but it seemed like a good idea to make it part of the game itself.
    I took inspiration from Flatland (of course) and Super Paper Mario for the aesthetic. I couldn't resist having the game contain its own Demake ;)
  • Fun anecdote: the code that handles movement in 2D/3D is the same as the code that handles movement in 3D/4D, but one axis is ignored ;) The display code is of course different, but it reuses a bunch of stuff.

This trailer took a long time to make! It will be the basis for talking about the game more.
The pixel art was done by G.P. Lackey and the music by Disasterpeace & Mateo Lugo. I love how it turned out!

Big thanks to Vi Hart [Youtube Channel], Chris Hecker [SpyParty], Brady Haran and others for their suggestions on how to improve this video.

Oh I forgot to post here that we will be showing the game at PAX Prime, as part of the Indie MEGABOOTH.

Consistency Boundary: What makes a logically and mathematically consistent system?

"The laws of the model's nature have to be logically and mathematically consistent with one another, but not with physics." -J.C.R. Licklider (see previous post)

What makes a logically and mathematically consistent system? It seems to me that every system has a boundary within which it is consistent, and outside the boundary it starts to break down. How do we define this boundary?

The process of describing reality using mathematics is not perfect. First, an abstraction layer needs to be selected. That means we have to decide how detailed a model will be. If the model is too detailed, it will take too much "computation" to predict anything. If the model is too coarse, it might not predict enough effects.

For example, we can simulate the motion of an object by simulating every atom inside of it, but that may be unnecessary, and if the object does not deform very much at the scale we care about we can approximate it by a rigid body, which gives a simpler model that could work very well for our case. But if too much force is applied the object would start to deform or fracture, and then the model breaks down, and might give nonsense results.

In physics, we may know if the assumptions we make are reasonable or not. If we know for sure that in the situation we want to model not enough force will be applied to the object and thus it will never break, then it does not matter that the model does not handle this case.

In games, we create a set or rules for the game that often approximates reality to some level. So already we have chosen an abstraction layer. But we may not know what the consistency boundary of that model is, i.e. find out the places where the approximations we made result in nonsense situations. We may need to discover the shape of this boundary by playing the game itself. Like in physics, we can adjust the rules of the system to increase or decrease the consistent area, but unlike physics we are not bound by having to approximate reality.

For example, imagine a game where a character moves in a 2D grid and the player can place an arrow on a square to redirect it. This model assumes that the character can only move horizontally of vertically within the grid, and so far this is consistent. But what happens if you let players place two arrows on top of each other? If the effects are additive, suddenly diagonal movement needs to be considered, but the model so far has not taken this case into account. This is the edge of the consistency boundary. At this point we need to either disallow the case of diagonal movement and potentially make the game less rich and interesting, or allow the case and grow the consistency boundary, but this may be hard to design as new rules need to be created, and problematic areas may still exist, just further away.

Sometimes, problematic cases can be avoided by level design instead of system design. You can design the levels of the game such that this situation can never occur (ex: what if no more that one arrow is ever given to the player?).

This process of stripping away problematic cases is a lot of what game design is, at least in my experience. There are ways in which it can be done elegantly and inelegantly. Inelegant ways often leave the problematic cases apparent. The player can see the parts that have been cut of, or worse, they are forced to understand details about it. Presumably the system is the interesting part, not its boundary (though games that explicitly explore this idea could be designed and may be interesting).

In the aesthetics of game design, I feel like a game that is very consistent is more beautiful than one that is not.

The thing I talked about before is that in games the goal is not to simulate reality, so we have an extra tool in our hands: generalization. For example if SimCity is a model of a city, then adding Godzilla to it is a generalization. Or taking an FPS (which is a simulation of a person walking in an environment) and adding portals.

This creates additional problems because while we are fairly certain that reality is consistent, we do not know about other, generalized, realities. (If the first part is like physics, this part is a bit like inventing new mathematics).

So for example, it appears to me that Portals are not very consistent. A lot of issues appear pretty quickly, as seen in these drawings I found online:

It might be interesting to try to design Portal in a way that would allow for more consistency, and gets closer to handling cases such as these. I am not sure this is possible.
Sometimes consistency can be improved by fixing things near the boundary, and sometimes the whole system needs to be rethought from first principles.

Another example: What happens in Fez when you get projected behind an object? This is an inconsistent situation that needs to be resolved with additional rules from the designers of the game. It comes from the many-to-one nature of projection.

Miegakure was surprisingly consistent. You can take almost any concept and generalize it to 4D. There are very few consistency problems in gameplay, mainly related to the rules of pushing blocks, which would happen in any number of dimensions. So because there is so much consistency in terms of the 4D generalization, the problem has been finding a good level of abstraction, especially graphically. In some way there are two types of consistency boundaries. The consistency problems that come from abstracting, and the consistency problems that come from exploring. The abstraction consistency boundary seems contained within the generalization consistency boundary.

Sometimes I want to make things clearer or more beautiful at the expense of correctness. Or rendering 4D properly is sometimes too slow (just like 3D graphics are just an approximation of reality!). For example I spent a long time on the extruded 3D trees and it was well worth it. The goal was to get a more detailed level of abstraction of what a four-dimensional universe would look like. But I will talk about it in a later post.

[Part 1] [Part 2] [Part 3] [Part 4] [Part 5] (Part 6)

Looking back (Do you think your conception of reality has changed from making this game?)

(Here's another expanded transcript of part of the talk I gave at NYU)
A question that people ask me once in while is: Do you think your conception of reality has changed from making this game?

I find that one of the things that excite me the most about Miegakure is that because it takes concepts familiar in three dimensions and generalizes them to four dimensions (or even n dimensions) it distills these concepts to a more fundamental core. There is something more truthful about a concept that has been distilled to a simpler form, but it also helps us understand the concept better.

A good example is rotation. A rotation is fundamentally a 2D thing; you need two dimensions to define a plane of rotation. We are so used to thinking about rotations in 3D that we think about the axis of rotation, but this doesn't generalize to nD: the 2D plane that is perpendicular to that axis is what is important. The focus on 3D space actually hides part of the truth. Transforming the plane of rotation into an axis loses some information about its semantics, which causes problems for example when you want to transform the rotation itself (ex: the idea of an axial vector in 3D is a hack). But you can't even transform the plane into an axis in any dimensions but three, so considering the general case helps us avoid making this mistake.

When trying to generalize a concept we often find that some parts of the representation do not generalize at all. Then, we search for a new way to think about things, a new viewpoint that allows us to throw away the parts that don’t generalize. What we are left with is often much more beautiful than what we started with originally, because it describes the same concept, but in a much more concise and precise way.

We can use n-dimensional space as a way to discard viewpoints that are not as fundamental as others because they do not generalize.

Rotating the view

In Miegakure, the player's main action is a rotation (when the space appears to deform, the player is actually just turning their head 90 degrees into the fourth dimension) and there are rotating objects in the game. So the player is learning about rotation in a more general context and it enlightens their concept of rotation in 3D.

I think this idea of distilling and generalizing concepts happens in other games, in some form.

Go board

We celebrate Go because of how rich its situations are compared to how small its rule set is. Go distills the rules of territory and life and death to a form that allows us to explore it, to a form where we can get glimpses of fundamental truths about our universe. Go is so simple that we feel that it has be fundamental.

Frank Lantz talked about similar ideas in this great talk.

This is a slightly different idea:
Portal takes the usual laws of physics and adds portals to them. There's some exploration of what having portals might mean by themselves, but what is especially interesting to me is the way that Portals affect things like momentum.

The idea of using gravity to increase momentum then rotating that momentum to cross a pit makes the player think about momentum in a different way and that reveals things about it. Momentum is seen through the more general lens of portals. There are actually problems with this because it is difficult to cleanly generalize, but I will talk about it more next time.

[Part 1][Part 2][Part 3] [Part 4] <Part 5> [Part 6]