Goombas as the antiparticles of mushrooms: New games for new physics

Games use physics. Since physics is all about how stuff interacts with other stuff, they can hardly avoid it.

Sometimes they use real physics. Often it is reality inspired, but optimized for fun. And sometimes they just make stuff up. Here we are going to do a bit of each.

We are going to look at the crazy kind of physics that is possible in 2D universes. It’s something that has gotten physicists pretty excited over the last decade, because we can actually make these universes using some strange new materials. We can play with the particles of these universes, which are called anyons, and make use of their exotic properties.

A particle/antiparticle pair

Apart from a few simple puzzle games that I made, this physics has not yet made the jump over to gaming. It’s time for that to end. It’s time to let game designers run wild with anyons. And give us scientists a new perspective on what they can do.

But we won’t constrain ourselves too much with reality. I won’t tell you about about all the mathematical consistency needed to see this stuff in the lab. Instead I’ll just focus on the essential ingredients. Then it’s over to you, to optimize for fun.

2D means 2D

The characters in our games are particles. Like balls, people, crickets, bats, cricket bats, monsters, tennis rackets and cat-like things that are cute by day and dangerous by night.

Parallel universes are fine.

The stage that they play on is 2D. That doesn’t just mean ‘Link to the Past’ rather than ‘Ocarina of Time’. Even in the NES and SNES days there were often mountains to explore with crisscrossing caverns, which told us that Hyrule was 3D after all.

We must instead be very constrained to the rigid two dimensions of Snake, Pac-Man or, as I expounded upon in a previous article, Oh Mummy.

You don’t get owt for nowt

Games are often perfectly happy to have things disappear. It’s often even part of the game mechanics. When Link smashes a pot, the shards of terracotta disappear instantly. And whatever goodies come out need to be collected quickly before they too pop out of existence.

For anyon games, this is not allowed. We are going to have to set a rule or three.

A particle cannot appear from nothingness unless accompanied by its antiparticle. A particle cannot disappear into nothingness unless combined by its antiparticle. If you have a particle in an area, and nothing else enters or leaves that area, it cannot change type.

Rule 3 basically means that a particle can’t change type unless it absorbs or emits something. But actually there is also weird thing that change it too.

This weird thing is something that only happens in 2D physics, and so our 3D English doesn’t really have a snappy way to explain it here. So we’ll spend a section on describing it later. For now, we’ll look at absorbing and emitting from the perspective of an Italian plumber.

Putting together and taking apart

Suppose we are playing Super Mario Bros. We can think of little Mario as one type of particle and big Mario as another. There is no way to turn from little to big out of nowhere. Instead you must absorb another particle: the mushroom. So this mechanic fits with our rules.

The transition back from big to small is not so anyonic, though. This happens when Mario touches any enemy. But touching things means nothing in anyonic game mechanics. To change they must not just touch, but absorb. If Mario absorbed a goomba when touching it, just as he absorbs a mushroom, then this would be fine.

In fact, it would be as if the goomba was the antiparticle of the mushroom. This makes sense: They have the opposite effects for Mario, so as far as he is concerned, they do act as the antiparticles of each other. If he absorbs one, something happens to him. If he then absorbs the other, it unhappens.

According to rule 2, since goombas and mushrooms are antiparticles they are able to annihilate each other. If they combine, they pull each other out of existence.

But their relationship isn’t completely destructive. Rule 1 tells us that they can team up to come into existence together. You can’t get something from nothing in an anyon game, but you can get something and its anti-something. As long as they pop into existence together, they maintain the balance of the force.

The trouble is, koopas have exactly the same effect, making them an antiparticle of the mushroom too. But this will cause trouble. So much so that we need a new rule.

4. The antiparticle of a particle must be unique.

For each particle, only one other can be called its antiparticle. Only the goomba or the koopa may undo the work of a mushroom. Not both.

Actually, this new rule is just a consequence of rule 3. If both the goomba and the koopa were antiparticles of the mushroom, it would be possible to turn a koopa into a goomba.

Suppose we have a koopa in a box. Then a goomba/mushroom pair pops into existence next to it. Since this is a particle antiparticle pair that would definitely annihilate if we combined them, as a whole it counts as nothing. So nothing has entered the box.

The mushroom and its only true antiparticle.

If the koopa is also an antiparticle of the mushroom, we can annihilate them. We would be left with just a goomba, who has assumed the koopa’s place in the universe. Rule 3 is broken

This is not the only thing to be careful of when making an anyonic game. When you decide how particles can combine to make other particles, you might decide to make it a bit random. But you must do so with care.

For example, let’s make it so there are two possibilities when a goomba absorbs a mushroom: Either they annihilate each other, or they combine to make a super goomba.

Now let’s introduce a new rule.

5. The rules for absorption and emission must be the same.

This means that whatever absorbing can do, emission can undo. And vice-versa. This is actually a consequence of rules 1 and 2, but I’ll leave that as an exercise to the reader.*

In our case, the fact that a goomba and mushroom can annihilate means that they can also be created as a pair from nothingness. And the fact that they could also form a super goomba means that a super goomba could emit a mushroom to become a normal goomba again.

So we can start from nothing, or from a super goomba, and end up with a goomba and a mushroom. Knowing this, consider the first parable of the super goomba.

In the beginning there was nothing. Then the random number generator said “Let there be a goomba and a mushroom!” And the game designer saw that it was good. Then the goomba and mushroom combined, and the random number generator said “Let there be a super goomba, for that’s allowed to happen too!” When once there was nothing, there was now a super goomba. Rule 1 had been violated. And the game designer saw that it was a bug.

The moral of the story is that the goomba and mushroom came from nothing, and so have to somehow remember to return to nothing if combined. Otherwise, rule 1 is not happy. And neither is rule 3.

Now for the second parable of the super goomba.

Then the random number generator turned its attention to a super goomba who walked the earth, and said “Let that super goomba decay into a normal goomba and a mushroom!” And then declared “Let them now combine and annihilate to nothing! For are they not each other’s antiparticle?” When once there was something, there now was nothing. Rule 2 had been violated. And the game designer saw that it was also a bug.

Similar story, with a similar moral. The goomba and mushroom again need to remember where they came from.

So, by all means do things randomly. But don’t let your random number generator get out of control. Sometimes the result of an absorption cannot be chosen freely, lest the magic circle of our rules get broken.

Going round in circles

In my previous article, I told the totally true story of how anyon physics was inspired by the 80s Pac-Man clone, Oh Mummy. There we discovered the basic power that anyonic give us. When one moves all the way around another, stuff happens. Now it’s time to define that stuff a bit better.

In this Oh Mummy screenshot, we see a few examples of where our brave explorer has walked around pairs of blocks, butreceived no information about their contents. This is not entirely anyonic, since a double loop like this should work the same way as going around each individually. I’m sure the game designers spent many sleepless nights on this issue.

One thing it can’t do is change the type of either particle involved. Both the circumambulator and circumambulatee remain the same particles before as they ever were.

What it can change is their relationships with others. For example, suppose we had two goomba/mushroom pairs that had both appeared from nothing. Recombining each pair in the same way as they entered the world, will cause them to leave it again. Ashes to ashes, dust to dust.

Do you prefer space-time diagrams to text? Then here you are, drawn with my own incompetent hands.

But suppose one of the goombas decides to go for a walk around the other, before submitting to the recombination. Then, when he combines with his mushroom, they might not annihilate. They might instead take the other option and form a super goomba! And if they do, the same thing will happen with the other goomba/mushroom pair.

So we started with nothing, and now we have two super goombas. According to rule 1, this means that super goombas must be the antiparticles of other super goombas, so that the pair can annihilate back to nothingness. So they’d better avoid each other!

Other than that, the rules have no problem with this process. It is the ‘weird thing’ mentioned before. It is called braiding, and it’s the secret power of anyons.

For example…

I have made some anyon games already. They are very simple puzzle games, and have none of the exciting braiding stuff. But nevertheless, they can serve as non-Mario examples to all the stuff I’ve been talking about. You can find out more about the flagship game, Decodoku, here.

The purple and green pairs are just plain old particle/antiparticle pairs. For the blue ones at the top we see 4 that decayed into a couple of 2’s. The red particles had even more messy stuff going on to get them in that state.

Decodoku comes in two flavours, called Z10 and Φ-Λ. In Z10 there are nine different types of particle, which simply take the form of the numbers from 1 to 9. Particle/antiparticle pairs always add up to 10. So 1 and 9 are each other’s antiparticles, as are 2 and 8 and so on. The 5 particles are the antiparticles of each other. Combining two particles, let’s call them a and b, gives a particle of type (a+b)%10. So they simply add up modulo 10.

Since these are pretty simple, with no randomness, ensuring that they obey the rules is pretty easy.

In Φ-Λ there are only two types of particle. Unsurprisingly, they are called Φ and Φ. The Λ particle is its own antiparticle. Combining two of these will always cause them to annihilate.

Here is exactly the same situation as above, but Φ-Λ style.

The Φ particle is more complex. They are also each others antiparticles, but combining them doesn’t always cause annihilation. Sometimes they will form a single Λ. Sometimes two Φs even combine to form a single Φ!

Because of the randomness, keeping track of the possibilities to ensure the rules are obeyed is a bit more tricky. But we take an easy route. Φ-Λ games are actually exactly the same as Z10 games, except that each 5 is replaced by a Λ. And everything else by a Φ. So if two Φs happen to be 2 and 8, they annihilate. For 2 and 3 they become a Λ. But 2 and 1 become a single Φ.

As I mentioned earlier, these games don’t have any nice braiding. For Z10 it is not really possible because there isn’t multiple possibilities when combining particles. For Φ-Λ we could have done it. For example, when one Φ encircles another, we could add 5 to both of their underlying numbers. The braiding would then cause both particles to absorb a Λ, which can then pop out later when things are combined. This process doesn’t change either one from being a Φ. It just affects what’ll happen when combined with other stuff. So the rules are obeyed.

A final rule

I could keep adding rules for ages, to ensure that the anyon physics of your games is exactly the same as the anyon physics of exotic topological materials in the real world. But that won’t do anyone any good. Instead, I will leave you with just one more rule.

6. Anything that clockwise braiding does should be undone by anticlockwsise braiding. And vice-versa.

This ensures that there is always a way to get out of any tangled mess that your particles might get into. If only our universe had this rule for us too!

Now it’s your turn

Now you know the basic rules of the 2D anyonic universes that physicists love. Now you can make games based on these weird little particles. And those games might tell us something about anyons that we didn’t realize before. Or they might just be fun. Either way, it’s a win.

* This is maths language for either “‘I can’t be bothered to tell you”, “I’ve never actually worked it out for myself, but I’m pretty sure it’s true”, or both. Usually both.

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