The benefits of naturality
The other day I had something quite special and educational happen that reaffirmed a feeling that has been at the back of my mind for a while. My feeling can be summarized as: "in art and in life, we should try to ask ourselves what the most mathematically natural course of action is, and see what benefits it gives us."
So, I was working on a pattern for a friend of mine. This was meant to be a mathematical pattern with a Steampunk theme. Here's how far I'd gotten:
I liked this. It took a fair amount of maths to get this nice tight, rigid look. And at this point I wanted to add colours. So how should I colour the gears? Well, what struck me as a rather beautiful/"natural" colouring would be one in which no two same-coloured gears are side-by-side. This makes sense right? It's like that Futurama skit where Kif orders Fry to "MIX THESE 'MIXED NUTS'! I see two almonds touching!"
So I colour one gold, colour the one next to it silver, and I see that that means there are is now a cog touching both the gold and the silver one, so that should be bronze... as should the one on the other side of the gold-silver pair... and now we kinda have a bronze-silver pair, and the cogs on either side of that pair should be gold, etc...
So now I have this, and it looks kinda nice. I sit back and admire my handiwork (the colours are badly chosen, but this was always going to be the case). One thing I had previously wondered doing this gear pattern is "is it possible for them to turn?" If I had arranged them in squares, then it would have been possible, but not this way. Hmm, perhaps they can turn if I remove a cog somewhere? Then ah, yes, if I was just looking at one "hexagon" of cogs, then removing the middle cog would allow the others to move and... hmmmm...
I realize that if you remove all the cogs of one colour, then the cogs that remain are able to move. But I decided on this colour scheme, and I didn't colour them with that in mind. It goes a little bit further, even: a friend of mine pointed out that with a gold cog turning clockwise in this picture, all other gold cogs would turn clockwise, and all silver cogs would turn anti-clockwise.
I believe this was not just a fortunate coincidence for me. This was a case of my aiming toward a mathematically natural space, and getting a cute benefit. This happens a lot, so it is something worth trying to consciously exploit.
It is hard to define what I mean by "mathematically natural" - it is an intuition you have to build up. But I might roughly define it as: "embodying some shape or pattern that is simple and rigid - although not completely boring".
I might offer the Hearst Tower, one of my favourite buildings, as a brief illustration. The triangular geometry is very simple - and confers the benefit of being able to support the structure using a minimal amount of steel. In programming you also often find that the most mathematically natural algorithm offers great efficiency.
Mathematical naturality in game design
So first let's look at Gears of War, a game about cover. The fundamental interaction is: you're behind cover, your enemy is behind cover, and you're both popping up and down to shoot one another. Also you're thinking about where you can go such that you have more cover, but your enemy has less. Many of the maps are concerned purely, then, with dots that move around behind lines (chest-high walls).
(There are doorways and pillars and higher-ground parts and things that mix things up a bit, but more often it is just the chest-high walls!)
And there are cool things that you can do with only lines at your disposal. What I think is a very mathematically natural piece of level design in the game is this part, where we just take one line and bend it into a circle. You see the coolness of this when you consider the cross-section below.
Here the green blob, our crouched avatar, is invulnerable to the red blob on the left but is a complete sitting duck for the one on the right. Bear in mind that this is stretched around in a circle, so if you were to move around the inside of the wall you'd swim in and out of the lines of sight of the many red blobs looking into it. Deciding whereabouts you want to be on the inside of the fountain wall is quite a strategically sophisticated consideration, then.
Bullet hell shmups often gravitate towards geometrically natural bullet patterns. I am inclined to think that this is because it allows patterns to be both rather complex and rather predictable.
Having said all this, there is an extreme I warn you not to go to. Alan Hazelden (These Robotic Hearts of Mine, Sokobond, above) told me that he designs levels by arranging objects in a geometrically nice way, then playing the resulting level. If he dislikes it he throws it away, if he likes it he keeps it.
Edit: I got Alan wrong, there is more to his process than this, and should have inquired with him. He DOES bear in mind this sentiment, which I put forward in opposition to him:
Exploration of mathematically natural setups can help you find something good, but that doesn't mean your job is done. You have to use it in combination with playtesting and streamlining of puzzles - that'll allow you to express the cool puzzle-tactic you discovered with conciseness and clarity.
Naturality as a way of life
"Naturalness", of course, has vastly important meanings in the context of politics and agriculture which should be acknowledged. We've probably all encountered some socially conservative spunkwad who says that gays are bad, and might add that women are inferior to men - and they argue for their position by appealing to "nature". There are a few ways I've seen people argue against this:
1."Actually it increasingly seems that homosexuality is quite natural."
2."Well, the benefits of naturalness are not sufficient to deny people the happiness"
3."You are mistaken to believe that there's anything good about natural things. If we only ever did natural things we couldn't use mobile phones, kidney dialysis machines, etc."
I see argument number 3 used a lot, but it's no good. I sometimes worry that people get into thinking "if I want to be socially progressive, I have to reject the idea of naturalness being good". Well it's not true! Plato wanted to run society in what he saw as a natural way, and he was a proto-feminist homosexual! Just because I think naturalness is an argument for something doesn't mean I believe it is the only argument.
Many people are getting used to the idea that biological naturalness might have become underrated. Permacultural farming, for example, offers benefits that were discovered by people with a commitment to biological naturality (having said that, GM crops are good too! These things are compatible). Medical science has given us a lot of powerful innovations - but prudent medical research often flags up "natural" processes that do things we can't do yet. An example is how the Zebrafish regenerates heart tissue.
If I've sold you on mathematical naturalness and you want to know more, I recommend reading more about the architecture of Norman Foster, and about tuning theory.
Finally I want to add that to me, homosexuality is more mathematically natural than heterosexuality. Gay sex is much more symmetrical - and here again, I can give an example of "mathematically natural setups lead to neat mathematical benefits". Specifically: increasing the number of people in a sexual encounter beyond two works much better if everyone is gay - you can really only have one heterosexual person in a threesome, at most.