Transliterated from java found here:

class ConvexHull
{
    public static double cross(Point O, Point A, Point B)
    {
        return (A.X - O.X) * (B.Y - O.Y) - (A.Y - O.Y) * (B.X - O.X);
    }

    public static List<Point> GetConvexHull(List<Point> points)
    {
        if (points == null)
            return null;

        if (points.Count() <= 1)
            return points;

        int n = points.Count(), k = 0;
        List<Point> H = new List<Point>(new Point[2 * n]);

        points.Sort((a, b) =>
             a.X == b.X ? a.Y.CompareTo(b.Y) : a.X.CompareTo(b.X));

        // Build lower hull
        for (int i = 0; i < n; ++i)
        {
            while (k >= 2 && cross(H[k - 2], H[k - 1], points[i]) <= 0)
                k--;
            H[k++] = points[i];
        }

        // Build upper hull
        for (int i = n - 2, t = k + 1; i >= 0; i--)
        {
            while (k >= t && cross(H[k - 2], H[k - 1], points[i]) <= 0)
                k--;
            H[k++] = points[i];
        }

        return H.Take(k - 1).ToList();
    }
}

Father Magnus Wenninger died last month.

The short answer is that in the hexagonal case the best analog of Conway’s Game of Life — in my opinion as someone who has been a CA hobbyist for 30 years or so — is an original creation which I will describe for the first time in detail in this blog post. (Note: The links to cellular automata in this post go to a custom web-based player which may not run correctly on mobile devices.)

Regarding a hexagonal Game of Life, a key thing to understand is that Conway Life isn’t just the rule; it is the rule (Life 2333) running on a square grid with an 8-cell neighborhood i.e. the neighborhood of four squares directly adjacent to a square plus those diagonally adjacent. You can, of course, apply the same rule on a hexagonal grid using the natural six hexagon neighborhood but what you will get won’t look anything like Life. It will look like this: Life 2333 on a hexagon grid.

So if the above does not qualify as a hexagonal Game of Life then what would? Well, at the very least we need a glider. Carter Bays, a professor at the University of South Carolina, presented a Conway-like rule that admits a glider on the hexagonal lattice in 2005, Life 3,5/2 in his notation. However, by Bays’ own admission “this rule is not as rich as Conway’s Life” and indeed when we run Bays’ Hex Life on random input we do not see gliders: Life 3,5/2 on random input. The problem is that its glider is too big to occur randomly. Its glider is in a sense artificial. Part of the beauty of Conway Life is that gliders are frequently spontaneously generated. The other thing you’ll notice about Bays’ Life 3,5/2 is that it descends into still-lifes and oscillators too quickly. Conway Life is dynamic. It sprawls and grows, descends into bounded chaos, before finally decaying into still-lifes and oscillators.

To summarize, we want a hexagonal cellular automaton that

1. Has a glider that is frequently generated by random initial input.
2. Frequently exhibits bounded growth from random initial input.

It is my contention that there is no simple totalistic rule over two states and the hexagonal grid using the natural 6-cell neighborhood that exhibits both 1. and 2.

In order to achieve what we want, we need to drop one of the constraints. Using my cellular automata breeding application Lifelike, I explored dropping the constraint that the rule must be a simple totalistic rule over two states. What I have come up with is a cellular automaton that uses a simple totalistic rule over three states,  states 0 to 2. One way to view this move is to view the live cells in Conway Life as counters — beans, pennies, whatever — and imagine dropping the constraint that a cell can only contain one counter. Anyway, my rule is as follows:

• Take the sum S of the 6-cell neighborhood where death=0, yellow=1, and red=2.
  • If the cell is currently dead it comes to life as yellow if S is exactly 4.
  • If the cell is yellow it goes to red if S is 1 to 4 inclusive or is exactly 6.
  • If the cell is red it stays red if S is 1 or 2 and goes to yellow if S is 4.
  • Otherwise it is dead in the next generation.

The above, which I call “Joe Life”, is as rich as Conway life: click this link to see it run. It exhibits bounded growth with about the same burn rate as Conway Life and features two gliders, the little fish and the big fish, which are frequently generated spontaneously.

The little fish

The big fish

I concede Conway’s rule is more elegant than Joe Life’s rule but if one thinks about it, Conway Life’s neighborhood is less natural than the six hexagon neighborhood in that it is kind of weird to include the the diagonally adjacent squares. So in my opinion, elegance that Joe Life lacks in its state table it makes up for in its neighborhood.

You can weave an icosahedron from 10 strips of paper — or anyway you can weave a construction that has icosahedral geometry; it is actually more of a snub icosahedron.

From about 11″ strips of printer paper folded lengthwise to have two layers with a little overlap to lock into rings, as pictured below, locks well and is rigid:

Construction follows the pattern implied by the following:

Here is what the game is looking like currently (click for fullscreen):

So, looking at the early entries of this blog, I must have started working on Syzygy around the beginning of the year 2012 because by March 2012 I had the Win32 prototype done. At that time, I didn’t own a Macintosh, didn’t own an iOS device, had never heard of cocos2d-x, and, professionally-wise, was still writing image processing code for Charles River Labs / SPC. Since then SPC was killed, and I moved from Seattle to Los Angeles … but anyway as of today, about a year later, I have the primary functionality of Syzygy running on my iPad, re-using the source code, mostly, from that prototype. I haven’t really been working on it the whole time — there was a lot of moving-to-California in there somewhere, but here’s a screenshot (click for full-size):

There’s a common question in the mobile games forum of gamedev.net, “How can I make a game for iOS only using Windows?”. The answer to this is either (1) you can’t or (2) write your game to Marmalade or cocos2d-x on Windows and then when you are done get a friend with a Mac to let you register as an Apple developer, build under Xcode, and submit to the App store. I always say (1) is the serious answer and if you are unserious, or want to develop a really simple game, then go with (2). Basically I say this because you need to run your game on a device frequently and early, and I’m seeing the truth to this now.

Now that I have Syzygy running on a device I’m seeing issues with input which are artifacts of running on an iPad. The prototype implemented mouse input as a stand-in for touch input. It turns out touch screens and mice aren’t the same thing. The game plays on the device, but when you drag tiles your finger is in the way of the tile visually. You can’t see the tile you are dragging — this seems like it wouldn’t be a big deal, but it kind of is. … This sort of thing is the reason, in my opinion, that if you are not testing on a device during primary development then you are not really serious…

So not sure what I’m going to do about this, I’m thinking of making the tile the user is dragging larger and offset to the upper-left while the user is dragging it. The problem with this is to make it look nice I’d have to have large versions of all the relevant art and some of it I don’t even really remember how I rendered in the first place…

Our picture round from trivia at the Lockspot this evening … Famous cats.
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