This is a very interesting, and very hard problem.

Unfortunately, there isn't any neat solution to turn a set of points into triangles. What exactly would the points represent? One option is the that they would be the corners of the triangles. But how do you assign which three form a triangle? One way would be a Delaunay triangulation though this is not straightforward to compute.

I'm going to consider other ways to get this effect, first.

In my discussion of the solution, I'm going to assume you have a function triangle(a, b, c) which makes a triangle with the corners a, b, and c. You can look up functions that do this on freemodels.

I'm also going to assume that we're working around the origin or in some other "nice" place, so that things are really simple. Moving it to somewhere else wouldn't ever be terribly challenging.

Simpler Solution

We can recursively break the rectangle up into smaller rectangles -- then the rectangles eventually get broken into two triangles: http://i.imgur.com/8woT1Tg.png

The result looks like this: http://i.imgur.com/LSZWv2X.png

Here is the code I used to make it:

function rectangle(left, bottom, right, top)
    wait();
    local stop = math.random() < 0.2 or right - left < 2 or top - bottom < 2
    -- 20% of the time
    -- OR when its too thing
    -- Stop -- turn into two triangles

    if stop then
        if math.random(2) == 1 then
            -- From bottom left to top right
            triangle.render(
                Vector3.new(left, bottom),
                Vector3.new(right, top),
                Vector3.new(right, bottom)
            );
            triangle.render(
                Vector3.new(left, bottom),
                Vector3.new(right, top),
                Vector3.new(left, top)
            );
        else
            -- From top left to bottom right
            triangle.render(
                Vector3.new(left, top),
                Vector3.new(right, bottom),
                Vector3.new(left, bottom)
            );
            triangle.render(
                Vector3.new(left, top),
                Vector3.new(right, bottom),
                Vector3.new(right, top)
            );
        end
    else
        -- Split it into 9 panels
        -- Horizontally, 3:
        local h1 = (right - left) * math.random() + left
        local h2 = (right - left) * math.random() + left
        -- h1 must be left of h2
        h1, h2 = math.min(h1, h2), math.max(h1, h2)
        -- Vertically, 3:
        local v1 = (top - bottom) * math.random() + bottom
        local v2 = (top - bottom) * math.random() + bottom
        -- v1 must be below v2
        v1, v2 = math.min(v1, v2), math.max(v1, v2)
        -- Dispatch the 9 new rectangles:
        rectangle(left, bottom, h1, v1); -- Bottom left
        rectangle(h1, bottom, h2, v1); -- Bottom middle
        rectangle(h2, bottom, right, v1); -- Bottom right

        rectangle(left, v1, h1, v2); -- Bottom left
        rectangle(h1, v1, h2, v2); -- Bottom middle
        rectangle(h2, v1, right, v2); -- Bottom right

        rectangle(left, v2, h1, top); -- Top left
        rectangle(h1, v2, h2, top); -- Top middle
        rectangle(h2, v2, right, top); -- Top right
    end
end

rectangle(-6, -6, 6, 6);

This turns out better than I expected it to! In the picture I included, you'll notice that most of the triangles aren't quite horizontal / vertical. That's because the triangle drawing routine I have splits all triangles along the altitude from their hypotenuse -- that wouldn't be necessary, and you could just cut all the rectangles into precisely 2 WedgeParts (instead of 4)

Unfortunately, that will make it look fairly regular, which probably isn't what you want. Depending on circumstance, it's possible no one would notice.

Voronoi Observation

There's something called a Voronoi diagram which groups a plane by which control point is closest.

The idea is based on using the control points, and the vertices of their regions.

... The problem is the edges, which I don't have an answer for. I can edit with a little more later.

Making Triangles

http://hastebin.com/mejoyaniwo.js