Random Tiling of a Sphere

Suppose we place an equilateral triangle with edge length 1 on the 
surface of a sphere of radius R >> 1  (say, 100).  Now randomly 
select one of the three edges of this triangle and construct an 
identical triangle on that edge (also on the surface of the sphere).
Now we have four exposed edges, each of which is eligible for the 
construction of another equilateral triangle.  We choose one of 
these edges at random and construct another triangle.  This leaves 
us with 5 exposed edges, all eligible, from which we randomly select 
one and construct another equilateral triangle, and so on.

Before long there will be some "exposed edges" on which it's not 
possible to construct an equilateral triangle (without overlapping 
an existing triangle), so we say those edges are not eligible, and 
we exclude them from our random selection list.  At the beginning of 
this process our list of eligible edges will obviously increase by 
1 at each step, but then it will occassionally not increase on some 
steps, and eventually it will start to decrease as we continue to 
adjoin more triangles.  Ultimately we will have no more eligible 
edges.

What fraction of the sphere's surface area would we expect to
cover in this way?  What is the asymptotic behavior of this 
fraction as R increases?  (Since this process would completely 
tile the plane, it might seem that the coverage ratio limit as R 
approaches infinity must be 1, but perhaps there's a tendancy for 
each local region to "waste" a certain amount of area due to certain 
patterns of inter- ference that are repeated throughout the surface 
for ANY finite R, even though all the interference vanishes when the 
surface is perfectly flat.)  How many ineligible exposed edges would 
be present at the end?  If, instead of randomly deciding where to 
add the next triangle, we are allowed to plan our moves, is there 
a simple algorithmic strategy that maximizes the covered area and/
or minimizes the final number of ineligible exposed edges?

Some people have suggested that, since it's possible to fit 5 but 
not 6 equilateral triangles to a common vertex, the coverage must 
be around 5/6, regardless of whether the pattern was random or not.  
However, I think the coverage may actually be less than that, and it 
can depend on the sequence in which the tiles are placed.  We could 
ask the opposite question, namely, what is the LEAST number of 
tiles that leave no eligible edges remaining.  Suppose we have a 
configuration like this on the surface of a very large sphere:
             
            ____A
           /\  /\
          /__\/__\C
         /\  /\  /\
        /__\/  \/__\
       /\  /   /\  /
      /__\/   /__\/
     /\  /   /\  /
    /__\/   /__\/
    \  /   /\  /
     \/   /__\/
     B   D

The lines AB and CD are nearly geodesics and essentially parallel as 
they cross the segment AC.  However, they don't remain parallel (as 
they would on a perfectly flat plane).  Instead they are slowly 
converging, I think, based on the fact that the centerlines of those 
two rows of tiles must be geodesics, and by symmetry the cross-wise 
row of tiles along AC must be the point of maximum separation between 
those geodesics.  As a result, not only is it impossible to fill in 
the 6th edge of the upper "hexagon", it's also impossible to place 
a triangle anywhere between the lines AB and CD anywhere below C. 
If we continue this pattern, the lines AB and CD will eventually 
intersect, but it will take a very long time if the curvature is 
very slight.

Also, we could repeat this pattern, creating a series of "stripes" 
alternating between tiled and untilable regions, like this:

            ____A
           /\  /\
          /__\/__\C
         /\  /\  /\
        /__\/  \/__\E
       /\  /   /\  /\
      /__\/   /__\/__\G
     /\  /   /\  /\  /\
    /__\/   /__\/  \/__\
    \  /   /\  /   /\  /
     \/   /__\/   /__\/
     B   /\  /   /\  /
        /__\/   /__\/
       D\  /   /\  /
         \/   /__\/
         F   /\  /
            /__\/
           H\  /
             \/

Here the lines EF and GH are also slightly converging, making the 
region between them unfillable.  In this way we might, by design, 
be able to exhaust our eligible edges while having tiled only 
about 2/3 of the surface, since the unfillable slice between 
every two rows of tiles would have about half the area of a row 
of tiles.

Of course, this pattern requires us to carefully arrange our 
construction.  For example, if we didn't tile all the way out to
DF on the central path, we could build a tile path from H to B,
which wouldn't quite connect perfectly, but would block the central
row of tiles.  In general I think we could construct some very
"maze-like" patterns, and if we just randomly selected our sites 
for each successive triangle, we would (or at least COULD) get
some fairly jagged arrangements with extensive unfillable 
"slices".   As a result, I'd expect to end up with less than 
5/6 coverage... and for it to be possible to get significantly
different coverage by design versus random selection.

In fact, I think it's possible to show that we can actually get 
arbitrarily low coverage simply by constructing one geodesic "belt" 
around a great cirlce of the sphere, and then constructing a triangle 
on each of the exposed faces of that belt.  We'll end up with an 
arrangement that looks like this

         /\  /\  /\  /\  /\  /\  /\
     __ /__\/__\/__\/__\/__\/__\/____
<--     \  /\  /\  /\  /\  /\  / geodesic belt -->
     ____\/__\/__\/__\/__\/__\/_____
         /\  /\  /\  /\  /\  /\
       \/  \/  \/  \/  \/  \/

Notice that it's not possible to construct any more triangles
on this configuration, since the only exposed edges are subtending
and angle just slightly less than what is required to accommodate
another triangle.  Thus, for an arbitrarily large sphere we have 
a saturated configuration of tiles that covers just an arbitrarily 
thin slice around a great circle.  Naturally we wouldn't expect to 
reach this configuration by chance, but I think it illustrates how 
this type of tiling can easily become saturated even while larges 
regions of the sphere's surface are totally uncovered.  Of course, 
this construction assumes the "belt" fits perfectly, but there are 
infinitely many values of R for which this is true.

Along these same lines, we could tile along any of the geodesic
projections of the edges of a regular solid, and then "saturate"
those geodesic segments by placing triangles on each exposed face,
just as we did with the geodesic belt.  In this way, we can get
just about any amount of coverage, from the max possible all the
way down to virtually zero.  

Speaking of the max possible, I'm wondering if it might be possible 
to achieve coverages close to 1 in many cases (aside from the obvious 
Pythagorean cases), because it might be possible to construct a spiral 
belt, with "dog legs" as needed in order to circle the sphere and end 
up just skimming the previous loop, without leaving much empty space. 
Anyway, it's clear that we can have saturated tilings of the sphere 
with a wide range of coverages, so this again raises the original 
question, i.e., what is the expected coverage for a random tiling?