Descriptions of some interesting matings:

Below are some descriptions of some matings I've made with the programs.
Try out the ones that sound interesting.

First, 1/7 is called the rabbit, and 1/3 is called the basillica.  I'll
use them a lot.

. 1/7, 1/3.  Also examine 1/3, 1/7.  Note the spherical
pictures are the same if you exchange black and white, and hold one paper
upside-down. In the flat versions you can really see the black julia set.

	. 1/7, 2/5.  Find 2/5 in the M-set.  You can think of it as the
1/3-bulb off of the main 1/3 bulb.  As a result, to get its julia set, you
take the 1/3 picture and replace each ball by a copy of the whole 1/3
julia set (basillicas in the basillica).  This is called a bifurcation of
1/3. Compare 1/7, 2/5 with 1/7, 1/3 and you'll see that, as expected, each
clear ball in 1/3 is replaced by a whole copy of 1/3.

	. 1/7, 22/63 is another bifurcation.  22/63 is obtained by
replacing the balls in 1/3 with copies of 1/7. (rabbits in the basillica). 


. 1/7, 1/7 is a self-mating

	. 1/7, 74/511 is a bifurcation.  74/511 is 1/7 with balls (i.e.
connected components of the filled julia set minus its boundary) replaced
by copies of 1/7 (rabbits in the rabbit).

	. 1/7, 10/63; where 10/63 is 1/7 with balls replaced by 1/3s
(basillicas in the rabbit).

		. 9/31, 10/63 is a variation on 1/7, 10/63, where 9/31 is like the
rabbit (1/7), but it has 3 ears instead of 2.  Julia sets exist for
rabbits with any # of ears, and we can make more complicated matings by
increasing # of ears in one Julia set. 

. 1/7, 3/7, rabbit mate aeroplane, is very exciting mathematically.  Note
that the julia sets themselves look nothing alike, but in the mating the
black and clear look just the same.  We say 1/7, 3/7 ~= 3/7, 1/7 and call
it a _shared mating_.  Note the points where 6 "petals" meet, alternating
black and clear.  What's happened here is that at the point in the rabbit
where 3 balls meet, in between each pair of rabbit balls we get to glue
the root point of an aeroplane ball onto the center point.  So we get
alternating rabbit and aeoroplane balls.  But since the aeroplane has
perfect circles, they conform to the shape of the rabbit, so the black and
clear look the same.

	. 1/7, 22/63 is a bifurcation of 1/7, 3/7, where 22/63 is the
aeroplane with basillicas(1/3) in the aeroplane balls.

	. 10/63, 28/63 is a double bifurcation.  10/63 is basillicas in
the rabbit, and 28/63 is basillicas in the aeroplane.  Note this is also a
shared mating, 10/63, 28/63 ~= 28/63, 10/63.

I should mention, the black and white dust in these pictures is because of
the 1-dim. parts of the aeroplane.  Since the mating covers the sphere,
these parts must wiggle around alot to fill space up.  More on that later.

	. 74/511, 220/511 is also a double bifurcation of 1/7, 3/7.
74/511 is rabbits in the rabbit and 220/511 is rabbits in the aeroplane.

	. 1/31, 15/31 is another kind of variation on rabbit mate
aeroplane.  1/31 is the 3-eared rabbit and 15/31 is like the aeroplane,
just with more (smaller) balls.  So here we have 8-petaled points instead
of 6.  

	. 1/63, 31/63 is the flat version of the 4-eared rabbit mate a
more complicated aeroplane.

	. 1/511, 255/511 uses the 9-eared rabbit.  Nice spiral, huh?  Look
back at 1/31, 15/31 and you can see the beginning of that spiral pattern.

	. 169/511, 169/511 is a self mating of a 9-eared rabbit, but not
the same 9-eared rabbit as 1/511.  The visual difference is in which ear
is the largest (that one creates the spiral).  In 1/511 it is the first
ear, in 169/511 it is an ear near the middle.  

	. 1/255, 1/255 is my favorite.  The self-mating of the 8-eared
rabbit.  Like 1/511, it's first ear is biggest, and the two big first ears
of each copy of 1/255 spiral around to touch each other's tips.  

	. 1/5, 2/5 is another shared mating.  It's not obvious, but this
is a kind of variation on 1/7, 3/7.

. 3/31, 15/31.  I like the way it looks like there are groups of black and
white balls floating in dust.

. 1/5, 1/5 has lots of symmmetry.  In any self-mating, you can mod out by
that symmetry and the result is actually ~= 1/2 mated with the original
julia set.  Why?  1/2 is just a straight line segment, and the map from
the circle to it send 0 to one endpoint and 1/2 to the other endpoint, in
the expected way that makes p(t)=p(1-t).  Then in gluing a julia set p/q 
to 1/2, you identify the points in it together that you would have
identified together had you mated p/q with itself and mod-ed out by the
symmetry.

. 1/2, 1/5 is shown next, since it was discussed above.

. 1/2, 1/5 with fewer iterations.  Notice that 1/2 is the black and it is
just a curve (an image of a line segment) twisting around trying to fill
up space around 1/5. 

. 1/6, 5/14.  These two julia sets both have no interior (they look like
feathery trees).  But they twist around each other to create a
space-filling curve on the sphere.

. 1/4, 1/4.  Another very rigid-looking space-filling curve (well, it's
not space-filling at this number of iterations)

. 1/4, 1/6.  The previous two pictures were very regular, but you see this
one has crazy splotches.  The math reason behind this has to do with an
invariant measure.

Well, if you are still reading, Congratulations!  I hope this has been
helpful and interesting.