For starters, the basic formula for the area of a cylindrical surface is just the circumference times the length:
A = pi x d x l
A = the area
pi = 3.141 (a constant)
d = the diameter
l = the length
The 60 degree thread case:
Several assumptions must be made. First, that 60 degree thread has a sharp crest and a sharp valley. Second, the thread depth or linear pitch is small in relation to the diameter. Finally, the OD (outside diameter) at the crest of the thread remains the same as the original OD of the cylinder.
A SHARP, 60 degree thread viewed in cross section forms a equilateral triangle: the pitch line from crest to crest and the two flanks of the thread form the triangle and all three of these sides are equal. So, each flank of the thread has the same width and length and therefore the same area as the original surface of the cylinder (which has been cut out). Therefore the area of the surface has been multiplied by a factor of 2, for the two thread flanks. And the modified formula is:
A = 2 x pi x d x l
In case you are wondering, the pitch of the thread does not enter into this calculation. If my first assumption is not true and there is a flat at the crest of some kind of fill in the thread's root, then the area will be somewhat less. The exact amount less will depend on the size and shape of those features.
Another flaw in this calculation is the second assumption: that the linear pitch of the thread is small compared to the cylinder's diameter. The formula is 100% accurate if the OD is infinite. Any smaller OD will mean that the parts of the thread flanks that are closer to the root of the thread sill be somewhat smaller due to the decreasing length of the thread at that, lower point. This decrease in length is due to the smaller diameter there. If the assumption is true, this error will be small and in many cases can be neglected. But if it is not true, then a more complicated calculation would be called for.
I had other things to do and had to interrupt my work on this at this point. I will post about the other case, with the diamond pyramids, later.
PS: If you are simply trying to maximize the surface area, among the three cases you cited, this is it. If you want even more, just decrease the included angle of the thread: the smaller, the better. Mathematically, there is no limit to the process: in theory, when the included angle of the thread approaches zero, the area approaches infinity.
PS #2: It is the included angle of the thread, not it's depth that counts. Changing the depth of the thread is equivalent to changing it's pitch and the area remains the same for a given included angle. Within the limitations of assumption #2 anyway.