Several mathmaticians recognize a ratio with a mathmatical basis in the shape of all icicles. Nomatter where they are formed in the world, they are all the same shape... thick at th etop and thinner along the shaft becoming nearly pointed at the end. The reason, they contend, is that as water drips onto an icicle and freezes, it releases heat. The warm air rises up the sides of the icicle. The scientists state that the warm air layer acts like a blanket that's an insulator, and so the blanket is very thin near the tip and thick at the top. That allows the top to grow very slowly and the tip to grow rapidly -- creating a long, thin icicle...
The impetus for roof eaves formed icicles is the warm air rising from the roof melting snow, which in turn starts the icicles formation... all very interesting.