There are some interesting patterns of things that come in three's, including the elven rings and the Silmarils. Here's my symbolic interpretation in tabular form--each column are things that go together across each category (unfortunately the formatting doesn't allow me to align the columns )
Domains of Arda: Air Fire/Earth(1) Water
Valar: Manwë Aulë Ulmo
Color: Blue Red Grey/White
Elven Kindreds: Vanyar Noldor Teleri
Noldorin Houses(2): Fingolfin Fëanor Finarfin
Silmaril: Eärendil's Maédhros's Maglor's
Elven Ring: Vilya Narya Nenya
Ringbearer(3): Gil-galad ??? Galadriel
Notes:
(1) It seems that in Arda, fire and earth are treated as being equivalent, unlike the aristotelian picture of the four elements of air, earth, fire, and water. For instance, the Silmarillion says that the silmaril that Maédhros took found its home "in the fires of the earth" and also that Eru put the Flame Imperishable in the heart of the world. Furthermore, Aulë, the vala having dominion over the substances of which Arda is made, is a smith, using fire.
(2) Fëanor's house was the only purely Noldorin one of Finwë's descendents. Fingolfin was half Vanyar, and his name was taken form a combination of Finwë's and Ingwë's. Finarfin is married to Eärwen (a telerin elf).
(3) Referring to the original ringbearers of the Three. Gil-galad was the original possessor of Vilya, which then passed to Elrond; both are descendents of Fingolfin. Galadriel (Finarfin's daughter) seems to have borne Nenya from the beginning. Narya is kind of a mystery, it was in Gil-Galad's possession, then kept by Cirdan, who according to one of the versions of the story in which he gives it to Gandalf in HoME X, never seems to have used it. My (purely speculative) thought is that Celebrimbor had originally intended to keep Narya for himself, thus the heir of each of the Finwë's sons would have one of the Three Rings.
In response to what someone said on whether the Three Rings were different in their powers, what Círdan says to Gandalf while giving him Narya suggests that they are different:
Quote:
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For this is the Ring of Fire, and herewith, maybe, thou shalt rekindle hearts to the valour of old in a world that grows chill. But as for me, my heart is with the Sea...
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Is this last statement in part an explanation of why Círdan never used Narya?