LA JOLLA — The Bird Rock community has been left without a well-known physical trademark in the wake of the collapse of its namesake. The rock formation that has stood as a monument in the region crumbled in the rain on Dec. 21. “It really is sad to see the end of our landmark namesake rock,” Philomène Offen, a Bird Rock resident for 37 years, wrote in an e-mail. “For those of us who have lived here a long time, it’s like losing an old friend.” The hole that formed the arch had gradually been getting bigger over the years as rough waves have pounded the formation and weakened the bridge. “Obviously, I won’t be sending tourists down to see it because there isn’t much for them to see,” wrote Offen, who created the Bird Rock History Museum in 2009. “Still, I think you’ll continue to see it on all things Bird Rock — logos, T-shirts, etc., along with the trademark birds that were usually perched upon it. So it will definitely live on as a symbol of the community.”