In honor of the 25th anniversary of the library fire, today's mosaic is a survivor of the blaze: the mosaic dome atop the 1926 structure.
Architect Bertram Grosvenor Goodhue designed the original Los Angeles Central Library to mimic the architecture of ancient Egypt. The central tower is topped with a tiled mosaic pyramid with suns on either side with a hand holding a torch representing the "Light of Learning" at the apex. Other elements include sphinxes, snakes, and celestial mosaics. It has similarities to the Nebraska State Capitol in Lincoln, Nebraska, also designed by Goodhue and which also featured sculpture by the architectural sculptor Lee Lawrie.
The library itself has a page on its art, but I could find nothing about the dome. However, they do have an incredible archive of dozens of pictures taken during construction in 1925 (top right) and 1926 (bottom right.)
And dang, all the color pictures I find of the mosaic are so washed out I don't want to use them. The exception at top is from the VigilantCitizen blog, which writes about symbolism on our public places. That post also includes extensive biographies of Goodhue and the artists who contributed to the library, as well as a discussion of the symbolism of the tower mosaic--mostly focused on the pyramid and the sun.