The “arrowhead” shape of the Starfleet logo was likely inspired by a similar shape seen on the real-world NASA logo, initially put into used in 1959 by President Eisenhower. The original logo featured a miniature space-scape, with a distant star, a nearby planet, and a small, comet-like moon circling the planet. There was also a large, floppy, arrow-like shape draped over the nearby planet, likely meant to invoke speedy travel through the stars. That floppy arrow shape was carried over into future iterations of the NASA logo, and is still in use to this day.
William Ware Theiss likely internalized the arrow shape’s association with space travel and made a subtler, simpler version of it for “Star Trek.” There has been no in-canon dialogue explaining why the Starfleet logo looks the way it does. The original Starfleet deltas were patches sewn onto the uniforms.
The original “Star Trek” series, recall, featured color-coded uniforms that indicated the department in which they worked. The gold uniforms indicated command, the red uniforms indicated engineering and security, and the blue uniforms indicated science and medical. If one looks closely at the respective insignias, however, one will see that they, too, had symbols to indicate their department. Command uniforms boasted five-pointed star shapes to indicate command, while engineering and security had a spiral-like “gear” shape. Sciences had a rudimentary globe in their deltas, while medical had old-fashioned Red Cross crosses. In the picture above, Scotty (James Doohan) is actually wearing the wrong symbol. It should be the engineering symbol.
In “Star Trek: The Motion Picture,” the logos would incorporate a circle behind the delta shape, and an officer’s department would also be indicated by a badge’s color. There were six colors for uniforms and patches in that movie, but most Trekkies haven’t bothered to memorize them.
For most of the “Star Trek” movies, the individual departmental symbols were all replaced by the command five-pointed star shape, and the patches were replaced with large, elaborate, bronze brooches.