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The Federation Starfleet insignia is a number of distinctive, slightly asymmetrical, arrowhead-shaped pennants adopted by Starfleet as its identifying emblems, used on starship hulls, installations and some uniforms worn by Starfleet personnel.
The United Earth Starfleet of the 22nd century used a simple forward-facing "pointer" shaped pennant, with a round backdrop, that seemed to be based on symbols used by NASA in the 20th and 21st centuries. The original design for the Starfleet uniform of that era did not include this symbol, unless it was a crewperson assigned to Starfleet Command, who would wear the insignia as their assignment patch on the left shoulder. In a later uniform revision of the 2160s, personnel wore the Starfleet patch on their right shoulder as well as the left shoulder assignment patch. (ENT episodes: "First Flight", "These Are the Voyages..."; TNG episode: "The Royale")
In the early 23rd century, a simplified pennant version of this symbol was used on Federation starships and installations. This symbol for Starfleet was not worn on uniforms, which again only displayed an individual crewperson's assignment patch. Some starships, such as the Defiant, had an assignment patch similar to this symbol. (TOS episode: "Court Martial"; ENT episodes: "In a Mirror, Darkly", "In a Mirror, Darkly, Part II")
By the early 2270s, while installations and starships (such as Epsilon IX station) maintained their individual assignment patches, Starfleet Command had adopted a new Starfleet insignia, a symbol previously used as the assignment patch of the USS Enterprise, and prior to that, the USS Kelvin. Specifically, the command division insignia, with the MACO five-pointed star contained in the arrowhead, was most often used, although a version omitting the internal star gained prominence as well. This arrowhead shape, with similar asymmetry as the previous incarnation, had roots in history as being used in a basic form on the United States armed forces space command ratings badge and unit insignia, and on UESPA exploration vessels, such as the Friendship 1, in the 21st century. This symbol was now pointed up when worn on the the chest or shoulder of uniforms, as well as displayed in signage. The arrowhead was faced forward when used as hull decoration. This symbol eventually phased out all assignment patches by the late 2270s, and remained in use for some Starfleet divisions well into the 24th century. The metal badge version of this symbol would become Starfleet's first combadge as well. (TNG episode: "Yesterday's Enterprise"; VOY episode: "Friendship One")
In the 2340s, a more stylized version of the symbol, the familiar arrowhead without any internal symbology within the shape, and mounted on an oval field, was incorporated into uniforms. A metal badge of this emblem, again serving as a combadge, was worn on the uniform chest. This symbol also became common in starship hull pennants. (TNG episodes: "Encounter at Farpoint", "Family")
By the 2370s, a newer version was created for use as a combadge, with a stylized trapezoidal background. Starship hull pennants of this era began using a simplified version of the arrowhead alone, without a background. Even with the newer insignia in use, Starfleet Command still used the 2270s circular-backed version with inscribed star shape in their signage and parts of flag officer uniforms. (DS9 episode: "The Search, Part I"; VOY episode: "Caretaker")