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The Vulcan IDIC is an acronym for Infinite Diversity in Infinite Combinations, the basis of Vulcan philosophy, celebrating the vast array of variables in the universe. In the Vulcan language, the term for IDIC is Kol-Ut-Shan. (VOY: "Gravity") When referring to the symbol in the English language, it, it was pronounced "ih-dihk". (TOS: "Is There in Truth No Beauty?")
IDIC symbols were in use at least as early as the 4th century AD, as one appeared on Surak's katric ark. According to Syrran, the symbol represented Surak's enlightenment atop Mount Seleya. (ENT: "The Forge")
In 2154, T'Les smuggled a holographic map to her daughter T'Pol inside an IDIC medallion pendant. (ENT: "The Forge") On January 22, 2155, T'Pol hung the same medallion on the side of an incubator containing her baby daughter, Elizabeth. Following the child's death, T'Pol was holding the medallion close to her in her quarters when the baby's father, Charles Tucker III, arrived. Moments later, they held hands over the medallion. (ENT: "Terra Prime")
On formal occasions, Spock wore his IDIC medallion (in pin form) as part of his dress uniform. During a dinner function aboard the Enterprise, Dr. Miranda Jones had noticed Spock's Vulcan IDIC pin on his dress uniform. She inquired if it was to serve as a reminder to Jones that as a Vulcan, Spock could mind-link with the Medusans far better than she could. Spock noted, on the contrary, that he donned it to honor Dr. Jones and her achievements. (TOS: "Is There in Truth No Beauty?")
In 2269, James T. Kirk reminded a giant clone of Spock of the concept, asking him whether an army of Spocks could defy this concept by attempting to impose peace and Phylosian philosophy on others. (TAS: "The Infinite Vulcan")
Vulcan IDIC pins were considered collectible merchandise in the retail markets of the 24th century, not easily acquired in the Bajor sector. In late 2370, Quark listed Vulcan IDIC pins as plausible items that he could sell over the monitors on board of Deep Space 9 if given permission by Commander Benjamin Sisko. As well as IDIC pins, other collectibles Quark listed that he could sell at a significant discount included Andorian jewelry and Bolian crystalsteel. (DS9: "The Jem'Hadar")
Background information Edit
The invention of the IDIC by Gene Roddenberry caused friction between him and Leonard Nimoy, who saw it as a cheap ploy to sell replica merchandise to fans. (I Am Spock) Regardless of its original introduction intent, the symbol, first introduced in "Is There in Truth No Beauty?", was designed by William Ware Theiss.  Michael and Denise Okuda have credited Roddenberry himself with designing the symbol, though Theiss' design sketches as published in Star Trek: The Original Series Sketchbook and The Art of Star Trek seem to contradict this. (ENT: "The Forge" text commentary, ENT Season 4 DVD & Blu-ray)
In the IDIC seen from the 4th century, the triangle portion is perfectly vertical and elongated, in contrast to the angle it is displayed at in the 22nd century forward.
In the Vulcan Jellyfish ship from Star Trek, the cockpit appears to form an IDIC with the chair, the controls, and the circular window. The IDIC as a design motif is continued in the Star Trek video game, where it forms the shape of every Vulcan airlock.
A wall decoration displayed in the Vulcan High Command in "Awakening" and "Kir'Shara" was sold in the It's A Wrap! auctions. The image, resembling a vertical IDIC, had the auction description "the panels form an image of the sun rising over Mount Seleya".