In the novelFederation, written by Judith and Garfield Reeves-Stevens, the shape was explained as indicative of the warp threshold boundary. The star in the middle of the badge represented the speed of light, the upper edge of the badge was a graph of normal energy versus speed (indicating that it was impossible to reach the speed of light without infinite energy) and the lower edge the graph with warp engines taken into account.
In the mid-23rd century, two types of distinctive Vulcan badges, including the IDIC, were worn by graduates of the Vulcan Science Academy on their graduation day. One of these badges was worn on the left breast of a white cloak and the other, the IDIC, was worn near the neck, clasping the front of the cloak together. (DIS: "Lethe")
These Vulcan badges were 3D printed and hand painted in Toronto, Canada. ("Star Trek Exhibition" at DIS: "Si Vis Pacem, Para Bellum" screening, Millbank Tower, London, UK, 5 November2017) One of each type of these badges was featured at Star Trek: The Exhibition in Blackpool, UK, and are now on a world tour, first shown at a screening of the episode "Si Vis Pacem, Para Bellum" at London's Millbank Tower on 5 November 2017. There, a placard describing the exhibit referred to them as "Vulcan IDIC badge and pendant".