A Pointy Hat, sometimes called a cornuthaum, is type of headgear associated with magic users, and in particular archmagi. At times in the past, sumptuary laws have restricted their wearing to certain kinds of legally-recognized magi, but such restrictions had ended before the Golden Age.
Use by the Harmonic Curacy
Archmagi are entitled to wear wide-brimmed pointed hats as part of their regalia. The canonical cornuthaum worn officially by an ensconced archmage is a wide-brimmed pointed hat, bent 45 degrees halfway up the cone and topped with an ornamental spherical metal bell. The hat is notionally bent because of the weight of responsibility borne by an ensconced archmage. Other officers of archmagical sconces also wear different kinds of (straight) pointy hats as part of their formal regalia, and, when Magistrates have been appointed by the archmage to decide cases at the court of the sconce, the magistrates wear a bent pointed hat identical to their archmage's own, though only when sitting in judgement and not at any other time. Recognized Archmagi who are not ensconced, e.g. the master or mistress of the Crooked Tower (which is not under the authority of the Ensconced Archmage of the Isle of Yore, but rather answers directly to the Procurator of the Oneiric Curacy), are further canonically legitimate pointy-hat wearers.
Procurators of the Harmonic Curacy wear a wide-brimmed pointed hat similar to that worn by archmagi, and also bent, at an angle of 80 degrees, and turned to the right.
The officers of a sconce or curacy (i.e. its magistrates, quaestores, and secretarii, and their deputies) wear un-bent pointed hats; its other employees (scribes and custodes) do not wear them. Like archmagi, magistrates have green and black regalia, quaestores mauve and black, and secretarii black. The respective non-court formal hats of magistrates, quaestores and secretarii are distinguished by their trim color, being otherwise identical wide-brimmed unbent pointy hats with tassels. Officers occupying their position at the level of a curacy have colored bells rather than tassels.
Historical Usage
The Loremasters of the Arcane Helix, though considered archmagi, did not wear pointy hats, because the association was not yet firmly established at the time of the Helix's foundation. Likely it was not later adopted because the wide brim of an archmage's hat would have interfered with, or concealed, the Obsidian Circlet of River Runes worn by the Loremasters.
Fungus Gnomes wear pointy hats, with their social dominance being correlated with hat height. Their pointy hats are sometimes brimless and held on by chin-straps, but a variety are seen.
It is thought by scholars that pointy hats might symbolize a horn.