Pileus (plural: pilei)

The pileus liberatis was a Phrygian soft woven wool cap often used as a symbol of freedom. The form of the pilei (plural for pileus) was varied; some were round, others resembled a helmet, some were shaped almost like a pyramid, and some were of a more depressed form. Pilei and pilei with stars above them also represent the Dioskouroi. The pileus was sometimes worn by the sick, old men and newly freed slaves (the latter in token of liberty). Freed slaves wearing pilei would follow their former master in his bier at his funeral (demonstrating his magnanimity and bringing him prestige). This liberty cap became a symbol of freedom for the Romans. It was a popular symbol during the French Revolution and was also depicted on some early U.S. coins, including the 1839 Seated Liberty dollar below.

DICTIONARY OF ROMAN COINS

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Pileus, a bonnet, or cap, of woven wool.  The form of the Roman pilei was varied; some were round, others resembled a helmet, some were shaped almost like a pyramid, and some were of a more depressed form.  The Romans for the most part went without hats, except on religious occasions, and the Saturnalia, or when on journeys or military service.  The pileus was sometimes worn by the sick, old men, and newly freed slaves (the latter in token of liberty).

The pileus was a symbol of liberty for the Romans, and was given to slaves upon emancipation.

On Roman coins, the pileus covers the heads of the Dioscuri, sometimes Liberty, and some personifications of provinces. The Phrygian Pileus, curved at the top, is seen on images of the god Mên, on Atys, the companion of Cybele, and on Iulus, the son of Aeneas. 

The pileus, or cap of Liberty, appears between two daggers on the celebrated denarius of Brutus, with the legend EID MAR, referring to the assassination of Julius Caesar.

Pileati Fratres, literally "the brothers who wear pilei," referring to Castor and Pollux, the Dioscuri.

Pileatum caput, the bonneted head of Liberty, appears on a Roman Republic denarius.

Pilei duo, refers to two caps surmounted by stars, which represent the Dioscuri. Castor and Pollux were marked by stars. They and their stars were believed help to guide mariners.