In art and symbolism, a crescent is generally the shape produced when a circular disk has a segment of another circle removed from its edge, so that what remains is a shape enclosed by two circular arcs of different diameters which intersect at two points (usually in such a manner that the enclosed shape does not include the center of the original circle).
In astronomy, a crescent is the shape of the lit side of a spherical body (most notably the Moon) that appears to be less than half illuminated by the Sun as seen by the viewer. Mathematically, assuming the terminator lies on a great circle, such a crescent will actually be the figure bounded by a half-ellipse and a half-circle, with the major axis of the ellipse coinciding with a diameter of the semicircle. The direction in which the “horns” (the points at the intersection of the two arcs) face indicates whether a crescent is waxing (also young, or increasing) or waning (also old, or decreasing). Eastward pointing horns (pointing to the left, as seen from the Northern hemisphere) indicate a waxing crescent, whereas westward pointing horns (pointing to the right, as seen from the Northern hemisphere) indicate a waning crescent. Note that the directions the horns point relative to the observer are reversed in the Southern hemisphere.
The crescent and star, while generally regarded as Islamic symbols today, have long been used in Asia Minor and by the ancient Turks, earlier than the advent of Islam. According to archaeological excavations, Göktürksused the crescent and star figure on their coins. The 1500-year-old coin includes three crescent moon figures and a star near a person.
The crescent was the symbol of the Sassanian Empire of Persia (Iran) and is prominently displayed on the crowns of its rulers. After the Arab conquest of that empire in 651 CE, it was gradually adopted by later caliphates and Muslim rulers as an established and recognized symbol of power in Western Asia. It was also a symbol of the Ottoman Empire. Though the crescent was originally a secular symbol of authority for Muslim rulers, it is now often used to symbolize the Islamic faith. However, it should be noted that the crescent was not a symbol used for Islam by Muhammad or any other early Muslim rulers, as the Islamic religion is, in fact, against appointing “Holy Symbols” (so that during the early centuries of Islam, Muslim authorities simply didn’t want any geometric symbols to be used to symbolize Islam, in the way that the cross symbolizes Christianity, themenorah was a commonly-occurring symbol of Judaism, etc.). This is why early Islamic coins were covered with Arabic writing, but contained no visual symbols.
Despite this mixed history, many Islamic nations and charities use the crescent symbol on their flags or logos (e.g. Pakistan, The Red Crescent, etc. — though currently none of the Arab states in Arabia or the Mashreq have crescents on their flags). Therefore it could be argued that this usage is meant to signify Islamic identity or Muslim brotherhood.