The word quatrefoil etymologically means "four leaves", and applies to general four-lobed shapes in various contexts.
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In heraldic terminology, a quatrefoil is a representation of a flower with four petals, or a leaf with four leaflets (such as a four-leaf clover). It is sometimes shown "slipped", i.e. with an attached stalk. However, it is not defined as a flower, but called a "foil".
In architecture and traditional Christian symbolism, a quatrefoil is a symmetrical shape which forms the overall outline of four partially-overlapping circles of the same diameter. Two common types of quatrefoil are shown in the first two images below (click on the images for larger illustrations, and explanation of the geometry involved):
The quatrefoil enjoyed its peak popularity during the Gothic Revival and Renaissance, but can still be seen on countless churches and cathedrals today. It is most commonly found as tracery, mainly in Gothic architecture, where a quatrefoil can often be seen at the top of a Gothic arch, sometimes with stained glass on the interior.
In art, the quatrefoil is a type of decorative framework (mainly used in engraving), consisting of an architectural quatrefoil combined with a square (just as the trefoil is often combined with an equilateral triangle), as seen in the image on the right above. Among the most famous works of art employing the quatrefoil are the bronze panels on the South Doors of the Baptistery in Florence by Andrea Pisano, the bronze panels of the North Doors of the Baptistery in Florence by Lorenzo Ghiberti, and also Filippo Brunelleschi's famous competition entry for the same doors, The Sacrifice of Isaac) as well as "Head of an Angel" by Piero della Francesca.
In the U.S. Marine Corps, quatrefoil refers to a four-pointed decoration on the top of a warrant or commissioned Marine officer's dress and service caps (see peaked caps, also known in the Marines as "barracks covers"). According to tradition, the design was first used with Marine officers on sailing ships so that Marine sharpshooters in the rigging did not shoot their own officers on the deck during close-quarters gun battles (as when crews of opposing ships attempted to board each other's ship).
An official part of U.S. Marine Corps officer uniforms since 1859, the quatrefoil was said to initially have been crossed pieces of rope sewed into officers' caps before becoming officially mandated as a uniform item.
The quatrefoil is the official symbol of the Bishop James Madison Society, est. 1812 at the College of William and Mary.
The quatrefoil is also the national symbol of the women's fraternity Phi Mu (a member of the National Panhellenic Conference.)
The four-leaf clover is a symbol of good luck.