Vesica piscis

This figure appears in the first proposition of Euclid's Elements, where it forms the first step in constructing an equilateral triangle using a compass and straightedge.

[4] Mathematically, the vesica piscis is a special case of a lens, the shape formed by the intersection of two disks.

The mathematical ratio of the height of the vesica piscis to the width across its center is the square root of 3, or 1.7320508... (since if straight lines are drawn connecting the centers of the two circles with each other and with the two points where the circles intersect, two equilateral triangles join along an edge).

Archimedes of Syracuse, in his Measurement of a Circle, uses these ratios as upper and lower bounds:[5] The area of the vesica piscis is formed by two equilateral triangles and four equal circular segments.

[3] In Christian art, some aureolas are in the shape of a vertically oriented vesica piscis, and the seals of ecclesiastical organizations can be enclosed within a vertically oriented vesica piscis (instead of the more usual circular enclosure).

[6][7] The vesica piscis has been used within Freemasonry, most notably in the shapes of the collars worn by officiants of the Masonic rituals.

The system was illustrated in Cesare Cesariano's 1521 version of Vitruvius's De architectura, which he called "the rule of the German architects".