Vesica Piscis

The vesica piscis is a shape that is the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other. The name literally means the "bladder of a fish" in Latin after the conjoined dual air bladders ("swim bladder") found in the bodies of most fishes. The shape is also called mandorla ("almond" in Italian).

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. The triangle has as its vertices the two disk centers and one of the two sharp corners of the vesica piscis.

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). The ratios 265:153 = 1.7320261... and 1351:780 = 1.7320513... are two of a series of approximations to this value, each with the property that no better approximation can be obtained with smaller whole numbers. Archimedes of Syracuse, in his On the Measurement of the Circle, uses these ratios as upper and lower bounds:

${\displaystyle {\tfrac {1351}{780}}>{\sqrt {3}}>{\tfrac {265}{153}}\,.}$

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