The vesica pisces (piscis) is a symbol made from two circles of the same radius, intersecting in such a way that the center of each circle lies on the circumference of the other. The name literally means the bladder of the fish in Latin. In the Christian tradition, it is a reference to Christ, as in ichthys. It is called a mandorla ("almond") in India and known in the early Mesopotamian, African, and Asian civilizations.
Geometry -- The symbol is formed from the almond-shaped area in the overlap between the circles, as shown in black in the diagram - for certain purposes also including the upper arcs as far as the edges of a rectangle whose sides coincide with the widest points of the almond (as shown in light blue in the diagram). The resulting figure looks like a stylized fish, or in the extended version like a flattened Greek letter alpha.

Mystical and Religious Significance - It has been the subject of mystical speculation at several periods of history, perhaps first among the Pythagoreans, who considered it a holy figure. The mathematical ratio of its width (measured to the endpoints of the "body", not including the "tail") to its height was reportedly believed by them to be 265:153. This ratio, equal to 1.73203, was thought of as a holy number, called the measure of the fish.

The geometric ratio of these dimensions is actually the square root of 3, or 1.73205... (since if you draw straight lines connecting the centers of the two circles with each other, and with the two points where the circles intersect, then you get two equilateral triangles joined along an edge, as shown in light red in the diagram).

The ratio 265:153 is an approximation to the square root of 3, with the property that no better approximation can be obtained with smaller whole numbers. The number 153 appears in the Gospel of John (21:11) as the exact number of fish Jesus caused to be caught in a miraculous catch of fish, which is thought by some to be a coded reference to Pythagorean beliefs. Ichthys a symbol used by early Christians, more popularly known as the fish symbol is created by the almond shape and the light blue extension as seen in the Construction Diagram of the Vesica Pisces above.

Uses of the shape -- Other uses of the shape include that by some early peoples of the almond-shaped central area as a representation of the female genitals, and the use of a similar (horizontally-oriented) fish symbol called the Ichthys by early Christians. 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). The most common modern object based on the vesica piscis is the American football, which resembles the interior almond-shaped area of the vesica piscis swept about its long "axis" to produce a 3D object with rotational symmetry.