What is the vesica piscis?

The geometry

Why the lens is √3 times as tall as it is wide

Put the two centres one radius apart and something exact happens. The two centres and the top crossing point are all one radius from each other, so they make an equilateral triangle. So do the two centres and the bottom crossing point. MathWorld: “The vesica piscis therefore has two equilateral triangles inscribed in it.”

That fixes the shape. The lens is one radius wide. Its height is two triangle heights, 2 × (√3 ÷ 2) = √3 radii. So height ÷ width = √3 ≈ 1.732, whatever size you draw it. Move the slider off 1.00 and the ratio drifts; only this spacing gives √3.

Its area is fixed too. MathWorld gives the lens area as (4π − 3√3) ÷ 6 × r², about 1.22837 r². The readout above shows the same number at 1.00.

The name is Latin for “fish bladder”. MathWorld also notes it “is the same figure as the Babylonian ox-eye.”

Page one of geometry

Euclid starts with it

The first proposition of Euclid’s Elements, written around 300 BCE, is a vesica piscis. The task: “On a given finite straight line to construct an equilateral triangle.” The method: “With centre A and distance AB let the circle BCD be described; again, with centre B and distance BA let the circle ACE be described.” Where the two circles cross is the third corner of the triangle.

So the most famous geometry book in history opens with two circles, each through the other’s centre. Press Euclid’s triangle above to see both triangles. Keep adding circles at the crossing points and you get the Seed of Life and then the Flower of Life.

What the traditions hold

The fish, the mandorla, seals and arches

The Christian fish. The early Christian fish sign, the ichthys, is drawn with two arcs crossing past one tip. The Greek word for fish was read as an acrostic, “Iesous Christos Theou Yios Soter, i.e. Jesus Christ, Son of God, Saviour,” and the Catholic Encyclopedia names Clement of Alexandria as the earliest writer to mention it.

The mandorla. Stand the lens upright and you get the almond-shaped glow around Christ in medieval art. Britannica: the mandorla, Italian for “almond,” “first appears in 5th-century mosaics decorating the Church of Santa Maria Maggiore in Rome” and “was abandoned by the painters of the Renaissance.”

Seals. The Norfolk Record Office: English seals after the Norman Conquest came in three shapes, “round, oval, or the pointed oval known as a vesica,” and “vesica seals were often (though not exclusively) used by ecclesiastics and women.”

The pointed arch. The Gothic “equilateral” arch is the top half of a vesica: each side is an arc drawn from the opposite base, “the radius being the whole width of the arch,” as the 1910 Britannica puts it.

Chalice Well, Glastonbury. The well lid was designed by Frederick Bligh Bond and Alice Buckton “incorporating the sacred geometry of the Vesica Piscis symbol” and dedicated on All Saints’ Day 1919. The Trust calls it “the ancient sacred geometry symbol of the union of complementary opposites.”

And the Venn diagram. The two overlapping circles every student draws come from John Venn’s 1880 paper “On the diagrammatic and mechanical representation of propositions and reasonings.” Venn’s circles can be any size and spacing; they only become a vesica piscis when each passes through the other’s centre.