Vesica Piscis

The construction is two circles. Take a segment AB. With centre A and radius AB draw a circle; with centre B and the same radius draw another. Each circle passes through the other's centre. The lens-shaped region where the two overlap is the vesica piscis.

This is not a decorative figure grafted onto geometry afterwards. It is Proposition 1 of Book I of the Elements, the first thing Euclid shows how to do. His purpose is not the lens but the two points where the circles cross: joining either crossing point to A and to B gives three segments all equal to the radius, and therefore an equilateral triangle on AB. The vesica is what the equilateral triangle is built out of.

Almost every elementary construction starts here. The two crossing points define a line, and that line is the perpendicular bisector of AB, which delivers the midpoint of a segment and a right angle in a single move. Apply the same trick from a point to a line and you have a perpendicular through it. Strike arcs about the arms of an angle and repeat, and you have the angle bisector. The vesica is the workhorse of compass work, and anyone who has done school constructions has drawn hundreds of them without being told the shape had a name.

The name itself is Latin for fish bladder and describes the outline. In Italian art history the same shape framing a figure is called a mandorla, almond. German medieval architectural vocabulary calls the related tracery form a Fischblase, fish bladder again. The names are late. The construction is as old as Greek geometry and older in practice, since striking two arcs to find a midpoint is what anybody laying out a building does.

The proportions of the vesica are exact and easy to check, which is worth doing, because the figure attracts a good deal of loose numerology.

Let the two circles have radius r, with centres A and B a distance r apart. The lens is symmetric about the line AB and about the perpendicular bisector of AB. Its width, measured along AB, spans a distance of r, since each circle reaches exactly r beyond the other's centre. Its height is the distance between the two crossing points. Each crossing point lies at distance r from both A and B, so it forms an equilateral triangle with them, and its perpendicular distance from AB is that triangle's altitude, r times sqrt 3 divided by 2. There are two such points, one on each side, so the full height is r times sqrt 3.

The ratio of height to width is therefore sqrt 3 to 1, roughly 1.732. That is the number, and it is not an approximation.

Two consequences follow directly. First, the crossing points and the two centres form a pair of equilateral triangles set back to back, so the vesica hands you the 60 degree angle for nothing, and six repetitions of that angle about a circle give the regular hexagon. Second, sqrt 3 is the same ratio that governs the equilateral triangle's own geometry, which is why medieval setting-out schemes working ad triangulum, by the equilateral triangle, keep producing it. The dispute recorded at Milan Cathedral in the 1390s over whether to raise the section ad quadratum or ad triangulum is the best-documented instance of masons arguing about exactly this choice.

Claims that the vesica contains the golden ratio, or encodes some particular significant number, are generally reached by measuring something other than the lens. What it demonstrably contains is sqrt 3, the 60 degree angle, the perpendicular bisector and the equilateral triangle.

The vesica has a long and genuinely documented life in Christian art and building, which is unusual among sacred-geometry claims and worth separating carefully from the claims that are not.

The mandorla is the almond-shaped frame of light drawn around an entire figure, as distinct from the nimbus around the head alone. It appears in Christian art from late antiquity and is standard by the Romanesque and Gothic periods, most often around Christ in Majesty above a west door, around Christ at the Transfiguration or the Ascension, and around the Virgin. The carved tympana at Moissac, Vezelay and Chartres are among the best known examples. The convention marks the figure as belonging to a different order of reality from the surrounding scene, rather than merely as being important.

Whether the mandorla is derived from the two-circle construction, or simply happens to be an almond shape, is a fair question, and art historians do not all answer it the same way. What is not in dispute is that Gothic masons drew with compasses constantly, and that circle-on-circle intersection is the generative move of window tracery. The pointed arch itself is two arcs struck from centres on the springing line, which is the vesica construction applied to an opening. Trefoils, quatrefoils and the late Fischblase forms are all built from arcs of the same kind.

Some claims go further than the evidence supports: that particular cathedrals were laid out on a vesica grid, or that the proportion encodes a specific doctrine. Those are argued rather than established, since medieval building accounts record dimensions and disputes but rarely the reasoning behind a geometric scheme. The safe statement is also the strong one. The construction is demonstrably the basis of the drawing technique, and the shape is demonstrably a standard Christian iconographic frame.