On January 2, 2025, I posted a piece called “My Introduction to Sacred Geometry.”

My next foray into “sacred geometry” began with the fish-shaped form created by the intersection of two circles. After drawing the first circle, a second circle with the same radius as the first is drawn, using a point on the first circle’s circumference as the center. The result is shown below.

Vesica Pisces

The shaded area in the center is central to Euclid’s classic geometry. The fish-like shape was also incorporated as a symbol of Christ in the early Christian church. This lens-like form is called the “vesica pisces”, or “bladder of a fish”.

The basic form of the “vesica pisces” can be extended into a complex, repeating form called the “flower of life”. My own drawing of the “flower of life” is given below. I won’t go into the exact methodology for constructing it here, but I will say it took many days of trial-and-error drawings.

Vesica Pisces and Euclidian Geometry

The vesica pisces form was used by Euclid to illustrate some important mathematical relationships.

First, a rectangle is drawn using the vertical diameters of both circles as the two sides.  Then, the top and bottom of the rectangle are drawn by connecting the intersection points of the vertical diameters on the two circles. The resulting rectangle has some interesting properties.

The Pythagorean theorem is central to Euclidian geometry. The theorem makes use of a universal property of any right-triangle (i.e., any triangle that has a “right” angle of 90 degrees).  In such a triangle, the longest side is called the hypotenuse. The theorem has to do with the relative lengths of the three sides of a right triangle. It says that the length of the hypotenuse, squared, is equal to the squares of the lengths of the other two sides.

The Pythagorean theorem can be used with the rectangle around the “vesica pisces” as shown in the figure above. First a line is drawn connecting opposite corners of the rectangle. This creates a right-triangle in which we already know the lengths of the two shorter sides.

We can define the radius of each circle as a length of 1 unit. Thus, we can easily see that the vertical lines of the rectangle are exactly 2 units tall (the length of two radii). Similarly, we can see that the top and bottom lines of the rectangle are the same length as the radius: 1 unit.

Using the Pythagorean theorem, we can calculate the length of the slanted line, the hypotenuse, as the square root of the squares of 1 and 2. The square of 1 (1X1) is 1 and the square of 2 (2X2) is 4. So, the length of the slanted line is the square-root of 5. You can use a calculator to estimate the square root of 5 as: 2.2360679775…

This is regarded as an irrational number – one whose succession of decimal digits follows no known pattern. The Greeks were intrigued by irrational numbers like this. The square root of 5 became an important number in the calculation of another important irrational number, Theta.

The value of Theta is calculated as half the value of the square root of 5, plus another half unit. Half the value of the square root of 5 is: 1.1180339887… Add .5 units to this and you get: 1.618033987… which is the value of Theta.

The Golden Rectangle

Theta is the value that defines what was known to the Greeks as the Golden Rectangle. A golden rectangle is defined as one in which the lengths of the sides are in proportion to the value of Theta.

The next figure shows a golden rectangle that is divided into a square and another rectangle.  The sides of the square can be identified as “a”, and the length of the whole rectangle can be given as: “a+b”.

In a golden rectangle, the following equation always holds:

a / b = (a + b) / a

This equation always yields the exact value of the irrational number Theta.

A Golden Rectangle Drawn from the Vesica Pisces

The rectangle enclosing the vesica piscis is NOT a golden rectangle. Its width is 1 radius unit, and its height is 2 such units. The proportion is simply 2 to 1, or 2.

But the hypotenuse of the right triangle drawn in Figure 2 can be used to construct a golden rectangle. Recall that the value of Theta can be expressed as the square root of 5, divided by 2, and adding .5.  Looking at Figure 2, it’s easy to see that the length of the hypotenuse, the square root of 5, is exactly bisected by the midline of the circles. Therefore, the length of half of the hypotenuse is the square root of 5, divided by 2. The additional .5 units can be drawn as half the length of the radius (1 unit). Knowing that this length represents the value of Theta, we can simply add one more radius unit and we have the known value of A and B.

Once this is known, it’s easy to draw the square with each side of length Theta, and an additional rectangle whose length is 1. The combination as a whole yields a golden rectangle. And the secondary rectangle contained within is also a golden rectangle, as shown below.

The Golden Rectangle became the design model for the Greek Parthenon and many other of the classical structures of the Greco-Roman era.  Many Renaissance artists, most notably Leonardo da Vinci, made extensive use of the Golden Rectangle in their paintings and architecture.

The Fibonacci Sequence and Theta

The value of Theta can also be calculated by using the arithmetic sequence known as the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…. The sequence starts with 0 and 1, with each succeeding value being the sum of the previous two. Divide each number in the sequence by its predecessor and the value will approximate Theta, getting closer and closer to it as the sequence continues. For example, 89/55 = 1.61818 and 144/89 = 1.61797.

Pursuing my interest in Sacred Geometry has opened in me a deeper curiosity and respect for some of the universal design principles used in western art and architecture.

John Bayerl, 3/11/2025