### The Mathematics of the Vesica Piscis

The vesica piscis is a type of lens, a mathematical shape formed by the intersection of two circles with the same radius.

For every angle we get two triangles as seen here. As you can see, these triangles have a very interesting relationship, for every angle the radius (r) = h1 + h2.

__Explore__ the mathematics of the vesica piscis (mandorla) in detail.

The symmetry of these triangles is shown below.

For every angle the radius (r) = h1 + h2.

Showing both symmetries at once.

### The Cosmic Identity

This mathematical identity works for ALL* numbers (x).

__Explore__ the mathematics of the cosmic identity (cosmic numbers) in detail.

### The Cosmic Triangle

We can link the cosmic identity with pythagoras theorem to get the following cosmic triangle. Substituting x = 4/π and 1/x = π/4 as shown below.

Because of the nature of the cosmic identity, the triangles above will work with any value x and π.

__Explore__ the mathematics of the cosmic triangle (cosmic triangles) in detail.

### Vesica Piscis and Cosmic Triangle

Using the cosmic triangle above in a vesica piscis, where both circles have radius (r) = 4/π + π/4.

As seen below the hypotenuse of the blue triangle = 8/π.

8/π = (4/π + π/4) + (4/π – π/4) and height (h1) = 4/π – π/4.

The small circle has a radius (h1) = 4/π – π/4 and a diameter (2 × h1) = 8/π – π/2.

Therefore the height (h2) = π/2 = (4/π + π/4) – (4/π – π/4).

Because of the symmetry of the vesica piscis.

Therefore the height (h2) = π/2 = (4/π + π/4) – (4/π – π/4).

### The Angle of Perfect Symmetry

At ONLY this special angle, the chord (c) = r + h1. As seen in the mandorla.

All symmetries together.

### The Pi Triangle

As we can see above, it seems the following two triangles must be correct.

8/π = (4/π + π/4) + (4/π – π/4) AND π/2 = (4/π + π/4) – (4/π – π/4)

If we use pythagoras theorem and the blue pi triangle to calculate pi (π), we get the following value.

This value is different to calculator pi.