The Vesica Piscis
A Rhombus and Cross The vesica piscis consists of two equal circles positioned so that the center of each of the circles lies on the circumference of the other. Given a radius of "r", the whole vesica figure is 3r wide. The width of the overlap is 1r. Inserting the diameter line (black) and the line of the chord of the circle that connects the intersection points (blue) produces a cross inside the overlap area. Connecting the ends of this chord to the points where the diameter crosses the circles produces a rhombus made of two equilateral triangles, and four 30-60-90 degree triangles. The ratio of the radius to this chord is 15:26. If we extend the sides of the rhombus to the edge of the circle we produce hexagons that figure 3D cubes. Note that the sides of the hexagon inscribed inside a circle are equal to the radius of that circle. This double cube is the template for Kircher's Tree of Life. The A-shaped Gates of Light Tree fits inside the rhombus. Consider the Milan Cathedral plan. Notice how the edge of a circle in an equilateral triangle, touches the triangle at the corners of the inscribed hexagon. Note especially how the edges of the hexagons are pointed at corners of the pentagon. Here we insert a circle in a rhombus and use two other circles to prove that the corners are correct. Now we remove those circles, leaving only the hexagon. Next we pinpoint the top corners of the pentagon by extending the top edges of the hexagon. Inside the rhombus, the corners of the pentagon and hexagon round out the eleven stations (including Daath) on the Tree of Life.

The Vesica Piscis

A Rhombus and Cross
The vesica piscis consists of two equal circles positioned so that the center of each of the circles lies on the circumference of the other. Given a radius of "r", the whole vesica figure is 3r wide. The width of the overlap is 1r.

Inserting the diameter line (black) and the line of the chord of the circle that connects the intersection points (blue) produces a cross inside the overlap area. Connecting the ends of this chord to the points where the diameter crosses the circles produces a rhombus made of two equilateral triangles, and four 30-60-90 degree triangles. The ratio of the radius to this chord is 15:26.
If we extend the sides of the rhombus to the edge of the circle we produce hexagons that figure 3D cubes. Note that the sides of the hexagon inscribed inside a circle are equal to the radius of that circle. This double cube is the template for Kircher's Tree of Life.

The A-shaped Gates of Light Tree fits inside the rhombus. Consider the Milan Cathedral plan. Notice how the edge of a circle in an equilateral triangle, touches the triangle at the corners of the inscribed hexagon. Note especially how the edges of the hexagons are pointed at corners of the pentagon.

Here we insert a circle in a rhombus and use two other circles to prove that the corners are correct.

Now we remove those circles, leaving only the hexagon.

Next we pinpoint the top corners of the pentagon by extending the top edges of the hexagon.

Inside the rhombus, the corners of the pentagon and hexagon round out the eleven stations (including Daath) on the Tree of Life.