Doubling the Circle
Ad Quadratum and ad Triangulum
In 1521 Cesariano published an image (below) that purported to show how the Milan Cathedral plan was designed. The main elements of the image that he presents are three concentric circles, three concentric triangles and a 60 degree grid pattern inside the central circle. The Milan Cathedral was laid out using the double the circle template as I will now demonstrate.
In the image above you can see that the top of the yellow equilateral triangle reminds us of the compass, and the bottom of the blue square reminds us of the square; the compass and square represent the triangulum and quadratum. A draftsman uses a 30-60-90 triangle and a 45-45-90 triangle that represent the same thing.
We begin with a vertical line (like the number 1 and the letter I); we cross that with a horizontal line and mark the point where they intersect, then we center a circle on the crossing point. The four points where the cross touches the circle mark the four corners of a square.
You can see this figure in the central circle in the cathedral plan above. This is the quadratum form used mainly for floors, which features 45 and 90 degree angles.
shows a rare quadratum elevation.
Looking at the DC map we can see the the District of Columbia is an oblique square (like above) that measures 10 x 10 miles, and that the streets are oriented N-S E-W like the crossing lines in the image.
If we center two more circles on the center line and on the circumference of circle one, we mark the boundaries of a circle twice the size of the first one, and we mark the corners of a hexagon and equilateral triangles inside circle one, just as in the cathedral plan. That is, the corners of the hexagon in the cathedral were located by three circles. [Note that the equilateral triangle and the grid inside the circle results from diagonals lines of the hexagon.]
Two large circles centered on the center line and on the circumference of circle two divide it into rhombus, triangles and hexagons as well. Note the red triangle in the catherdal image. This combination of circle, hexagon, rhombus and triangle is the triangulum form and features 30, 60 and 90 degree angles.
Metatron's Cube is a doubled hexagram. Extending the top sides of the hexagon to the sides of the triangle (above right) locates the top points of a pentagon and places the Tree of Life inside Metatron's Cube. John Mitchell's Glastonbury Plan (below) is based on the same figure (the double circle/hex, but without the quadratum or the pentagon).
Returning to the cathedral image, we add two more small circle and one large one. Next we project the diagonals of the hexagon to the edge of the large circle locating the points of the large hexagram, part of which matches the roof line. The width of the nave is indicated by the blue lines, what you see that sticks past that is the crossing part of the building (the building is shaped like a cross). The nave is as wide as the hexagon that inscribes the central circle.
In terms of the (ideal, pre-shrunk) map grid, you can see that the horizontal part of that is marked by five points on the center line, meaning that the hexagram quarters the diameter of the circle. Vertical axes are the center line, the lines through the corners of the central hexagon and the lines through two of the corners of the large hexagram. [Note that the grid is as tall as the circle but only as wide as the hexagon, meaning it is taller than it is wide.]
The DC Map
New Hampshire Ave forms the left side of what is the large red (upright) triangle in the cathedral plan, but that triangle is not equilateral in the DC map, where the base angle is ~ 52 degrees like the Great Pyramid. The template was reduced in height to about 75% of the original to produce a triangle that matched the pyramid cross-section. Looking at the pentagram in the map, you can see that it is short and wide. South of the White House there is an ellipse, which is a short wide circle.
Scott Circle (seen below), located north of the WH, is a short wide hexagram with an ellipse and point inscribed within it. The diagonal streets at Scott Cr form 23.5 degree angles.
The DC map template was adjusted so that the triangle (New Hampshire) has 52 degree base angles and the diagonals formed by the streets connecting the points of the pentagram make 23.5 degree angles.
Above, John Michell shows his interpretation of the Stonehenge plan, based on the doubled circle. Below, I show how the shortened hexagram matches the locations of the stations stones.