Continueing the above (1), a new geometry is found in Vitruvian Man by Leonardo da Vinci as the following in June 06, 2016.
Specifically, it is found that the canon of da vinci is built of both of a pentagon (the side = 2X) and a octagon (the side= X).
The height of the square of the canon of da Vinci is X / tangent18, and the height of circle of the canon of da Vinci is 2X / tangent18 minus X / tangent22.5.
The height of square written above is derives as described below.
The ratio between the length of a side of the regular pentagon and the height of the same pentagon is 3.077, because tangent 18 is 1/3.077 where 18 is the angle (in degrees) between the holizontal line and right-upper side of pentagon as displayed in the following figure.
Then, if half side of the pentagon is X, the heigth of the pentagon is 3.077 X.
The ratio between the height of the da vinci square and the radius of the da vinci circle must be closer to 0.6035 (derived of the square rotation) than 0.618 (golden proportion) as seen in the figure.
Here, we intend to treat the ratius as 0.6077 in the middle value.
Then the diameter of the circle is the double of 0,6077 (1.2155) times the height of the square.
The ratio between the length of a side of the regular octagon and the height of the same octagon is 2.414, because tangent 22.5 is 1/2.414 where 22.5 is the angle formed with the vertical and the straigth line that cuts the one of upper vertexes and one of the lower vertexes of the octagon.
Then, if a side of the octagon is X too (like half side of the pentagon), the ratio between the heigt of the pentagon and the height of the octagon is 0.7844.
As the height of the pentagon is equals the height of the da vinci square and equals to 1, the first equacion is 3.077 X =1.
Then, the diameter of the circle is 1.2155 and the height of the octagon is 0.7844, (and two times the height of pentagon minus one time the height of the octagon is 2- 0.7844 = 1.2155 which equals the diameter of the circle of da vinci)
Two times the height of pentagon minus one time the height of the octagon is 2*3.077X - 2.414X (where X always is a side of the octagon and half side of the pentagon), then, the height of circle of the canon of da Vinci is 2X / tangent18 - X / tangent 22.5 = 2*3.077X - 2.414X = 1.2155 where tangent18 = 1/3.077 and tangent 22.5 = 20414.
If b=0.6967a on the egg curve (approximated from b=0.7a in the original site of the egg shaped curve), the side of the square of da vinci is 2/(1+square root of 3) (= X/ tangent18).
Then, we reach the height of the egg (=1) with a square and a hexagon of side 1/(1+square root of 3).