For this 5'1" Arc-Swallowtail, I wanted to find what might be the closest thing to "sacred geometry" in this design. So I first built the foundation dimensions of the tail with an equilateral triangle which defines the width of the tail at its widest point. Since all sides of this equiangular triangle are equal in length, this length also defines the radius of the circle which transcribes the arctail. The length of the equilateral triangle (and the radius of the circle) is 12".
The foam removed for the swallowtail is an isosceles triangle whose vertex is one half the angle of the equilateral or equiangular triangle, one half of 60º, which is 30º. The length of the base of this isosceles triangle is one half its height, in this case 4" wide and 8" deep.
There is a relationship between the numbers here: 1, 2, 4, 8...
"In mathematics, 1 + 2 + 4 + 8 + … is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2.
As a series of real numbers it diverges to infinity, so in the usual sense it has no sum. In a much broader sense, the series is associated with another value besides ∞, namely -1.
These relationships are proportions which can be used to scale the design up or down depending on the length of the board.