Vesica Piscis : inspection of square root of 3.

The creation of Vesica Piscis, consists of having two points in the 2-dimentional space first, A and B. They both can make a full rotation each on another( i.e A as center and AB as diameter and vice-versa).

In the following figure we inspect the created shape so far.

Figure 1.

We can then connect with straight lines A, B together as well as with C and D the points where the 2 circles intersect.

Figure 2

Assuming then as the radius of the two same circles to be x , we get that: AB = AC = AD = BC = BD = x. And triangles ACB, and ABD are equilateral thus A1+A2 = B1+B2 = 120 degrees while C1+C2 = 60 = D1+ D2.

By applying the laws for same triangles we can also see that CD intersects AB exactly in the middle of its length.

Irrational number square root of 3 will appear when we try to calculate the length of CD (the width of the Vesica Piscis).

By applying the Pythagorean theoreum in any of the 4 inner triangles we get:

x²/4 +CD²/4 = x² . Which leads to CD²/4 = (3/4)*x² and eventually in the result that CD = sqrt(3) * x².

So if x is equal to one, then we can see that the length of CD will be equal to the irrational number of the square root of 3.