Vesica piscis
The Cathedral of Burgos has mathematical foundations. The architects were aware of mathematics. The book "Teseros Matematicos de la Catedral de Burgos" explains the mathematics behind its architecture. The vesica pisces, for example, is the intersection of two circles with the same radius, intersecting in such a way that the center of each circle lies on the perimeter of the other. In Latin, "vesica piscis" means "bladder of a fish".
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See Wikipedia: Vesica Piscis
Exercise 0
Start an image search on "vesica pisces", collect and describe the results.
Exercise 1
Proof that the angle between the tangent lines in the intersection point is 120°.
Exercise 2
The overlapping area has a shortest diameter and a longest diameter. Say the circle radius is . So the length of the shortest diameter is . Proof that the length of the longest diameter is .
Exercise 3
Inside the vesica piscis, there is a rhombus.
Compute the length of its sides and its angles.
Exercise 4
The overlapping area has a perimeter.
Say the circle radius is .
Proof that the length of the perimeter is .
Exercise 5
The general formula of a the area of a circle segment with radius is and angle is
because the area of a circle sector is and the area of the triangle is .
Proof the general formula of a the area of a circle segment.
Exercise 6
Now, we have got a general formula of the area of a circle segment with radius is and angle . The area is .
One circle segment of is only half of the vesica piscis.
Proof that the overlapping area of the vesica piscis is .
