Search

Vesica Piscis is a figure generated intersecting two circles having the same radius and such that the center of each circle lies on the exterior circumference of the other circle. The origin of the Vesica Piscis is ancient and unknown and this symbol has been present in all the ages up to nowadays. It has been proved that if two segments are traced joining particular points of the Vesica Piscis, these segments form a cross and their ratio is equal to the square root of three. One segment joins the two intersecting points of the two circumferences and the other segment joins the two intersecting points between the segment joining the two centers with the exterior circumferences themselves. In this article, we prove that the converse is also true: if we consider two intersecting circles and the aforementioned segments such that their ratio is equal to the square root of three, then the two circles form a Vesica Piscis. We prove this results through two alternative proofs. The first proof uses Euclidean geometry and the second proof uses analytic geometry.

In this paper we will consider a regular pentagon and discuss three of its properties, which are linking side, radius, diagonal and apothem to the golden ratio. One of the properties, that regarding the ratio between the diagonal and the radius of the circumscribed circumference is strictly connected to the construction of the regular pentagon with compass and straightedge.

Regular pentagon is a geometric figure which had a relevant symbolic meaning in Pythagorean and Platonic philosophies and a subsequent important role in the Western thought, appearing also in arts and architecture. A property of the pentagon, which was probably found by Pythagoreans, is that the ratio between diagonal and side is the golden ratio. Here, we will study some links between regular pentagon and golden ratio. We will focus first on the group of five-fold rotation symmetry, to find the position of the vertices of this geometric figure in the complex plane; then, we will give an analytic method to solve the same problem in the Cartesian plane. In the analytic approach, we find the golden ratio without any specific geometric consideration. This study can be interesting also for a general education in mathematics, because it can convey and link several concepts, requiring only a general pre-college education.

In this paper, we are proposing a study of the Vesica Piscis, a symbol of Sacred Geometry, starting from mathematics. This study aims to give a deeper comprehension of the hidden meanings of this icon. The paper is also showing that scientific and philosophical studies can be integrated, leading to very interesting results., Suggested citation: Sparavigna, Amelia Carolina and Baldi, Mauro Maria, A Mathematical Study of a Symbol: The Vesica Piscis of Sacred Geometry (January 23, 2016). PHILICA, article number 560, Available at SSRN: https://ssrn.com/abstract=2748242

The work reported in this thesis is about one of the many branches of convex optimization: the truss structures design. A truss is a collection ofbars which meet each other at certain points - the nodes - to form a planar or spatial lattice. Truss structures are ubiquitous and widespread inthe industrialized and civil world; they appear as bridges, towers, roof supports, building exoskeletons, cranes and so on everywhere. The most famousand popular example of truss structure is the Eiffel tower, which has morethan 15000 girders connected at over 30000 points. These numbers givethe straightforward idea that it would be almost impossible (particularlyin complex cases) and tedious to design a truss structures by hand; hencedesign algorithms are required. The philosophy at the basis of this thesis is to handle a robust or probabilistic truss structures design towards uncertainty associated to the dataof the problem by providing to the processor the lattice geometry (i.e. position of nodes where the bars meet each other), the fixed nodes holding thestructure and, finally, the external forces acting on the truss.