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A Pythagorean triple is an ordered triple of integers (a,b,c) = (0,0,0) such that a² +b² = c². It is well known that the set P of all Pythagorean triples has an intrinsic structure of commutative monoid with respect to a suitable binary operation, (P,*). In this article, we will introduce the "commensurability" relation R among Pythagorean triples, and we will see that it induces a group quotient, P/R, which is isomorphic with the direct product of infinite (countable) copies of C∞, the infinite cyclic group, and a cyclic group of order 4. As an application, we will see that the acute angles of Pythagorean triangles are irrational when measured in degrees. [ABSTRACT FROM AUTHOR]

In this article, we introduce the notion of continue quadrilateral, that is a quadrilateral whose sides are in geometric progression. We obtain an extension of the principal result referring to the growth of continue triangles. Precisely, we will see that the growth of a continue quadrilateral belongs to the interval (1/Φ2, Φ2), where Φ2 is the Silver mean. The main result is that in any circle a continue quadrilateral of growth µ can be inscribed for every µ belonging to the interval (1/Φ2, Φ2). Our investigation is supported by dynamical software. [ABSTRACT FROM AUTHOR]

In this article we will use some special triangles, to construct polygonal chains that describe the families of logarithmic spirals, among which are the celebrated Golden Spiral, Spira solaris and Pheidia Spiral. [ABSTRACT FROM AUTHOR]