MATHEMATICAL PROPERTIES OF THE GOLDEN RATIO – A FASCINATING NUMBER

Abstract

The Golden Ratio appears in many situations: in geometry, in lists with special numbers drawn up by mathematicians, in art history lectures, in rose petals arrangement or spiral shells of mollusks. This paper presents definitions and algebraic properties of Golden Section and their applications in geometry, in the calculation pentagon and decagon sides and area, and volume of Platonic solids. Golden Ratio occurrences are analyzed in geometric shapes as Golden Triangle, Golden Rectangle and Golden Gnomon. Different expressions involving radicals that continue indefinitely and continued fractions with Golden Section value are given and discussed. It is also revealed the connection between the Fibonacci series and the Golden Ratio. Are presented two patterns that have symmetry of order five and that have incorporated in them the Golden Section.

Keywords

• proportion, pentagram, pentagon, Golden Triangle, Golden Gnomon, Golden Ratio.