# Gabriele Gelatti

Despite the mystery on the nature of the human thought, still mind and nature share something: numbers. The primitive aspect of numbers as figures and patterns creates “sense” to the mind, and even “beauty”.

Any creation of human thought faces the idea of beauty, even to refuse it. If a universal definition of beauty is still missing, it remains the most effective link between art and mathematics, demonstrating that “true is beautiful”.

Golden section and related Fibonacci numbers express the convergence of both mathematical and symbolic thoughts: the beauty of their contents can be experienced by the two points of view.

The research of Gabriele Gelatti explores the archetypes of the symbolic thought to find new mathematical objects.

Definitions: α = (√(5) + 1) / 2, and β = (√(5) – 1) / 2.

The image shows an original construction of the pentagon by mean of a rectangle with sides α and β.

The interest is also given by the idea that this particular rectangle is produced by colours. These colours are the “translation” of positive integers, using the digital root operation. The pattern displays all possible Fibonacci-like sequences produced by the initial rule of “adding 3 to numbers from 1 to 9”. The structure is made by rectangles of 6 colours in a grid of black-grey-white.

The possible sequences are only 4, and they make a cycle of 24, divided in complementary cycles of 12. The full idea is shown in Bridges 2014 paper “On Colouring Sequences of Digital Roots”.

Definitions: α = (√(5) + 1) / 2, and β = (√(5) – 1) / 2.

The image shows an original design of the golden section with circles.

Three different circles (in blue) divide the diameter of a given circle. The biggest blue circle measures half diameter; the medium one is β value of the biggest one; the smallest circle is the β value of the medium one.

Another sequence of circles (in gold) is constructed by dividing the biggest blue circle in two halves: if the diameter of the first gold circle is β, then the sequence of gold circles is the succession of powers of β. The entire sequence develops from the center of the biggest blue circle to the center of the smallest blue circle.

The sum of β with all its successive powers is equal to α.