# 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.