OPINII COMUNE/ Common points of views

In nature we are surrounded by symmetry and order. People are constantly trying to understand nature and its laws, to feel the cosmic rhythms, to actually understand life more deeply, in order to reach a harmony with the environment.

The applications of the golden number, in fact of the ratio as such, are found in the proportion of works in architecture, painting, sculpture, aesthetics and art in general, which confirms the interest shown over time for this number.

The gold (divine) ratio led to the construction of the "golden rectangle", in which the ratio of the sides is equal to the gold number. This type of rectangle is considered to be particularly aesthetic and as a result it has been and is intensely used in architecture and art.

As a geometric application of the Fibonacci sequence, golden rectangles are constructed:

Among the plane curves researched in mathematics are the algebraic spirals (Archimedes' spiral, hyperbolic, etc.) and pseudo spirals (logarithmic, circle evolution, etc.). Spirals have a wide application in technology, the machine building industry, telecommunications and in the execution of construction and geodetic works.

For the logarithmic spiral, you can follow the material: https://www.geogebra.org/m/rsbvEuG2

and the tutorial in Italian https://www.youtube.com/watch?v=MMhqgQf68Jg

Feedback from the students

(R.S., Xth grade) We found other useful information in our search:

Theodorus' spiral (from Cyrene) (also called Einstein's spiral or Pythagorean spiral). It is composed of right triangles.

The spiral begins with an isosceles right triangle, with legs of length = 1. On the hypotenuse with the length of , another rectangular triangle is constructed in which the legs have the dimensions of and 1. The hypotenuse of the new rectangle is equal to . The procedure can be repeated to obtain a hypotenuse of length starting from a right triangle that has the lengths of the legs of , respectively 1.

In Geogebra, I practiced the above constructions very easily.

https://www.geogebra.org/m/QCJPrZeF

https://www.geogebra.org/m/m7eCDmdM

(T.D, VIIth grade) We really liked the activity in which we were asked to draw on a cardboard a logarithmic spiral starting from a different number of rays (depending on the number extracted, between 12 and 18).

(L.I., VIth grade) I liked the most the team work in which I coloured the logarithmic spirals.

(J. D., VIth grade) I really liked the Youtube movie: https://www.youtube.com/watch?app=desktop&v=4XN8LSfrEpU