Connections between mathematics and the arts

The relationship between mathematics and the arts is perceived as one of the most fascinating aspects of math. Although mathematics and the arts may appear to be two completely distinct topics, there are many ways in which they intersect. According to the Engineering and Physical Sciences Research Council, "Without mathematics, we couldn't measure anything, make anything, or build anything..."1
and in consonance with Thomas Aquinas who defined art as „ars est recta ratio factibilium"2- right reason in relation to the making of things, given more than common definitions, we can see that the linkages between mathematics and the arts are surprisingly substantial.

Mathematics has been used as an integral part of art creation from antiquity to the most modern era, which is visible in various forms 3;4;5. Nevertheless, one of the strongest associations between mathematics and the arts can be proved on the basis of the golden ratio principle. The golden ratio, which derives from the Fibonacci sequence, is also known as the "divine proportion" or the letter Phi φ. The first definition of the golden ratio was provided by Euclid
in his Elements. It is based on a simple mathematical equation, in which a line is divided into two parts and the ratio of the longer part to the shorter part is equal to the ratio of the sum of
the two parts to the longer part. The value of this ratio is 1.61803398875(...).

The golden ratio comprises: the golden spiral, which is a derivative of the golden rectangle, which is derived from the golden ratio. The golden rectangle can be divided into a square and
a rectangle with sides in the golden ratio, that is, (a + b)/a = a/b = φ, where a is the width of the rectangle and a + b is its length;φ = (1 + 5)/2.

The Golden Spiral is a type of logarithmic spiral that grows wider by a factor of phi = 1.618(...) for every 1/4 turn.

Because of its abilities, the golden ratio is extremely important for artists. By taking the dimensions of an object or painting and dividing them into two parts according to the ratio, it is easier to determine the placement of a figure or object, resulting in a harmony that is aesthetically pleasing to the human eye and gives the composition its sense of balance. Additionally, symmetrical patterns and shapes can be made using this ratio, as well as a work's sense of depth and movement. Leonardo Da Vinci was one of the greatest artists in history and, despite cooperation with Pacioli6, most likely drew inspiration from mathematics for his compositions, as evidenced by the geometric model found in the "Vitruvian man", Fibonacci spiral in the composition of "Mona Lisa" or linear perspective in "The last supper"7;8;9;10

As all the above points have demonstrated, mathematics and the arts are deeply intertwined, as they both rely on structure and patterns to create something that is meaningful, and both use mathematical analysis to provide a deeper understanding, and improve upon their work. The Golden Ratio's use in art is a confirmation of the timelessness of beauty and the power of mathematics. In general, it can be said that math is the art of human understanding, and without mathematics, there is no art.