Samantha Pezzimenti
As a mathematics graduate student and future educator, I am interested in the ways artistic representations and physical models can help students understand abstract mathematical concepts. This painting provides a visual representation for an infinite continued fraction. Furthermore, it exhibits the paradoxical nature of the square root of two. The painting positions the simplicity of the hypotenuse of an isosceles right triangle against the fractal-like complexity of its infinite continued fraction representation.
Any real number has a continued fraction representation of the form a+1/(b+1/(c+1/(d+...))), denoted by [a;b,c,d,...]. The continued fraction of any square root has a repeating pattern. In particular, the continued fraction of sqrt(2) is [1;2,2,2,...]. In the painting, the ratio between the two sides of the largest rectangle is sqrt(2):1. The blue square is the largest square that fits inside the rectangle. Since only one square fits, the first term of the continued fraction is 1. Similarly, the green square is the largest that fits in the remainder rectangle, but this time two squares fit. This process continues infinitely with exactly two squares fitting in each subsequent remainder rectangle.