Here is a graphic showing the sequence q *|q ϕ-round(q ϕ)|. For instance, it is the maximally badly approximable irrational number: Its continued fraction representation is as simple and beautiful as a mathematical expression can get:Īnother similar form is the following iterated square root:Īlthough just a simple square root, mathematically the golden ratio is a special number. Its base 2 or base 10 digit sequences are more or less random digit sequences: The golden ratio ϕ=(1+ )/2≈1.618033988… is a special number in mathematics. Prelude: The golden ratio, a beautiful construction in mathematics We will discover that the trend over the last century or so is to become more rationalist. We can find out more about Fechner with the following piece of code:īy 1876 standards, Fechner did amazing work, and we can redo some of his analysis in today’s world of big data, infographics, numerical models, and the rise of digital humanities as a scholarly discipline.Īfter a review of the golden ratio and Fechner’s findings, we will study the distribution of the height/width ratios of several large painting collections and the overall distribution, as well as the most common aspect ratios for paintings. To validate his experimental observations, Fechner also analyzed the aspect ratios of more than ten thousand paintings. In 1876, the German scientist Gustav Theodor Fechner studied human responses to rectangular shapes, concluding that rectangles with an aspect ratio equal to the golden ratio are most pleasing to the human eye.

But what about their outer shape-the ratio of a painting’s height to its width? Their composition, colors, details, and themes can fascinate us for hours.

They are treasured and admired, carefully preserved, sold for hundreds of millions of dollars, and, perhaps not coincidentally, are the prime target of art thieves. Paintings of the great masters are among the most beautiful human artifacts ever produced.

Aspect ratio calculator art