Pacioli and Divine Proportion | by Ruchika Matto

What makes a single number so captivating that the ancient Greeks, the most acclaimed painter and inventor Leonardo da Vinci, a modern historical novelist Dan Brown and many others write about it? The answer lies within the name given to this number which is known by many names such as Golden Ratio, Golden Number, Golden Proportion, Divine Proportion, etc. The “Golden” number is 1.618 commonly represented by the 21st Greek letter Phi named after Phidias, a Greek sculptor who made statues using the Golden Ratio. It is revered as ‘divine’ due to its ubiquity in every sphere such as mathematics, biology, psychology, architecture and even in arts, music, etc.

Some of the greatest minds in history like Pythagoras, Euclid, Fibonacci, Johannes Kepler, etc. have spent endless hours over this simple ratio and its properties. Although Euclid defined it around 300 BC, and the followers of Pythagoras probably knew of it two centuries earlier, it was Pacioli’s three volume treatise, “De Divina Propaortione” (The Divine Proportion), that was crucial in disseminating the Golden Ratio beyond the world of mathematics. This book was written by Luca Pacioli himself but illustrated by the great Leonardo da Vinci, composed around 1498 in Milan and first printed in 1509. Pacioli believed that there's only one value for the Divine Proportion and only one Christian God, that's why he renamed the extreme and mean ratio by Euclid, the Divine Proportion which shows his supernatural interests.

According to Pacioli, the Golden Ratio results when a line is divided so that the short portion is to the longer portion as the longer is to the whole, and is expressed numerically as 1.61830339887... ‘The Father of Accounting and Bookkeeping’ has described the mathematics of the Divine Proportion and its relation to the heavenly spheres. He has actually explained all the material forms of geometric bodies which have been unknown to the living at that time. These geometric bodies - a collection of regular solids and semi-regular solids - which had never before been visually represented. The representation’s credit goes to the art-master Da Vinci whose spatial imagination to represent the 3-D figures in 2-D space, made them more like real objects.

“Some of the greatest minds in history like Pythagoras, Euclid, Fibonacci, Johannes Kepler, etc. have spent endless hours over this simple ratio and its properties.”

He has explored the relation of the Golden Ratio to the five Platonic (regular) solids; cube, tetrahedron, octahedron, icosahedron and dodecahedron (used and developed by the Pythagoreans). He has also explained how complex geometrical figures were used in architecture, for example, the 26-sided rhombi cuboctahedron and its stellated truncated form with 72 sides were used in the construction of the Pantheon in Rome and the Santa Maria delle Grazie in Milan.

Thus, Pacioli has changed the world of mathematics by introducing the Divine Proportion and 3-dimensional figures with the help of Da Vinci, which are now the building blocks of the world.