Golden Ratio cover

Race Point Publishing
Oct 23, 2018
91/4 x 113/8 in., 224 pages
ISBN: 9781631064869

I have often used the golden ratio in my design work. I have to confess that this sometimes reflected a kind of laziness: the golden ratio is always pretty sure to look good, and it is easy to implement. But why does it look good? Partly because its proportions are found throughout nature and in the human face and body. But devotees of a mystical bent in the tradition of Pythagorus — those who, like Johannes Kepler, find the signature of divinity in the mathematics of the natural world — will feel that the importance of the proportion goes beyond that.

Gary B. Meisner’s The Golden Ratio: The Divine Beauty of Mathematics, a handsome, richly illustrated oversized hardcover book, attempts to explain the ratio and demonstrate its occurrence in art and nature. He begins by defining the ratio and considering its unique properties.

In my work I tend to think of the ratio in terms of rectangles, which are often the building blocks of book pages. In Robert Bringhurst’s Elements of Typographic Style, for example, he describes a method for setting a type block in the shape of a golden rectangle in an ISO-sized page:

Bringhurst on golden section

But in its simplest form the ratio can be expressed in just a line. In any line there is a point at which it is divided into two segments, of which the longer is to the shorter as the whole is to the longer. The ratio defined by that point, known in mathematics as phi, was formerly described as marking “the mean and the extreme.” It’s value is 1.618033988749895…, conventionally rounded to 1.618 for many practical purposes. According to Meisner, it only became termed “golden” in the nineteenth century.

Line divided into golden ratio segments.
Illustration from Gary B. Meisner, The Golden Ratio.

Phi is a number with many curious attributes. Adding one gives its square, subtracting one gives its reciprocal, multiplying it by itself given an endless series with the same proportion — the list goes on. It also has an affinity for the number five, and for pentagons, a form “basic to many things,” as Bringhurst says, “from roses and forget-me-nots to sea urchins and starfish.” Meisner notes that several phi relationships can be found in a five-pointed star, which may, he says, have been the symbol of Pythagorus and his school.

Illustration from Gary B. Meisner, The Golden Ratio.

If we connect the points of the star we construct a pentagram. The triangular pointed and outer areas surrounding the interior pentagram can then be assembled in various ways in what are called Penrose tiles, after mathematical physicist Sir Roger Penrose (b. 1931).

Illustration from Gary B. Meisner, The Golden Ratio.

I found Meisner’s discussion of the relation between pentagram and the golden ratio particularly interesting. But he loses me when he moves on to discover instances of the ration seemingly everywhere he looks in art and nature, and this makes up about three-quarters of the book. For example, he finds several instances of the golden ratio in Georges Seurat’s Bathers in Asnières (1884).

Illustration from Gary B. Meisner, The Golden Ratio.

To me some of these overlays seem a bit arbitrarily placed. There is always the question of where to begin to draw the rectangles. I suspect that if other proportions — 1:3 or 1:4, say — were searched for they might also be found. My dissertation director at the University of Wisconsin, David Hayman, called this “the blueberry principle.” When you go gathering blueberries the main thing you are going to notice is blueberries, and you are not going to notice other things as much as you otherwise would.

And what is one to make of these proportions in this painting? Do they help us to better understand the work or to make our own artworks? I’m not so sure.

Meisner’s book is written in a popular tone, which may appeal to those who otherwise find math off-putting. It also has a promotional quality. Meisner urges the reader to visit his website,, and to download PhiMatrix, his “design and analysis” software, which includes “dozens of customizable grids and templates,” and allows you to “overlay any image on your screen” to “analyze and design in seconds.”

Although I remain largely unpersuaded by Meisner’s survey of the golden ratio in artworks, his book is well produced and rich with images. It makes an enjoyable casual read for nonspecialists intrigued by this fascinating ratio.