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Posts Tagged ‘Pythagoras

“The golden ratio is the key”*…

… in any case, to good design. So, how did it come into currency? Western tradition tends to credit the Greeks and Euclid (via Fibonacci), while acknowledging that they may have been inspired by the Egyptians. But recent research has surfaced a a more tantalizing prospect:

Design remains a largely white profession, with Black people still vastly underrepresented – making up just 3% of the design industry, according to a 2019 survey

Part of the lack of representation might have had to do with the fact that prevailing tenets of design seemed to hew closely to Western traditions, with purported origins in Ancient Greece and the schools out of Germany, Russia and the Netherlands deemed paragons of the field. A “Black aesthetic” has seemed to be altogether absent.

But what if a uniquely African aesthetic has been deeply embedded in Western design all along? 

Through my research collaboration with design scholar Ron Eglash, author of “African Fractals,” I discovered that the design style that undergirds much of the graphic design profession today – the Swiss design tradition that uses the golden ratio – may have roots in African culture

The golden ratio refers to the mathematical expression of “1: phi,” where phi is an irrational number, roughly 1.618. 

Visually, this ratio can be represented as the “golden rectangle,” with the ratio of side “a” to side “b” the same as the ratio of the sides “a”-plus-“b” to “a.” 

The golden rectangle. If you divide ‘a’ by ‘b’ and ‘a’-plus-‘b’ by ‘a,’ you get phi, which is roughly 1.618

Create a square on one side of the golden rectangle, and the remaining space will form another golden rectangle. Repeat that process in each new golden rectangle, subdividing in the same direction, and you’ll get a golden spiral [the image at the top of this post], arguably the more popular and recognizable representation of the golden ratio.

This ratio is called “golden” or “divine” because it’s visually pleasing, and some scholars argue that the human eye can more readily interpret images that incorporate it.

For these reasons, you’ll see the golden ratio, rectangle and spiral incorporated into the design of public spaces and emulated in the artwork in museum halls and hanging on gallery walls. It’s also reflected in naturearchitecture, and design – and it forms a key component of modern Swiss design.

The Swiss design style emerged in the 20th century from an amalgamation of Russian, Dutch and German aesthetics. It’s been called one of the most important movements in the history of graphic design and provided the foundation for the rise of modernist graphic design in North America.

The Helvetica font, which originated in Switzerland, and Swiss graphic compositions – from ads to book covers, web pages and posters – are often organized according to the golden rectangle. Swiss architect Le Corbusier famously centered his design philosophy on the golden ratio, which he described as “[resounding] in man by an organic inevitability.”

An ad for Swiss Air by graphic designer Josef Müller-Brockmann incorporates the golden ratio. Grafic Notes

Graphic design scholars – represented particularly by Greek architecture scholar Marcus Vitruvius Pollo – have tended to credit early Greek culture for incorporating the golden rectangle into design. They’ll point to the Parthenon as a notable example of a building that implemented the ratio in its construction.

But empirical measurements don’t support the Parthenon’s purported golden proportions, since its actual ratio is 4:9 – two whole numbers. As I’ve pointed out, the Greeks, notably the mathematician Euclid, were aware of the golden ratio, but it was mentioned only in the context of the relationship between two lines or figures. No Greek sources use the phrase “golden rectangle” or suggest its use in design.

In fact, ancient Greek writings on architecture almost always stress the importance of whole number ratios, not the golden ratio. To the Greeks, whole number ratios represented Platonic concepts of perfection, so it’s far more likely that the Parthenon would have been built in accordance with these ideals.

If not from the ancient Greeks, where, then, did the golden rectangle originate? 

In Africa, design practices tend to focus on bottom-up growth and organic, fractal forms. They are created in a sort of feedback loop, what computer scientists call “recursion.” You start with a basic shape and then divide it into smaller versions of itself, so that the subdivisions are embedded in the original shape. What emerges is called a “self-similar” pattern, because the whole can be found in the parts… 

Robert Bringhurst, author of the canonical work “The Elements of Typographic Style,” subtly hints at the golden ratio’s African origins:

“If we look for a numerical approximation to this ratio, 1: phi, we will find it in something called the Fibonacci series, named for the thirteenth-century mathematician Leonardo Fibonacci. Though he died two centuries before Gutenberg, Fibonacci is important in the history of European typography as well as mathematics. He was born in Pisa but studied in North Africa.”

These scaling patterns can be seen in ancient Egyptian design, and archaeological evidence shows that African cultural influences traveled down the Nile river. For instance, Egyptologist Alexander Badaway found the Fibonacci Series’ use in the layout of the Temple of Karnak. It is arranged in the same way African villages grow: starting with a sacred altar or “seed shape” before accumulating larger spaces that spiral outward.

Given that Fibonacci specifically traveled to North Africa to learn about mathematics, it is not unreasonable to speculate that Fibonacci brought the sequence from North Africa. Its first appearance in Europe is not in ancient Greece, but in “Liber Abaci,” Fibonacci’s book of math published in Italy in 1202. 

Why does all of this matter?

Well, in many ways, it doesn’t. We care about “who was first” only because we live in a system obsessed with proclaiming some people winners – the intellectual property owners that history should remember. That same system declares some people losers, removed from history and, subsequently, their lands, undeserving of any due reparations. 

Yet as many strive to live in a just, equitable and peaceful world, it is important to restore a more multicultural sense of intellectual history, particularly within graphic design’s canon. And once Black graphic design students see the influences of their predecessors, perhaps they will be inspired and motivated anew to recover that history – and continue to build upon its legacy.

The longer-than-we’ve-acknowledged history of the Golden Ratio in design; Audrey Bennett (@audreygbennett) unpacks “The African roots of Swiss design.”

For more on Fibonacci‘s acquisitive habits, see this earlier post.

* Sir Edward Victor Appleton, Nobel Laureate in physics (1947)


As we ruminate on relationships, we might send careful-calculated birthday greetings to Mary Jackson; she was born on this date in 1921. A mathematician and aerospace engineer, she worked at Langley Research Center in Hampton, Virginia (part of the National Advisory Committee for Aeronautics [NACA], which in 1958 was succeeded by the National Aeronautics and Space Administration [NASA]) for most of her career. She began as a “computer” at the segregated West Area Computing division in 1951; in 1958, she became NASA’s first black female engineer.

Jackson’s story features in the 2016 non-fiction book Hidden Figures: The American Dream and the Untold Story of the Black Women Who Helped Win the Space Race. She is one of the three protagonists in Hidden Figures, the film adaptation released the same year. In 2019, she was posthumously awarded the Congressional Gold Medal; in 2020 the Washington, D.C. headquarters of NASA was renamed the Mary W. Jackson NASA Headquarters.


“In so far as the mind sees things in their eternal aspect, it participates in eternity”*…



David Ramsay Hay’s mapping of color onto musical notes, a diagram from his The Laws of Harmonious Colouring (1838)


“All art constantly aspires towards the condition of music.” So wrote the Victorian art critic Walter Pater in 1888. Earlier in the century, Scottish artist David Ramsay Hay composed a series of fifteen books published between 1828 and 1856 that attempted to develop a theory of visual beauty from the basic elements of music theory. Anticipating Pater but also fin-de-siècle attempts to unite the arts via spiritual or synesthetic affinities, Hay’s writings mapped colors, shapes, and angles onto familiar musical constructs such as pitches, scales, and chords. While these ideas might appear highly eccentric today, an understanding of them offers a glimpse of the remarkable importance of music to the Victorian Zeitgeist…

Hay’s approach to visual aesthetics was equally applicable to architecture, color theory, the ornamental arts, and the human face and figure. It can be understood as a psychological account of beauty, as opposed to other contemporary theories that anchored beauty in notions of the picturesque, the mimetic, or the sublime. Though analogies between music and the fine arts certainly do not originate with Hay, his application of music theory to an extensive array of visual experiences including color, shapes, figures, and architecture broke new ground. Rather than locating musical properties in the objects themselves, as earlier thinkers ranging from Plato to Newton had done, Hay worked in the post-Kantian tradition, regarding these features as immanent to our own minds, where they create our experience of beauty by determining the very structure of our perceptions…

Throughout his writings, Hay consistently links the claim that a single fundamental law of nature determines aesthetic perception to the work of the philosopher and mathematician Pythagoras…

Understanding the same laws to apply to both visual and aural beauty, David Ramsay Hay thought it possible not only to analyze such visual wonders as the Parthenon in terms of music theory, but also to identify their corresponding musical harmonies and melodies: “Music of the Squares: David Ramsay Hay and the Reinvention of Pythagorean Aesthetics.”

* Baruch de Spinoza


As we excavate the essential, we might send elegantly-composed birthday greetings to Mary Cassatt; she was born on this date in 1844.  An American printmaker and painter, she moved to Paris as an adult, where she developed a friendship with Edgar Degas and became, as  Gustave Geffroy wrote in 1894, one of “les trois grandes dames” of Impressionism (with Marie Bracquemond and Berthe Morisot).

Self-portrait, c. 1878




Written by (Roughly) Daily

May 22, 2019 at 1:01 am

“All practical jokes, friendly, harmless or malevolent, involve deception, but not all deceptions are practical jokes”*…



When you think of the ancient Greeks, practical jokes might not be the first thing that comes to mind. But along with art, architecture, and philosophy, you can add trick cups to their list of accomplishments.

The Pythagorean cup is so-named because it was allegedly invented by Pythagoras of Samos (yes, the same guy who gave us theories about right triangles). It’s a small cup with a column in its center. It doesn’t look like much, but when an unsuspecting drinker fills it past a designated level, the liquid mysteriously drains out. Legend has it that Pythagoras used it as a way to punish greedy drinkers who poured themselves too much wine…

A timeless practical joke, brought to you by the ancient Greeks: more merriment at “Pythagorean Cup.”

* W. H. Auden, The Dyer’s Hand


As we ponders pranks, we might send a “Alles Gute zum Geburtstag” to the polymathic Gottfried Wilhelm Leibniz, the philosopher, mathematician, and political adviser, who was important both as a metaphysician and as a logician, but who is probably best remembered for his independent invention of the calculus; he was born on this date in 1646.  Leibniz independently discovered and developed differential and integral calculus, which he published in 1684;  but he became involved in a bitter priority dispute with Isaac Newton, whose ideas on the calculus were developed earlier (1665), but published later (1687).

As it happens, Leibnitz was no mean humorist.  Consider, e.g…

If geometry conflicted with our passions and our present concerns as much as morality does, we would dispute it and transgress it almost as much–in spite of all Euclid’s and Archimedes’ demonstrations, which would be treated as fantasies and deemed to be full of fallacies. [Leibniz, New Essays, p. 95]

28134677537_d79a889e6a_o source


Written by (Roughly) Daily

July 1, 2018 at 1:01 am

This is cool, but I’m holding out for a disease…


Hankering for a little immortality?  New Scientist has the answer:

While most mathematical theorems result from weeks of hard work and possibly a few broken pencils, mine comes courtesy of TheoryMine, a company selling personalised theorems as novelty gifts for £15 a pop.

Its automated theorem-proving software can churn out a theoretically infinite number of theorems for customers wishing to join the ranks of Pythagoras and Fermat. “We generate new theorems and let people name them after themselves, a friend, a loved one, or whoever they want to name it after,” explains Flaminia Cavallo, managing director of TheoryMine, based in Edinburgh, UK…

“We’re inventing totally novel theorems, and the tradition is you have the right to name these theorems,” explains Alan Bundy, professor of automated reasoning at the University of Edinburgh and another member of the TheoryMine team. “There are 10 star companies out there, and none of them have any affiliation to the International Astronomical Union.”

He’s got a point. Automated theorem proving is a well-respected mathematical field, used by manufacturers to guarantee that the algorithms in computer processors will work correctly. Bundy and his colleagues have worked in this area for a number of years, and Cavallo came up with the idea for TheoryMine during her final year of an undergraduate degree in artificial intelligence and mathematics at the University of Edinburgh, where she wrote a program to generate novel theorems for her dissertation.

From its library of mathematical knowledge, the program generates a set of mathematical axioms, then combines them in different ways to produce a series of conjectures. It then uses the library to discard a portion of these on the basis that there are already counter-examples, showing they can’t be true. Overly complex conjectures are also ignored. Then it applies a technique known as “rippling”, in which it tries out various sequences of logical statements until one of these sequences turns out to be a proof of the theorem…

“It’s a clever idea,” says Lawrence Paulson, a computational logician at the University of Cambridge and the creator of Isabelle, a theorem prover that Cavallo’s program uses. He is more interested in the theory behind the new program though, adding that “some of the technology here is quite impressive, and I would hope that it finds other applications apart from selling certificates”.

It may well do. Lucas Dixon, another TheoryMiner, is investigating the possibility of using the same techniques to elucidate the rules of algebra in quantum computing systems, which follow different mathematical rules to classical systems.

Don’t prepare your Fields medal acceptance speech just yet though, as TheoryMine’s theorems are unlikely to break drastically new ground. “We can’t say that we’ll never do that, but having looked at the things that come out, they’re not typically things that are going to change the world,” says Dixon.

Your correspondent just purchased “Eleanor’s Equation” for his daughter; reader’s can score their own mathematical monument at TheoryMine.

As we search for the “rum” in theorum,” we might wish a Buon Compleanno to Count Francesco Algarotti, the philosopher, critic, and popularizer of complex scientific ideas; he was born in Venice on this date in 1712– and wrote Neutonianismo per le dame (Newtonism for Ladies) when he was 21.



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