## Posts Tagged ‘**Euclid**’

## “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.”

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 nature, architecture, 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.”

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)

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**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.

## “The laws of nature are but the mathematical thoughts of God”*…

2,300 years ago, Euclid of Alexandria sat with a reed pen–a humble, sliced stalk of grass–and wrote down the foundational laws that we’ve come to call geometry. Now his beautiful work is available for the first time as an interactive website.

Euclid’s

Elementswas first published in 300 B.C. as a compilation of the foundational geometrical proofs established by the ancient Greek. It became the world’s oldest, continuously used mathematical textbook. Then in 1847, mathematician Oliver Byrne rereleased the text with a new, watershed use of graphics. While Euclid’s version had basic sketches, Byrne reimagined the proofs in a modernist, graphic language based upon the three primary colors to keep it all straight. Byrne’s use of color made his book expensive to reproduce and therefore scarce, but Byrne’s edition has been recognized as an important piece of data visualization history all the same…

Explore elemental beauty at “A masterpiece of ancient data viz, reinvented as a gorgeous website.”

* Euclid, *Elements*

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**As we appreciate the angles,** we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978. A logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history. Gödel had an immense impact upon scientific and philosophical thinking in the 20th century. He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.

Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement. — John von Neumann

## Everything goes better with sharks…

Sharks!

Given the successes of “Shark Week” and *Sharknado*, it’s a sure bet that Hollywood will move to remake the classics to feature those creepily-cartilaginous predators. See what to expect at Sharks Make Movies Better.

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**As we decide that it isn’t yet, perhaps, safe to go back into the water,** we might might send carefully-calculated birthday greetings to Giovanni Girolamo Saccheri; he was born on this date in 1667. A Jesuit priest and Scholastic philosopher, Saccheri is probably best remembered for his attempt to disprove the fifth postulate of Euclid (“through any point not on a given line, one and only one line can be drawn that is parallel to the given line”). In fact, Saccheri’s thinking closely mirrored that of Omar Khayyám’s 11th Century *Discussion of Difficulties in Euclid* (*Risâla fî sharh mâ ashkala min musâdarât Kitâb ‘Uglîdis*)– though it’s not clear that Saccheri knew the earlier work.

In any case, Saccheri’s *Euclides ab omni naevo vindicatus* (*Euclid Freed of Every Flaw*, 1733) helped lay the foundation for what we now call Non-Euclidean Geometry.

## Got your number…

Euler’s Number (e): 2.7182…Euler’s number is also known as the exponential growth constant. It is the base for natural logarithms and is found in many areas of mathematics.

Application: In finance, Euler’s number is used to determine compound interest, which is extremely vital in understanding the time value of money — the backbone of finance.Moreover, Euler’s number is crucial when describing any decaying relationship – think Carbon 14 dating.

… and then there are the other 9 “

10 Most Important Numbers in the World.”

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**As we activate the abaci,** we might send carefully-calculated birthday wishes to Stephen Smale; the Wolf Prize-winning mathematician was born on this date in 1930. Among many other accomplishments, Smale helped develop the logistic model for population growth– one of the foundational insights that allowed the development of chaos theory (and thus, enhanced our understanding of the way in which natural systems actually work)– one of the **17 Equations That Changed the World**: