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Posts Tagged ‘Euclid

“Sex and death are the only things that can interest a serious mind”*…

As Greg Woolf observed, “The Epic of Gilgamesh is one of the earliest examples of what is sometimes termed a “Mirror of Princes,” a book that illustrates the conduct of both bad and good rulers, and makes clear the difference between them.”

Nicolas Liney reviews a new verse translation of the 4,000-year-old text by Simon Armitage and considers its remarkable power, its extraordinary history, and its profound relevance to our moment…

There are two stories of Gilgamesh, the ancient Mesopotamian epic written in the second millennium BCE. First, there’s the story of Gilgamesh himself, the semidivine king of Uruk. He is 11 cubits tall and four cubits from nipple to nipple (roughly 16 by six feet). He is hyperactive and priapic. He is not a good ruler. The gods create the wild Enkidu out of clay to keep him in check. The pair clash mightily, and then become inseparable. Restless and hungry for glory, they journey to the Forest of Cedar to defeat the monster Humbaba. Then they slay the Bull of Heaven sent by Ishtar, the god of sex and war whose advances Gilgamesh rejects. The gods deem that Enkidu must die, and so he does, slowly and unheroically. Gilgamesh watches over Enkidu’s body until a maggot falls from his nostril, a fantastically intense image that drives home death’s finality.

At this point, the register of the poem shifts, and Gilgamesh’s triumphs are replaced by sorrow and an overwhelming awareness of his own mortality. Alone and anguished, he journeys to the underworld to visit Uta-napishti, the immortalized survivor of a cataclysmic flood, intent on unlocking the secret to eternal life. Inevitably, he is disappointed and returns to Uruk. Gilgamesh is an epic about power, about self-knowledge, about passionate companionship and the unquenchable pain of its loss. Fundamentally, it is an epic about death. Rilke labeled it “das Epos der Todesfurcht”—the epic of the fear of death—and this is what gave it its vital appeal: “It concerns me,” he confessed. “Thousands of years later death is no less bewildering to humankind,” the poet Simon Armitage says in the introduction to his new translation of the epic; “there is no more relatable subject.”

The second story of Gilgamesh is about the text itself, one of the world’s oldest surviving long-form poems. Like Homeric epic, its roots are most likely oral, and questions of authorship are futile. The earliest version was a Sumerian cycle of five poems from around 2100 BCE, probably part of a larger group of stories about the heroic dynasty of Uruk. Sumerian eventually died out, and the five episodes were replaced by one unified version in Akkadian. This was recorded in cuneiform script, often carved in clay tablets, and spread throughout Mesopotamia and the Levant. Sometime between 1300 and 1000 CE, a man called Sin-leqi-unninni created a heavily revised edition organized into 11 “tablets”—referred to now as the Standard Version—which was copied widely and included in the great library of Ashurbanipal, the Assyrian king, built in Nineveh in the seventh century.

And then … silence. By the new millennium, Akkadian was a defunct language, and Uruk and Nineveh were in ruins. As far as we know, Gilgamesh was not translated into other writing systems, so when cuneiform fell out of use, the epic seemed to go with it. For centuries it slept, until the Library of Ashurbanipal was discovered by Austen Henry Layard and Hormuzd Rassam in 1850, and what documents could be recovered were transported to the British Museum. Cuneiform was eventually deciphered, and in 1872, George Smith, an assistant curator working on the archive, came across a fragment of the epic describing a great flood—similar to the one in the Book of Genesis,but in a work significantly older than the Bible. This was too much for Smith, who began stripping his clothes off in excitement: “I am the first man to read that after more than two thousand years of oblivion.”

Critics like to say that Gilgamesh is both incredibly old and refreshingly young. Its sheer age staggers—for comparison, just try to imagine a current novel being rediscovered in the year 5120 CE. As a quasi-historical figure, Gilgamesh was considered by Babylonians to be even older: the Sumerian King List,a chronographic record,hyperbolically places his reign in 7800 BCE. Within the world of the epic itself, time reaches back further still: when Gilgamesh meets Uta-napishti, the Noah-type figure who survived the flood long before Gilgamesh, even he can speak of an “ancient city,” Shuruppak, on the banks of the Euphrates. The epic constantly forces us into these dizzying loops of deep time, forces us both to drastically exceed the limits of our brief lifespan and to be persistently reminded of them.

But Gilgamesh’s comparatively recent reentry into the modern imagination makes it feel fresh, not overburdened by centuries of interpretation and adaptation, like Homer or Virgil, and firmly outside Western literary traditions. There is no first looking into Chapman’s Gilgamesh.This can be dangerous for translators and adapters: there’s an urge to treat the epic like a blank canvas, to make it say something relevant to contemporary concerns, which can strip it of its strangeness and also cut it loose from its Iraqi heritage. But the subject matter of Gilgamesh also seems undeniably contemporary: how could a story about ecological destruction, poor leaders, and misogynist alphas not concern us here and now?…

Eminently worth reading in full. A classic which has survived, against all odds, and what it offers us today: “The Epic of the Fear of Death” from @lareviewofbooks.bsky.social.

* William Butler Yeats

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As we reach back, we might recall that it was on this date in 2004 that the discovery of what was (and is) believed to be the world’s oldest seat of learning (dating from 295 BCE), the Library of Alexandria, was announced by Zahi Hawass, president of Egypt’s Supreme Council of Antiquities during a conference at the University of California. A Polish-Egyptian team had uncovered 13 lecture halls featuring an elevated podium for the lecturer. Such a complex of lecture halls had never before been found on any Mediterranean Greco-Roman site. Alexandria may be regarded as the birthplace of western science, where Euclid discovered the rules of geometry, Eratosthenes measured the diameter of the Earth and Ptolemy wrote the Almagest, the most influential scientific book about the nature of the Universe for 1,500 years.

See also: “Oldest University Unearthed in Egypt

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“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)

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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.

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

 

euclid

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 Elements was 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

kurt_gödel source

 

Written by (Roughly) Daily

January 14, 2019 at 1:01 am

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.

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Written by (Roughly) Daily

September 5, 2013 at 1:01 am

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:

Written by (Roughly) Daily

July 15, 2012 at 1:01 am