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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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“Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”*…

No scripture is as old as mathematics is. All the other sciences are younger, most by thousands of years. More than history, mathematics is the record that humanity is keeping of itself. History can be revised or manipulated or erased or lost. Mathematics is permanent. A² + B² = C² was true before Pythagoras had his name attached to it, and will be true when the sun goes out and no one is left to think of it. It is true for any alien life that might think of it, and true whether they think of it or not. It cannot be changed. So long as there is a world with a horizontal and a vertical axis, a sky and a horizon, it is inviolable and as true as anything that can be thought.

As precise as mathematics is, it is also the most explicit language we have for the description of mysteries. Being the language of physics, it describes actual mysteries—things we can’t see clearly in the natural world but suspect are true and later confirm—and imaginary mysteries, things that exist only in the minds of mathematicians. A question is where these abstract mysteries exist, what their home range is. Some people would say that they reside in the human mind, that only the human mind has the capacity to conceive of what are called mathematical objects, meaning numbers and equations and formulas and so on—the whole glossary and apparatus of mathematics—and to bring these into being, and that such things arrive as they do because of the way our minds are structured. We are led to examine the world in a way that agrees with the tools that we have for examining it. (We see colors as we do, for example, because of how our brains are structured to receive the reflection of light from surfaces.) This is a minority view, held mainly by neuroscientists and a certain number of mathematicians disinclined toward speculation. The more widely held view is that no one knows where math resides. There is no mathematician/naturalist who can point somewhere and say, “That is where math comes from” or “Mathematics lives over there,” say, while maybe gesturing toward magnetic north and the Arctic, which I think would suit such a contrary and coldly specifying discipline.

The belief that mathematics exists somewhere else than within us, that it is discovered more than created, is called Platonism, after Plato’s belief in a non-spatiotemporal realm that is the region of the perfect forms of which the objects on earth are imperfect reproductions. By definition, the non-spatiotemporal realm is outside time and space. It is not the creation of any deity; it simply is. To say that it is eternal or that it has always existed is to make a temporal remark, which does not apply. It is the timeless nowhere that never has and never will exist anywhere but that nevertheless is. The physical world is temporal and declines; the non-spatiotemporal one is ideal and doesn’t.

A third point of view, historically and presently, for a small but not inconsequential number of mathematicians, is that the home of mathematics is in the mind of a higher being and that mathematicians are somehow engaged with Their thoughts. Georg Cantor, the creator of set theory—which in my childhood was taught as a part of the “new math”—said, “The highest perfection of God lies in the ability to create an infinite set, and its immense goodness leads Him to create it.” And the wildly inventive and self-taught mathematician Srinivasa Ramanujan, about whom the movie “The Man Who Knew Infinity” was made, in 2015, said, “An equation for me has no meaning unless it expresses a thought of God.”

In Book 7 of the Republic, Plato has Socrates say that mathematicians are people who dream that they are awake. I partly understand this, and I partly don’t.

Mathematics has been variously described as an ideal reality, a formal game, and the poetry of logical ideas… an excerpt from “What is Mathematics?” from Alec Wilkinson— eminently worthy of reading in full.

* Albert Einstein

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As we sum it up, we might send carefull-calcuated birthday greetings to Georgiy Antonovich Gamov; he was born on this date in 1904. Better known by the name he adopted on immigrating to the U.S., George Gamow, he was a physicist and cosmologist whose early work was instrumental in developing the Big Bang theory of the universe; he also developed the first mathematical model of the atomic nucleus.

But mid-career Gamow began to shift his energy to teaching and to writing popular books on science… one of which, One Two Three… Infinity, inspired legions of young scientists-to-be and kindled a life-long interest in science in an even larger number of other youngsters (like your correspondent).

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“Facts alone, no matter how numerous or verifiable, do not automatically arrange themselves into an intelligible, or truthful, picture of the world. It is the task of the human mind to invent a theoretical framework to account for them.”*…

PPPL physicist Hong Qin in front of images of planetary orbits and computer code

… or maybe not. A couple of decades ago, your correspondent came across a short book that aimed to explain how we think know what we think know, Truth– a history and guide of the perplexed, by Felipe Fernández-Armesto (then, a professor of history at Oxford; now, at Notre Dame)…

According to Fernández-Armesto, people throughout history have sought to get at the truth in one or more of four basic ways. The first is through feeling. Truth is a tangible entity. The third-century B.C. Chinese sage Chuang Tzu stated, ”The universe is one.” Others described the universe as a unity of opposites. To the fifth-century B.C. Greek philosopher Heraclitus, the cosmos is a tension like that of the bow or the lyre. The notion of chaos comes along only later, together with uncomfortable concepts like infinity.

Then there is authoritarianism, ”the truth you are told.” Divinities can tell us what is wanted, if only we can discover how to hear them. The ancient Greeks believed that Apollo would speak through the mouth of an old peasant woman in a room filled with the smoke of bay leaves; traditionalist Azande in the Nilotic Sudan depend on the response of poisoned chickens. People consult sacred books, or watch for apparitions. Others look inside themselves, for truths that were imprinted in their minds before they were born or buried in their subconscious minds.

Reasoning is the third way Fernández-Armesto cites. Since knowledge attained by divination or introspection is subject to misinterpretation, eventually people return to the use of reason, which helped thinkers like Chuang Tzu and Heraclitus describe the universe. Logical analysis was used in China and Egypt long before it was discovered in Greece and in India. If the Greeks are mistakenly credited with the invention of rational thinking, it is because of the effective ways they wrote about it. Plato illustrated his dialogues with memorable myths and brilliant metaphors. Truth, as he saw it, could be discovered only by abstract reasoning, without reliance on sense perception or observation of outside phenomena. Rather, he sought to excavate it from the recesses of the mind. The word for truth in Greek, aletheia, means ”what is not forgotten.”

Plato’s pupil Aristotle developed the techniques of logical analysis that still enable us to get at the knowledge hidden within us. He examined propositions by stating possible contradictions and developed the syllogism, a method of proof based on stated premises. His methods of reasoning have influenced independent thinkers ever since. Logicians developed a system of notation, free from the associations of language, that comes close to being a kind of mathematics. The uses of pure reason have had a particular appeal to lovers of force, and have flourished in times of absolutism like the 17th and 18th centuries.

Finally, there is sense perception. Unlike his teacher, Plato, and many of Plato’s followers, Aristotle realized that pure logic had its limits. He began with study of the natural world and used evidence gained from experience or experimentation to support his arguments. Ever since, as Fernández-Armesto puts it, science and sense have kept time together, like voices in a duet that sing different tunes. The combination of theoretical and practical gave Western thinkers an edge over purer reasoning schemes in India and China.

The scientific revolution began when European thinkers broke free from religious authoritarianism and stopped regarding this earth as the center of the universe. They used mathematics along with experimentation and reasoning and developed mechanical tools like the telescope. Fernández-Armesto’s favorite example of their empirical spirit is the grueling Arctic expedition in 1736 in which the French scientist Pierre Moreau de Maupertuis determined (rightly) that the earth was not round like a ball but rather an oblate spheroid…

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One of Fernández-Armesto most basic points is that our capacity to apprehend “the truth”– to “know”– has developed throughout history. And history’s not over. So, your correspondent wondered, mightn’t there emerge a fifth source of truth, one rooted in the assessment of vast, ever-more-complete data maps of reality– a fifth way of knowing?

Well, those days may be upon us…

A novel computer algorithm, or set of rules, that accurately predicts the orbits of planets in the solar system could be adapted to better predict and control the behavior of the plasma that fuels fusion facilities designed to harvest on Earth the fusion energy that powers the sun and stars.

he algorithm, devised by a scientist at the U.S. Department of Energy’s (DOE) Princeton Plasma Physics Laboratory (PPPL), applies machine learning, the form of artificial intelligence (AI) that learns from experience, to develop the predictions. “Usually in physics, you make observations, create a theory based on those observations, and then use that theory to predict new observations,” said PPPL physicist Hong Qin, author of a paper detailing the concept in Scientific Reports. “What I’m doing is replacing this process with a type of black box that can produce accurate predictions without using a traditional theory or law.”

Qin (pronounced Chin) created a computer program into which he fed data from past observations of the orbits of Mercury, Venus, Earth, Mars, Jupiter, and the dwarf planet Ceres. This program, along with an additional program known as a ‘serving algorithm,’ then made accurate predictions of the orbits of other planets in the solar system without using Newton’s laws of motion and gravitation. “Essentially, I bypassed all the fundamental ingredients of physics. I go directly from data to data,” Qin said. “There is no law of physics in the middle.”

The process also appears in philosophical thought experiments like John Searle’s Chinese Room. In that scenario, a person who did not know Chinese could nevertheless ‘translate’ a Chinese sentence into English or any other language by using a set of instructions, or rules, that would substitute for understanding. The thought experiment raises questions about what, at root, it means to understand anything at all, and whether understanding implies that something else is happening in the mind besides following rules.

Qin was inspired in part by Oxford philosopher Nick Bostrom’s philosophical thought experiment that the universe is a computer simulation. If that were true, then fundamental physical laws should reveal that the universe consists of individual chunks of space-time, like pixels in a video game. “If we live in a simulation, our world has to be discrete,” Qin said. The black box technique Qin devised does not require that physicists believe the simulation conjecture literally, though it builds on this idea to create a program that makes accurate physical predictions.

This process opens up questions about the nature of science itself. Don’t scientists want to develop physics theories that explain the world, instead of simply amassing data? Aren’t theories fundamental to physics and necessary to explain and understand phenomena?

“I would argue that the ultimate goal of any scientist is prediction,” Qin said. “You might not necessarily need a law. For example, if I can perfectly predict a planetary orbit, I don’t need to know Newton’s laws of gravitation and motion. You could argue that by doing so you would understand less than if you knew Newton’s laws. In a sense, that is correct. But from a practical point of view, making accurate predictions is not doing anything less.”

Machine learning could also open up possibilities for more research. “It significantly broadens the scope of problems that you can tackle because all you need to get going is data,” [Qin’s collaborator Eric] Palmerduca said…

But then, as Edwin Hubble observed, “observations always involve theory,” theory that’s implicit in the particulars and the structure of the data being collected and fed to the AI. So, perhaps this is less a new way of knowing, than a new way of enhancing Fernández-Armesto’s third way– reason– as it became the scientific method…

The technique could also lead to the development of a traditional physical theory. “While in some sense this method precludes the need of such a theory, it can also be viewed as a path toward one,” Palmerduca said. “When you’re trying to deduce a theory, you’d like to have as much data at your disposal as possible. If you’re given some data, you can use machine learning to fill in gaps in that data or otherwise expand the data set.”

In either case: “New machine learning theory raises questions about nature of science.”

Francis Bello

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As we experiment with epistemology, we might send carefully-observed and calculated birthday greetings to Georg Joachim de Porris (better known by his professional name, Rheticus; he was born on this date in 1514. A mathematician, astronomer, cartographer, navigational-instrument maker, medical practitioner, and teacher, he was well-known in his day for his stature in all of those fields. But he is surely best-remembered as the sole pupil of Copernicus, whose work he championed– most impactfully, facilitating the publication of his master’s De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres)… and informing the most famous work by yesterday’s birthday boy, Galileo.

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“The seen is the changing, the unseen is the unchanging”*…

Pharmacist’s Spatula, by William Toogood Ltd, English

We start 2021 with three big milestones for the Science Museum Group Collection.

100,000 incredible objects now have a photograph online, the online collection regularly receives 100,000 views each month and we’ve just recorded 3,000,000 visitors since launching the website in late 2016.

Each time you visit our online collection you can see more than ever before. Almost a quarter of the remarkable objects we care for (24.9% or 105,715 objects to be exact) have a photograph online, with hundreds of new photographs added each month as we digitise our vast collection.

You can explore photographs of artworks, tools and video games, or items from astronomy, firefighting and printing to give a few examples from the collection…

n the past we’ve released digital tools to help you explore the collection, including our Random Object Generator, Museum in a Tab (a Google Chrome extension) and What the machine saw (a machine learning experiment). You can even add our objects to the popular game Animal Crossing.

However, it can be difficult to spot recently photographed objects in the collection. So today we have published a new tool to help you explore these new items.

Never Been Seen shows objects from the Science Museum Group Collection that have never been seen online before. Each time you refresh this webpage an object with zero views is shown, making you the very first person to see it…

The spatula at the top of this post is no longer in that category, as your correspondent has seen (and now shared) it. But there’s so much more! Explore as yet unnoticed items in the collection of the Science Museum (London): “Never Been Seen.”

* Plato

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As we uncover the unobserved, we might spare a thought for a man who saw much that had hitherto been unseen: Frank Plumpton Ramsey, a philosopher, mathematician, and economist who made major contributions to all three fields before his death (at the age of 26) on this date in 1930.

For more on Ramsey and his thought, see “One of the Great Intellects of His Time,” “The Man Who Thought Too Fast,” and Ramsey’s entry in the Stanford Encyclopedia of Philosophy.

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Written by LW

January 19, 2021 at 1:01 am

“Mathematics is the art of giving the same name to different things”*…

Freeman Dyson was a translator: he turned physics into math, and those subjects into English for the general public.

Mathematics, like life, is complicated. But, for those who do mathematics, it is a source of joy. “The main thing is just astonishment that there’s such a rich world out there—a wonderful, abstract, very beautiful, simple world,” [John] Conway said. “It’s like Pizarro standing on the shores of the Pacific or whatever. . . . I can sit here in this chair and go on a voyage of exploration. A very different voyage of exploration, but, still, there are things to be discovered, things to be seen, that you can quite easily be the first person ever to see.”

So many of us now sit in our rooms, bound in space while time drips away. It can be a bit of a comfort to know that, as long as you are able to sit still and think, your creative spirit can be an engine of exploration. On their journeys, these playful, curious mathematicians discovered Monsters and numbers so large that they can hardly be written down. We’re grateful for the lively stories of their expeditions, and for the thinkers who led them. They’ll be missed…

John Conway, Ronald Graham, and Freeman Dyson all explored the world with their minds. Dan Rockmore (@dan_rockmore) celebrates “Three Mathematicians We Lost in 2020.”

Special bonus: an interview with an heir to Dyson– that’s to say, an important mathematician who’s also a gifted “translator”– Steven Strogatz.

* Henri Poincaré

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As we share their amazement, we might we might spare a thought for Max Born; he died on this date in 1970.  A German physicist and Nobel Laureate, he coined the phrase “quantum mechanics” to describe the field in which he made his greatest contributions.  But beyond his accomplishments as a practitioner, he was a master teacher whose students included Enrico Fermi and Werner Heisenberg– both of whom became Nobel Laureates before their mentor– and  J. Robert Oppenheimer.

Less well-known is that Born, who died in 1970, was the grandfather of Australian phenom and definitive Sandy-portrayer Olivia Newton-John.

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