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Thales: Ancient Greeks built the cosmos with right triangles
The ancient Greeks were obsessed with geometry, which may have formed the basis of their philosophical cosmology.
- Every triangle inscribed inside a circle on its diameter is a right triangle.
- Upon this discovery, Thales is said to have performed a great ritual sacrifice.
- Might Thales have believed that the entire cosmos was constructed of right triangles?
Thales is credited by the late commentator Proclus, on the authority of Aristotle's student Eudemus, with "discovering" geometrical propositions, some of them more generally and others more practically. Consider some of the diagrams expressing practical examples of right-angled triangles.
From left to right, we have Thales' measurement of (i) the height of a pyramid when its shadow is equal to its height; (ii) the height of a pyramid when its shadow is unequal but proportional to its height; (iii) the distance to a ship at sea from the shoreline; and (iv) the distance to a ship at sea from a tower. Note that, when rotated, they are all the same diagram!
The more general propositions also seem to be relevant to practical geometry:
We have a report about a special accomplishment of Thales. Originating with Diogenes Laertius of the 3rd century BCE on the authority of the mathematician Pamphila, it says that Thales made a splendid ritual sacrifice upon inscribing a right triangle in a circle. Obviously, he thought this was a pretty big deal. More on that a bit later.
The first thing Thales had to know is that the angles of every triangle sum to two right angles. (The angles inside every triangle sum to 180°. Two right angles, each of which is 90°, also sum to 180°.) We have an ancient report that credits Thales' generation of geometers with having grasped this fact in all species of triangles — equilateral, isosceles, and scalene. How might Thales and his geometers have done it? Consider the following diagrams:
By dropping a perpendicular from a vertex to the opposite side in each species of triangle, and then completing the two rectangles formed, one can see immediately that each rectangle (containing four right angles) is halved by the diagonal created by each side of the triangle. Therefore, each half-triangle contains two right angles. And if the two right angles at the base are removed, leaving the three angles of one large triangle, the angles sum to two right angles.
Now, consider how Thales may have proved that every triangle inscribed inside a circle on its diameter must be a right triangle. To show this, he relied on the isosceles triangle proposition and proved that the angle at A [α + β] is right-angled.
Perhaps he did it this way: Based upon the isosceles triangle proposition, Thales knows that segments BD and AD (left diagram) are equal in length because they are both radii of the circle BAC. Thus, their opposite angles — α and α — must be equal. Since every triangle is 180° (that is, contains the equivalent of two right angles) and the angle BDA at the base is a right angle, α + α must also equal one right angle. By itself, α is half of a right angle.
Next, CD and AD are both equal in length since they, too, are both radii of the circle BAC, and so the angles opposite each must also be equal — that is, β equals β. If we acknowledge that the angle at the base ADC is a right angle, and there is the equivalent of two right angles in every triangle, then β + β must equal one right angle. By itself, β is half of a right angle.
Finally, the angle at A is divided into two equal parts, α and β. Because each is half of a right angle, together (α + β) they equal one right angle.
That explains the right angle for an isosceles triangle inscribed inside of a circle. But what about all the varieties of the scalene? More or less, it's the same argument.
Consider triangle ABC (right diagram). It is composed of two triangles ABD and ACD. In ABD, AD must be equal to BD because both are radii of the circle BAC, and so the angles opposite those sides also must be equal. The same argument applies for triangle ADC. Thus, the three angles of triangle ABC are α + β + (α + β). Since we already know that the angles of every triangle sum to 180° (that is, the equivalent of two right angles), then α + β + (α + β) equals two right angles. Thus, α + β must equal one right angle.
Perhaps these lines of proof persuaded Thales and his companions that every triangle inscribed in a circle on its diameter is right. But why the great ritual sacrifice?
The ancient traditions do not give us more insight, and we are left only to speculate. Aristotle claims that Thales posited an underlying unity, water, that alters without changing. Although things look different, water is the substrate of all appearances. Water is merely altered without changing substantially. Had Thales been looking into geometry to try to discover the underlying structure of water, perhaps he followed a similar line of thought as Plato did when he identified the four elements (fire, air, water, and earth) with geometric shapes.
Thales may have identified the right triangle as the fundamental structure of water. Moreover, he now had a way to produce an unlimited number of them for further investigation simply by making a circle, drawing its diameter, and inscribing a triangle inside it.
But there is perhaps another reason for his splendid sacrifice, seen in this metaphysical light. I can imagine one of his compatriots objecting, upon hearing Thales' idea that water was the underlying nature or unity of all things and that the right triangle was its structure. The objection may have gone like this: right triangles may form the basis of every rectilinear figure, but they certainly don't form the basis of the circle. The circle is not constructed out of right triangles, is it? Thus, the right triangle is not the fundamental figure of all appearances.
Thales' reply must have been as astonishing to his compatriots as it is to many of us today. Indeed, the circle too is built out of right triangles! If we plot on the circle's diameter all the possible triangles inscribable inside a circle — starting from one end of the diameter, touching the circle, and then finishing at the other end of the diameter — we produce what modern mathematicians call a "geometrical loci." The circle itself is constructed out of right triangles!
Prof. Robert Hahn has broad interests in the history of ancient and modern astronomy and physics, ancient technologies, the contributions of ancient Egypt and monumental architecture to early Greek philosophy and cosmology, and ancient mathematics and geometry of Egypt and Greece. Every year, he gives "Ancient Legacies" traveling seminars to Greece, Turkey, and Egypt. His latest book is The Metaphysics of the Pythagorean Theorem.
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"You dream about these kinds of moments when you're a kid," said lead paleontologist David Schmidt.
- The triceratops skull was first discovered in 2019, but was excavated over the summer of 2020.
- It was discovered in the South Dakota Badlands, an area where the Triceratops roamed some 66 million years ago.
- Studying dinosaurs helps scientists better understand the evolution of all life on Earth.
David Schmidt, a geology professor at Westminster College, had just arrived in the South Dakota Badlands in summer 2019 with a group of students for a fossil dig when he received a call from the National Forest Service. A nearby rancher had discovered a strange object poking out of the ground. They wanted Schmidt to take a look.
"One of the very first bones that we saw in the rock was this long cylindrical bone," Schmidt told St. Louis Public Radio. "The first thing that came out of our mouths was, 'That kind of looks like the horn of a triceratops.'"
After authorities gave the go-ahead, Schmidt and a small group of students returned this summer and spent nearly every day of June and July excavating the skull.
Credit: David Schmidt / Westminster College
"We had to be really careful," Schmidt told St. Louis Public Radio. "We couldn't disturb anything at all, because at that point, it was under law enforcement investigation. They were telling us, 'Don't even make footprints,' and I was thinking, 'How are we supposed to do that?'"
Another difficulty was the mammoth size of the skull: about 7 feet long and more than 3,000 pounds. (For context, the largest triceratops skull ever unearthed was about 8.2 feet long.) The skull of Schmidt's dinosaur was likely a Triceratops prorsus, one of two species of triceratops that roamed what's now North America about 66 million years ago.
Credit: David Schmidt / Westminster College
The triceratops was an herbivore, but it was also a favorite meal of the Tyrannosaurus rex. That probably explains why the Dakotas contain many scattered triceratops bone fragments, and, less commonly, complete bones and skulls. In summer 2019, for example, a separate team on a dig in North Dakota made headlines after unearthing a complete triceratops skull that measured five feet in length.
Michael Kjelland, a biology professor who participated in that excavation, said digging up the dinosaur was like completing a "multi-piece, 3-D jigsaw puzzle" that required "engineering that rivaled SpaceX," he jokingly told the New York Times.
Morrison Formation in Colorado
James St. John via Flickr
The Badlands aren't the only spot in North America where paleontologists have found dinosaurs. In the 1870s, Colorado and Wyoming became the first sites of dinosaur discoveries in the U.S., ushering in an era of public fascination with the prehistoric creatures — and a competitive rush to unearth them.
Since, dinosaur bones have been found in 35 states. One of the most fruitful locations for paleontologists has been the Morrison formation, a sequence of Upper Jurassic sedimentary rock that stretches under the Western part of the country. Discovered here were species like Camarasaurus, Diplodocus, Apatosaurus, Stegosaurus, and Allosaurus, to name a few.
|Credit: Nobu Tamura/Wikimedia Commons|
As for "Shady" (the nickname of the South Dakota triceratops), Schmidt and his team have safely transported it to the Westminster campus. They hope to raise funds for restoration, and to return to South Dakota in search of more bones that once belonged to the triceratops.
Studying dinosaurs helps scientists gain a more complete understanding of our evolution, illuminating a through-line that extends from "deep time" to present day. For scientists like Schmidt, there's also the simple joy of coming to face-to-face with a lost world.
"You dream about these kinds of moments when you're a kid," Schmidt told St. Louis Public Radio. "You don't ever think that these things will ever happen."
In ancient Greece, the Olympics were never solely about the athletes themselves.
Because of a dramatic rise in COVID-19 cases, the opening and closing ceremonies of the 2021 Olympics will unfold in a stadium absent the eyes, ears and voices of a once-anticipated 68,000 ticket holders from around the world.
Events during the intervening days will likewise occur in silent arenas missing the hundreds of thousands of spectators who paid US$815 million for their now-useless tickets.
After 48 years teaching classics, I can't help but wonder what the Greeks – who invented the Games nearly 3,000 years ago, in 776 B.C. – would make of such a ghostly version of their Olympic festival.
In many ways, they'd view the prospect as absurd.
In ancient Greece, the Olympics were never solely about the athletes themselves; instead, the heart and soul of the festival was the experience shared by all who attended. Every four years, athletes and spectators traveled from far-flung corners of the Greek-speaking world to Olympia, lured by a longing for contact with their compatriots and their gods.
In the shadow of dreams
For the Greeks, during five days in the late-summer heat, two worlds miraculously merged at Olympia: the domain of everyday life, with its human limits, and a supernatural sphere from the days superior beings, gods and heroes populated Earth.
Greek athletics, like today's, plunged participants into performances that pushed the envelope of human ability to its breaking point. But to the Greeks, the cauldron of competition could trigger revelations in which ordinary mortals might briefly intermingle with the extraordinary immortals.
The poet Pindar, famous for the victory songs he composed for winners at Olympia, captured this sort of transcendent moment when he wrote, “Humans are creatures of a day. But what is humankind? What is it not? A human is just the shadow of a dream – but when a flash of light from Zeus comes down, a shining light falls on humans and their lifetime can be sweet as honey."
However, these epiphanies could occur only if witnesses were physically present to immerse themselves – and share in – the spine-tingling flirtation with the divine.
Simply put, Greek athletics and religious experience were inseparable.
At Olympia, both athletes and spectators were making a pilgrimage to a sacred place. A modern Olympics can legitimately take place in any city selected by the International Olympic Committee. But the ancient games could occur in only one location in western Greece. The most profoundly moving events didn't even occur in the stadium that accommodated 40,000 or in the wrestling and boxing arenas.
Instead, they took place in a grove called the Althis, where Hercules is said to have first erected an altar, sacrificed oxen to Zeus and planted a wild olive tree. Easily half the events during the festival engrossed spectators not in feats like discus, javelin, long jump, foot race and wrestling, but in feasts where animals were sacrificed to gods in heaven and long-dead heroes whose spirits still lingered.
On the evening of the second day, thousands gathered in the Althis to reenact the funeral rites of Pelops, a human hero who once raced a chariot to win a local chief's daughter. But the climactic sacrifice was on the morning of the third day at the Great Altar of Zeus, a mound of plastered ashes from previous sacrifices that stood 22 feet tall and 125 feet around. In a ritual called the hecatomb, 100 bulls were slaughtered and their thigh bones, wrapped in fat, burned atop the altar so that the rising smoke and aroma would reach the sky where Zeus could savor it.
No doubt many a spectator shivered at the thought of Zeus hovering above them, smiling and remembering Hercules' first sacrifice.
Just a few yards from the Great Altar another, more visual encounter with the god awaited. In the Temple of Zeus, which was erected around 468 to 456 B.C., stood a colossal image, 40 feet high, of the god on a throne, his skin carved from ivory and his clothing made of gold. In one hand he held the elusive goddess of victory, Nike, and in the other a staff on which his sacred bird, the eagle, perched. The towering statue was reflected in a shimmering pool of olive oil surrounding it.
During events, the athletes performed in the nude, imitating heroic figures like Hercules, Theseus or Achilles, who all crossed the dividing line between human and superhuman and were usually represented nude in painting and sculpture.
The athletes' nudity declared to spectators that in this holy place, contestants hoped to reenact, in the ritual of sport, the shudder of contact with divinity. In the Althis stood a forest of hundreds of nude statues of men and boys, all previous victors whose images set the bar for aspiring newcomers.
“There are a lot of truly marvelous things one can see and hear about in Greece," the Greek travel writer Pausanias noted in the second century B.C., “but there is something unique about how the divine is encountered at … the games at Olympia."
Communion and community
The Greeks lived in roughly 1,500 to 2,000 small-scale states scattered across the Mediterranean and Black Sea regions.
Since sea travel in summertime was the only viable way to cross this fragile geographical web, the Olympics might entice a Greek living in Southern Europe and another residing in modern-day Ukraine to interact briefly in a festival celebrating not only Zeus and Heracles but also the Hellenic language and culture that produced them.
Besides athletes, poets, philosophers and orators came to perform before crowds that included politicians and businessmen, with everyone communing in an “oceanic feeling" of what it meant to be momentarily united as Greeks.
Now, there's no way we could explain the miracle of TV to the Greeks and how its electronic eye recruits millions of spectators to the modern games by proxy. But visitors to Olympia engaged in a distinct type of spectating.
The ordinary Greek word for someone who observes – “theatês" – connects not only to “theater" but also to “theôria," a special kind of seeing that requires a journey from home to a place where something wondrous unfolds. Theôria opens a door into the sacred, whether it's visiting an oracle or participating in a religious cult.
Attending an athletic-religious festival like the Olympics transformed an ordinary spectator, a theatês, into a theôros – a witness observing the sacred, an ambassador reporting home the wonders observed abroad.
It's hard to imagine TV images from Tokyo achieving similar ends.
No matter how many world records are broken and unprecedented feats accomplished at the 2020 games, the empty arenas will attract no gods or genuine heroes: The Tokyo games are even less enchanted than previous modern games.
But while medal counts will confer fleeting glory on some nations and disappointing shame on others, perhaps a dramatic moment or two might unite athletes and TV viewers in an oceanic feeling of what it means to be “kosmopolitai," citizens of the world, celebrants of the wonder of what it means to be human – and perhaps, briefly, superhuman as well.
The ancient Greeks wouldn't recognize some aspects of the modern Olympics.
Vincent Farenga, Professor of Classics and Comparative Literature, USC Dornsife College of Letters, Arts and Sciences
A new brain imaging study explored how different levels of the brain's excitatory and inhibitory neurotransmitters are linked to math abilities.
- Glutamate and GABA are neurotransmitters that help regulate brain activity.
- Scientists have long known that both are important to learning and neuroplasticity, but their relationship to acquiring complex cognitive skills like math has remained unclear.
- The new study shows that having certain levels of these neurotransmitters predict math performance, but that these levels switch with age.
Why do roughly one in five people find math especially difficult?
You might blame teaching methods, which some argue explains why the U.S. lags behind other countries in standardized math test scores. You could point to math anxiety, which affects about 20 percent of students and 25 percent of teachers, according to surveys. And there are also medical conditions that make math difficult, such as dyscalculia, a learning disability that disrupts the normal development of arithmetic skills.
But another explanation centers on neurotransmitters. In a new study published in PLOS Biology, researchers explored how the brain's levels of GABA and glutamate relate to math abilities over time in students of varying ages. The results showed that levels of these neurotransmitters can predict students' performance on math tests. However, this relationship seems to flip as people get older.
GABA and glutamate are responsible for regulating brain activity. In the mature brain, GABA is the brain's main inhibitory neurotransmitter, helping to block impulses between nerve cells in the brain, which can calm feelings of stress, anxiety, or fear. GABA is made from glutamate, the brain's major excitatory neurotransmitter that helps send signals throughout the central nervous system.
Researchers have long known that these neurotransmitters play crucial roles in learning, development, and neuroplasticity. That is partly because they are thought to help trigger developmental windows (or "sensitive periods") during which neural systems become more plastic and better able to acquire certain cognitive skills.
"Importantly, sensitive periods vary for different functions, with relatively simple abilities (e.g., sensorimotor integration) occurring earlier in development, while the sensitive period for acquiring more complex cognitive functions extends into the third decade of life," the researchers wrote.
GABA, glutamate, and math
Still, the exact relationship between GABA, glutamate, and complex cognitive functions has remained unclear. The new study explored that relationship by focusing on associations between the neurotransmitters and math abilities, which "provides a unique cognitive model to examine these questions due to its protracted skill acquisition period that starts already from early childhood and can continue for nearly two decades," the researchers wrote.
For the study, the researchers measured levels of GABA and glutamate in the left intraparietal sulcus (IPS) of 255 students, ranging from primary school to college. The participants completed a math test as their brains were imaged. About a year and a half later, the participants repeated the same process.
"The longitudinal design allowed us to further examine whether neurotransmitter concentration is linked to MA [mathematical abilities] as well as predict MA in the future," the researchers wrote. "Crucially, adopting this design allowed us to discern the selective effect of glutamate and GABA in response to natural (i.e., learning in school) rather than artificial environmental stimulation, thus allowing us to test the knowledge gained from lab-based experiments in high ecological settings."
The results suggest that GABA and glutamate play an important role in math abilities, but that the dynamic switches with age. For the young participants, higher GABA levels in the IPS were associated with higher scores on math tests. The opposite was observed among older students: higher glutamate levels correlated with higher scores. Both results held true on subsequent math tests.
Although the study sheds light on how neurotransmitter levels at different stages of development contribute to learning some cognitive skills, like math, the researchers noted that acquiring other skills may involve different processes.
"Our findings may also highlight a general principle that the developmental dynamics of regional excitation and inhibition levels in regulating the sensitive period and plasticity of a given high-level cognitive function (i.e., MA) may be different compared to another high-level cognitive function (i.e., general intelligence) that draws on similar, albeit not identical, cognitive and neural mechanisms," they wrote.