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Posts Tagged ‘Ptolemy

“Werner Heisenberg once proclaimed that all the quandaries of quantum mechanics would shrivel up when 137 was finally explained”*…

One number to rule them all?

Does the Universe around us have a fundamental structure that can be glimpsed through special numbers?

The brilliant physicist Richard Feynman (1918-1988) famously thought so, saying there is a number that all theoretical physicists of worth should “worry about”. He called it “one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man.”

That magic number, called the fine structure constant, is a fundamental constant, with a value which nearly equals 1/137. Or 1/137.03599913, to be precise. It is denoted by the Greek letter alpha – α.

What’s special about alpha is that it’s regarded as the best example of a pure number, one that doesn’t need units. It actually combines three of nature’s fundamental constants – the speed of light, the electric charge carried by one electron, and the Planck’s constant, as explains physicist and astrobiologist Paul Davies to Cosmos magazine. Appearing at the intersection of such key areas of physics as relativity, electromagnetism and quantum mechanics is what gives 1/137 its allure…

The fine structure constant has mystified scientists since the 1800s– and might hold clues to the Grand Unified Theory: “Why the number 137 is one of the greatest mysteries in physics,” from Paul Ratner (@paulratnercodex) in @bigthink.

* Leon M. Lederman, The God Particle: If the Universe Is the Answer, What Is the Question?


As we ruminate on relationships, we might spare a thought for Georg von Peuerbach; he died on this date in 1461. A mathematician, astronomer, and instrument maker, he is probably best remembered for his streamlined presentation of Ptolemaic astronomy in the Theoricae Novae Planetarum (which was an important text for many later-influential astronomers including Nicolaus Copernicus and Johannes Kepler).

But perhaps as impactful was his promotion of the use of Arabic numerals (introduced 250 years earlier in place of Roman numerals), especially in a table of sines he calculated with unprecedented accuracy.

Georg von Peuerbach: Theoricarum novarum planetarum testus, Paris 1515 [source]
Page from Peurbach’s sine table [source]

“No part of mathematics is ever, in the long run, ‘useless’.”*…

The number 1 can be written as a sum of distinct unit fractions, such as 1/2 + 1/3 + 1/12 + 1/18 + 1/36…

Number theorists are always looking for hidden structure. And when confronted by a numerical pattern that seems unavoidable, they test its mettle, trying hard — and often failing — to devise situations in which a given pattern cannot appear.

One of the latest results to demonstrate the resilience of such patterns, by Thomas Bloom of the University of Oxford, answers a question with roots that extend all the way back to ancient Egypt.

“It might be the oldest problem ever,” said Carl Pomerance of Dartmouth College.

The question involves fractions that feature a 1 in their numerator, like 1/2, 1/7 or 1/122. These “unit fractions” were especially important to the ancient Egyptians because they were the only types of fractions their number system contained: With the exception of a single symbol for 23, they could only express more complicated fractions (like 3/4) as sums of unit fractions (1/2 + 1/4).

The modern-day interest in such sums got a boost in the 1970s, when Paul Erdős and Ronald Graham asked how hard it might be to engineer sets of whole numbers that don’t contain a subset whose reciprocals add to 1. For instance, the set {2, 3, 6, 9, 13} fails this test: It contains the subset {2, 3, 6}, whose reciprocals are the unit fractions 1/2, 1/3 and 1/6 — which sum to 1.

More exactly, Erdős and Graham conjectured that any set that samples some sufficiently large, positive proportion of the whole numbers — it could be 20% or 1% or 0.001% — must contain a subset whose reciprocals add to 1. If the initial set satisfies that simple condition of sampling enough whole numbers (known as having “positive density”), then even if its members were deliberately chosen to make it difficult to find that subset, the subset would nonetheless have to exist.

“I just thought this was an impossible question that no one in their right mind could possibly ever do,” said Andrew Granville of the University of Montreal. “I didn’t see any obvious tool that could attack it.”…

Bloom, building on work by Ernie Croot, found that tool. The ubiquity of ways to represent whole numbers as sums of fractions: “Math’s ‘Oldest Problem Ever’ Gets a New Answer,” by Jordana Cepelewicz (@jordanacep) in @QuantaMagazine.

* “No part of mathematics is ever, in the long run, ‘useless.’ Most of number theory has very few ‘practical’ applications. That does not reduce its importance, and if anything it enhances its fascination. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role.” – C. Stanley Ogilvy, Excursions in Number Theory


As we recombine, we might send carefully-calculated birthday greetings to Ulugh Beg (or, officially, Mīrzā Muhammad Tāraghay bin Shāhrukh); he was born on this date in 1394. A Timurid sultan with a hearty interest in science and the arts, he is better remembered as an astronomer and mathematician.

The most important observational astronomer of the 15th century, he built the great Ulugh Beg Observatory in Samarkand between 1424 and 1429– considered by scholars to have been one of the finest observatories in the Islamic world at the time and the largest in Central Asia. In his observations he discovered a number of errors in the computations of the 2nd-century Alexandrian astronomer Ptolemy, whose figures were still being used. His star map of 994 stars was the first new one since Hipparchus. Among his contributions to mathematics were trigonometric tables of sine and tangent values correct to at least eight decimal places.

Ulugh Beg’s Statue in Samarkand, Uzbekistan


“Happy accidents are real gifts”*…

Fresco by Bertini, “Galileo and the Doge of Venice”

On the morning of July 25, 1610, Galileo pointed his telescope at Saturn and was surprised to find that it appeared to be flanked by two round blobs or bumps, one on either side. Unfortunately, Galileo’s telescope wasn’t quite advanced enough to pick out precisely what he had seen (his observations are now credited with being the earliest description of Saturn’s rings in astronomical history), but he nevertheless presumed that whatever he had seen was something special. And he wanted people to know about it.

Keen to announce his news and thereby secure credit for whatever it was he had discovered, Galileo sent letters to his friends and fellow astronomers. This being Galileo, the announcement was far from straightforward:


Each message that Galileo sent out contained little more than that jumbled string of letters, which when rearranged correctly spelled out the Latin sentence, “altissimum planetam tergeminum observavi”—or “I have observed that the highest planet is threefold.”

As the outermost planet known to science at the time, Saturn was the “highest planet” in question. And unaware that he had discovered its rings, Galileo was merely suggesting to his contemporaries that he had found that the planet was somehow divided into three parts. Announcing such a discovery in the form of an anagram might have bought Galileo some time to continue his observations, however, but there was a problem: Anagrams can easily be misinterpreted.

One of those to whom Galileo sent a letter was the German scientist Johannes Kepler. A keen astronomer himself, Kepler had followed and supported Galileo’s work for several years, so when the coded letter arrived at his home in Prague he quickly set to work solving it. Unfortunately for him, he got it completely wrong.

Kepler rearranged Galileo’s word jumble as “salve, umbistineum geminatum Martia proles,” which he interpreted as “be greeted, double-knob, children of Mars.” His solution was far from perfect (umbistineum isn’t really a grammatical Latin word, for one thing), but Kepler was nevertheless convinced that, not only had he correctly solved the riddle, but Galileo’s apparent discovery proved a theory he had been contemplating for several months.

Earlier in 1610, Galileo had discovered the four so-called “Galilean moons” of Jupiter: Io, Europa, Ganymede and Callisto. Although we now know that Jupiter has several dozen moons of varying shapes, sizes, and orbits, at the time the announcement of just four natural satellites had led Kepler to presume that there must be a natural progression in the heavens: the Earth has one moon; Jupiter, two places further out from the Earth, has four; and sat between the two is Mars, which Kepler theorized must surely have two moons, to maintain the balanced celestial sequence 1, 2, 4 and so on (his only question was whether Saturn had six or eight).

Kepler got the anagram wrong, and the presumption that Jupiter only had four moons had been wrong. Yet as misguided as both these facts were, the assumption that Kepler made based on both of them—namely, that Mars had two moons—was entirely correct. Unfortunately for Kepler, his theory would not be proved until long after his death, as the two Martian moons Phobos and Deimos (named after Ares’s sons in Greek Mythology) were not discovered until 1877, by the American astronomer Asaph Hall.

Nevertheless, a misinterpretation of the anagram had accidentally predicted a major astronomical discovery of the 19th century, nearly 300 years before it occurred…

Serendipity in science: “How A Misinterpreted Anagram Predicted The Moons of Mars.”

(For an account of Isaac Newton’s use of anagrams in his scientific communications, see here.)

* David Lynch


As we code and decode, we might recall that it was on this date in 1781 that English astronomer William Herschel detected every schoolboy’s favorite planet, Uranus, in the night sky (though he initially thought it was a comet); it was the first planet to be classified as such with the aid of a telescope.  In fact, Uranus had been detected much earlier– but mistaken for a star: the earliest likely observation was by Hipparchos, who (in 128 BC) seems to have recorded the planet as a star for his star catalogue, later incorporated into Ptolemy’s Almagest.  The earliest definite sighting was in 1690 when John Flamsteed observed it at least six times, cataloguing it as the star 34 Tauri.

Herschel named the planet in honor of his King: Georgium Sidus (George’s Star), an unpopular choice, especially outside England; argument over alternatives ensued.  Berlin astronomer Johann Elert Bode came up with the moniker “Uranus,” which was adopted throughout the world’s astronomical community by 1850.

 Uranus, photographed by Voyager 2 in 1986.


Written by (Roughly) Daily

March 13, 2021 at 1:01 am

“I have known uncertainty: a state unknown to the Greeks”*…


Octadrachm, reverse: jugate portrait Ptolemy I and Berenice I, Alexandria, 260–240 BCE

The Ptolemies who ruled Egypt for nearly three centuries, from about 320 to 31 BCE, had a difficult dual part to play: that of Hellenistic monarchs, in the mold of Alexander the Great, and, simultaneously, Egyptian pharaohs. The founding father of their line, Ptolemy I Soter (“Savior”), a Macedonian general in Alexander’s army of conquest, secured rule over Egypt amid the confusion following his king’s death, crowned himself monarch in 306 BCE. But he bequeathed to his heirs—the fourteen other Ptolemies who would succeed him, not to mention several Cleopatras—a difficult demographic and geopolitical position. The Ptolemies’ palace complex, staffed by a European elite, stood in Alexandria, one of the world’s original Green Zones, a Greek-style city founded on a strongly fortified isthmus facing the Mediterranean. To the south, nearly cut off by the vast marshes of Lake Mareotis, lived most of their Egyptian subjects. Some scholars have reckoned the country’s ratio of Egyptians to Greco-Macedonians at ten to one…

Find out how the Greeks did it at “When the Greeks Ruled Egypt.” (Spoiler alert:  it involved respect for and tolerance of Egyptian religious and social beliefs.  Genghis Khan operated in a similar fashion; more modern empires, not so much…)

* Jorge Luis Borges


As we go native, we might spare a thought for Aristophanes; he died on this date in 386 BCE (or so many scholars deduce; the exact date has not been documented).  A poet and dramatist, Aristophanes– whose works are the sole surviving examples of what is known as “Old Comedy”– is widely known as as “the Father of Comedy.”  His eleven surviving plays essentially laid the foundation for satire as we know it, and have a significance that goes beyond this artistic value: Aristophanes acute observations of classical Athens are perhaps as important as historical documents as the writings of Thucydides.  They had impact in their own time, as well.  His powers of ridicule were feared and acknowledged by influential contemporaries; Plato singled out Aristophanes’ play The Clouds as slander that contributed to the trial and subsequent condemning to death of Socrates (although other satirical playwrights had also caricatured the philosopher). His second play, The Babylonians (now lost), was sufficiently scathing to be denounced by the demagogue Cleon as a slander against the Athenian polis.  Aristophanes survived The Peloponnesian War, two oligarchic revolutions, and two democratic restorations– evidence that he was not himself actively involved in politics; rather, an objective “commentator.”  In this, he agreed with Socrates (as “reported” by Plato in The Apology): “he who will fight for the right, if he would live even for a brief space, must have a private station and not a public one.”

Illustration from a bust found near Tusculum (likely altogether imaginary, as Aristophanes was reportedly bald)



Written by (Roughly) Daily

November 11, 2014 at 1:01 am

“Creativity is that marvelous capacity to grasp mutually distinct realities and draw a spark from their juxtaposition”*…


Artist Bill Domonkos:

I view my work as a collision and recombination of ideas. My process unfolds gradually and spontaneously—using found materials such as archive film footage, photographs, and the internet. I experiment by combining, altering, editing and reassembling using digital technology, special effects and animation to create a new kind of experience. I am interested in the absurd, as well as moments of sublime beauty—to renew and transform materials, experiences and ideas. The extraordinary thing about cinema is its ability to suggest the ineffable—something that cannot or should not be expressed in words, only hinted at through sounds and images. It is this elusive, dreamlike quality that informs my work.


Check out more of the work that The Boston Globe‘s Michael Hardy calls “Spooky. Hypnotic. Lush. Witty. Sublime.” at his site, and on his Tumblr, devoted to GIFs like those above.

* Max Ernst


As we amuse ourselves with animation, we might recall that it was on this date in 1972 that the first leap second was added to a day.  The modern definition of a “second” was settled in 1874 by European scientists working from Muslim scholars’ improvement on Ptolemy’s Second Century calculations.  But in the early 1960’s astronomers realized that the rotation of the earth is irregular– fundamentally, it is slowing.  Coordinated Universal Time (CUT), calculated with an atomic clock, was systematically slowed each year, for a decade, to compensate…. But that meant that CUT and UTC (the time standard used by broadcasters, transportation providers, and other commercial and military users, a standard still fixed on the original definition of the “second”) were diverging.  To true them up, the leap second was added to the UTC.  Since 1972, a total of 25 seconds have been added– that’s to say, the Earth has slowed down 25 seconds compared to atomic time since then.  (But this does not mean that days are 25 seconds longer nowadays: only the days on which the leap seconds are inserted have 86,401 instead of the usual 86,400 seconds.)




Written by (Roughly) Daily

June 30, 2014 at 1:01 am

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