## My All 2020 Mathematics A to Z: Fibonacci

Dina Yagodich suggested today’s A-to-Z topic. I thought a quick little biography piece would be a nice change of pace. I discovered things were more interesting than that.

# Fibonacci.

I realized preparing for this that I have never read a biography of Fibonacci. This is hardly unique to Fibonacci. Mathematicians buy into the legend that mathematics is independent of human creation. So the people who describe it are of lower importance. They learn a handful of romantic tales or good stories. In this way they are much like humans. I know at least a loose sketch of many mathematicians. But Fibonacci is a hard one for biography. Here, I draw heavily on the book Fibonacci, his numbers and his rabbits, by Andriy Drozdyuk and Denys Drozdyuk.

We know, for example, that Fibonacci lived until at least 1240. This because in 1240 Pisa awarded him an annual salary in recognition of his public service. We think he was born around 1170, and died … sometime after 1240. This seems like a dismal historical record. But, for the time, for a person of slight political or military importance? That’s about as good as we could hope for. It is hard to appreciate how much documentation we have of lives now, and how recent a phenomenon that is.

Even a fact like “he was alive in the year 1240” evaporates under study. Italian cities, then as now, based the year on the time since the notional birth of Christ. Pisa, as was common, used the notional conception of Christ, on the 25th of March, as the new year. But we have a problem of standards. Should we count the year as the number of full years since the notional conception of Christ? Or as the number of full and partial years since that important 25th of March?

If the question seems confusing and perhaps angering let me try to clarify. Would you say that the notional birth of Christ that first 25th of December of the Christian Era happened in the year zero or in the year one? (Pretend there was a year zero. You already pretend there was a year one AD.) Pisa of Leonardo’s time would have said the year one. Florence would have said the year zero, if they knew of “zero”. Florence matters because when Florence took over Pisa, they changed Pisa’s dating system. Sometime later Pisa changed back. And back again. Historians writing, aware of the Pisan 1240 on the document, may have corrected it to the Florence-style 1241. Or, aware of the change of the calendar and not aware that their source already accounted for it, redated it 1242. Or tried to re-correct it back and made things worse.

This is not a problem unique to Leonardo. Different parts of Europe, at the time, had different notions for the year count. Some also had different notions for what New Year’s Day would be. There were many challenges to long-distance travel and commerce in the time. Not the least is that the same sun might shine on at least three different years at once.

We call him Fibonacci. Did he? The question defies a quick answer. His given name was Leonardo, and he came from Pisa, so a reliable way to address him would have “Leonardo of Pisa”, albeit in Italian. He was born into the Bonacci family. He did in some manuscripts describe himself as “Leonardo filio Bonacci Pisano”, give or take a few letters. My understanding is you can get a good fun quarrel going among scholars of this era by asking whether “Filio Bonacci” would mean “the son of Bonacci” or “of the family Bonacci”. Either is as good for us. It’s tempting to imagine the “Filio” being shrunk to “Fi” and the two words smashed together. But that doesn’t quite say that Leonardo did that smashing together.

Nor, exactly, when it did happen. We see “Fibonacci” used in mathematical works in the 19th century, followed shortly by attempts to explain what it means. We know of a 1506 manuscript identifying Leonardo as Fibonacci. But there remains a lot of unexplored territory.

If one knows one thing about Fibonacci though, one knows about the rabbits. They give birth to more rabbits and to the Fibonacci Sequence. More on that to come. If one knows two things about Fibonacci, the other is about his introducing Arabic numerals to western mathematics. I’ve written of this before. And the subject is … more ambiguous, again.

Most of what we “know” of Fibonacci’s life is some words he wrote to explain why he was writing his bigger works. If we trust he was not creating a pleasant story for the sake of engaging readers, then we can finally say something. (If one knows three things about Fibonacci, and then five things, and then eight, one is making a joke.)

Fibonacci’s father was, in the 1290s, posted to Bejaia, a port city on the Algerian coast. The father did something for Pisa’s duana there. And what is a duana? … Again, certainty evaporates. We have settled on saying it’s a customs house, and suppose our readers know what goes on in a customs house. The duana had something to do with clearing trade through the port. His father’s post was as a scribe. He was likely responsible for collecting duties and registering accounts and keeping books and all that. We don’t know how long Fibonacci spent there. “Some days”, during which he alleges he learned the digits 1 through 9. And after that, travelling around the Mediterranean, he saw why this system was good, and useful. He wrote books to explain it all and convince Europe that while Roman numerals were great, Arabic numerals were more practical.

It is always dangerous to write about “the first” person to do anything. Except for Yuri Gagarin, Alexei Leonov, and Neil Armstrong, “the first” to do anything dissolves into ambiguity. Gerbert, who would become Pope Sylvester II, described Arabic numerals (other than zero) by the end of the 10th century. He added in how this system along with the abacus made computation easier. Arabic numerals appear in the Codex Conciliorum Albeldensis seu Vigilanus, written in 976 AD in Spain. And it is not as though Fibonacci was the first European to travel to a land with Arabic numerals, or the first perceptive enough to see their value.

Allow that, though. Every invention has precursors, some so close that it takes great thinking to come up with a reason to ignore them. There must be some credit given to the person who gathers an idea into a coherent, well-explained whole. And with Fibonacci, and his famous manuscript of 1202, the Liber Abaci, we have … more frustration.

It’s not that Liber Abaci does not exist, or that it does not have what we credit it for having. We don’t have any copies of the 1202 edition, but we do have a 1228 manuscript, at least, and that starts out with the Arabic numeral system. And why this system is so good, and how to use it. It should convince anyone who reads it.

If anyone read it. We know of about fifteen manuscripts of Liber Abaci, only two of them reasonably complete. This seems sparse even for manuscripts in the days they had to be hand-copied. This until you learn that Baldassarre Boncompagni published the first known printed version in 1857. In print, in Italian, it took up 459 pages of text. Its first English translation, published by Laurence E Sigler in 2002(!) takes up 636 pages (!!). Suddenly it’s amazing that as many as two complete manuscripts survive. (Wikipedia claims three complete versions from the 13th and 14th centuries exist. And says there are about nineteen partial manuscripts with another nine incomplete copies. I do not explain this discrepancy.)

He had other books. The Liber Quadratorum, for example, a book about algebra. Wikipedia seems to say we have it through a single manuscript, copied in the 15th century. Practica Geometriae, translated from Latin in 1442 at least. A couple other now-lost manuscripts. A couple pieces about specific problems.

So perhaps only a handful of people read Fibonacci. Ah, but if they were the right people? He could have been a mathematical Velvet Underground, read by a hundred people, each of whom founded a new mathematics.

We could trace those hundred readers by the first thing anyone knows Fibonacci for. His rabbits, breeding in ways that rabbits do not, and the sequence of whole numbers those provide. Fibonacci did not discover this sequence. You knew that. Nothing in mathematics gets named for its discoverer. Virahanka, an Indian mathematician who lived somewhere between the sixth and eighth centuries, described the sequence exactly. Gopala, writing sometime in the 1130s, expanded on this.

This is not to say Fibonacci copied any of these (and more) Indian mathematicians. The world is large and manuscripts are hard to read. The sequence can be re-invented by anyone bored in the right way. Ah, but think of those who learned of the sequence and used it later on, following Fibonacci’s lead. For example, in 1611 Johannes Kepler wrote a piece that described Fibonacci’s sequence. But that does not name Fibonacci. He mentions other mathematicians, ancient and contemporary. The easiest supposition is he did not know he was writing something already seen. In 1844, Gabriel Lamé used Fibonacci numbers in studying algorithm complexity. He did not name Fibonacci either, though. (Lamé is famous today for making some progress on Fermat’s last theorem. He’s renowned for work in differential equations and on ellipse-like curves. If you have thought what a neat weird shape the equation $x^4 + y^4 = 1$ can describe you have tread in Lamé’s path.)

Things picked up for Fibonacci’s reputation in 1876, thanks to Édouard Lucas. (Lucas is notable for other things. Normal people might find interesting that he proved by hand the number $2^{127} - 1$ was prime. This seems to be the largest prime number ever proven by hand. He also created the Tower of Hanoi problem.) In January of 1876, Lucas wrote about the Fibonacci sequence, describing it as “the series of Lamé”. By May, though in writing about prime numbers, he has read Boncompagni’s publications. He says how this thing “commonly known as the sequence of Lamé was first presented by Fibonacci”.

And Fibonacci caught Lucas’s imagination. Lucas shared, particularly, the phrasing of this sequence as something in the reproduction of rabbits. This captured mathematicians’, and then people’s imaginations. It’s akin to Émile Borel’s room of a million typing monkeys. By the end of the 19th century Leonardo of Pisa had both a name and fame.

We can still ask why. The proximate cause is Édouard Lucas, impressed (I trust) by Boncompagni’s editions of Fibonacci’s work. Why did Baldassarre Boncompagni think it important to publish editions of Fibonacci? Well, he was interested in the history of science. He edited the first Italian journal dedicated to the history of mathematics. He may have understood that Fibonacci was, if not an important mathematician, at least one who had interesting things to write. Boncompagni’s edition of Liber Abaci came out in 1857. By 1859 the state of Tuscany voted to erect a statue.

So I speculate, without confirming that at least some of Fibonacci’s good name in the 19th century was a reflection of Italian unification. The search for great scholars whose intellectual achievements could reflect well on a nation trying to build itself.

And so we have bundles of ironies. Fibonacci did write impressive works of great mathematical insight. And he was recognized at the time for that work. The things he wrote about Arabic numerals were correct. His recommendation to use them was taken, but by people who did not read his advice. After centuries of obscurity he got some notice. And a problem he did not create nor particularly advance brought him a fame that’s lasted a century and a half now, and looks likely to continue.

I am always amazed to learn there are people not interested in history.

And now I can try to get ahead of deadline for next week. This and all my other A-to-Z topics for the year should be at this link. All my essays for this and past A-to-Z sequences are at this link. And I am still taking topics to discuss in the coming weeks. Thank you for reading and please take care.

## Reading the Comics, February 11, 2017: Trivia Edition

And now to wrap up last week’s mathematically-themed comic strips. It’s not a set that let me get into any really deep topics however hard I tried overthinking it. Maybe something will turn up for Sunday.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 7th tries setting arithmetic versus celebrity trivia. It’s for the old joke about what everyone should know versus what everyone does know. One might question whether Kardashian pet eating habits are actually things everyone knows. But the joke needs some hyperbole in it to have any vitality and that’s the only available spot for it. It’s easy also to rate stuff like arithmetic as trivia since, you know, calculators. But it is worth knowing that seven squared is pretty close to 50. It comes up when you do a lot of estimates of calculations in your head. The square root of 10 is pretty near 3. The square root of 50 is near 7. The cube root of 10 is a little more than 2. The cube root of 50 a little more than three and a half. The cube root of 100 is a little more than four and a half. When you see ways to rewrite a calculation in estimates like this, suddenly, a lot of amazing tricks become possible.

Leigh Rubin’s Rubes for the 7th is a “mathematics in the real world” joke. It could be done with any mythological animals, although I suppose unicorns have the advantage of being relatively easy to draw recognizably. Mermaids would do well too. Dragons would also read well, but they’re more complicated to draw.

Mark Pett’s Mr Lowe rerun for the 8th has the kid resisting the mathematics book. Quentin’s grounds are that how can he know a dated book is still relevant. There’s truth to Quentin’s excuse. A mathematical truth may be universal. Whether we find it interesting is a matter of culture and even fashion. There are many ways to present any fact, and the question of why we want to know this fact has as many potential answers as it has people pondering the question.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th is a paean to one of the joys of numbers. There is something wonderful in counting, in measuring, in tracking. I suspect it’s nearly universal. We see it reflected in people passing around, say, the number of rivets used in the Chrysler Building or how long a person’s nervous system would reach if stretched out into a line or ever-more-fanciful measures of stuff. Is it properly mathematics? It’s delightful, isn’t that enough?

Scott Hilburn’s The Argyle Sweater for the 10th is a Fibonacci Sequence joke. That’s a good one for taping to the walls of a mathematics teacher’s office.

Bill Rechin’s Crock rerun for the 11th is a name-drop of mathematics. Really anybody’s homework would be sufficiently boring for the joke. But I suppose mathematics adds the connotation that whatever you’re working on hasn’t got a human story behind it, the way English or History might, and that it hasn’t got the potential to eat, explode, or knock a steel ball into you the way Biology, Chemistry, or Physics have. Fair enough.

## The End 2016 Mathematics A To Z: Yang Hui’s Triangle

Today’s is another request from gaurish and another I’m glad to have as it let me learn things too. That’s a particularly fun kind of essay to have here.

## Yang Hui’s Triangle.

It’s a triangle. Not because we’re interested in triangles, but because it’s a particularly good way to organize what we’re doing and show why we do that. We’re making an arrangement of numbers. First we need cells to put the numbers in.

Start with a single cell in what’ll be the top middle of the triangle. It spreads out in rows beneath that. The rows are staggered. The second row has two cells, each one-half width to the side of the starting one. The third row has three cells, each one-half width to the sides of the row above, so that its center cell is directly under the original one. The fourth row has four cells, two of which are exactly underneath the cells of the second row. The fifth row has five cells, three of them directly underneath the third row’s cells. And so on. You know the pattern. It’s the one that pins in a plinko board take. Just trimmed down to a triangle. Make as many rows as you find interesting. You can always add more later.

In the top cell goes the number ‘1’. There’s also a ‘1’ in the leftmost cell of each row, and a ‘1’ in the rightmost cell of each row.

What of interior cells? The number for those we work out by looking to the row above. Take the cells to the immediate left and right of it. Add the values of those together. So for example the center cell in the third row will be ‘1’ plus ‘1’, commonly regarded as ‘2’. In the third row the leftmost cell is ‘1’; it always is. The next cell over will be ‘1’ plus ‘2’, from the row above. That’s ‘3’. The cell next to that will be ‘2’ plus ‘1’, a subtly different ‘3’. And the last cell in the row is ‘1’ because it always is. In the fourth row we get, starting from the left, ‘1’, ‘4’, ‘6’, ‘4’, and ‘1’. And so on.

It’s a neat little arithmetic project. It has useful application beyond the joy of making something neat. Many neat little arithmetic projects don’t have that. But the numbers in each row give us binomial coefficients, which we often want to know. That is, if we wanted to work out (a + b) to, say, the third power, we would know what it looks like from looking at the fourth row of Yanghui’s Triangle. It will be $1\cdot a^4 + 4\cdot a^3 \cdot b^1 + 6\cdot a^2\cdot b^2 + 4\cdot a^1\cdot b^3 + 1\cdot b^4$. This turns up in polynomials all the time.

Look at diagonals. By diagonal here I mean a line parallel to the line of ‘1’s. Left side or right side; it doesn’t matter. Yang Hui’s triangle is bilaterally symmetric around its center. The first diagonal under the edges is a bit boring but familiar enough: 1-2-3-4-5-6-7-et cetera. The second diagonal is more curious: 1-3-6-10-15-21-28 and so on. You’ve seen those numbers before. They’re called the triangular numbers. They’re the number of dots you need to make a uniformly spaced, staggered-row triangle. Doodle a bit and you’ll see. Or play with coins or pool balls.

The third diagonal looks more arbitrary yet: 1-4-10-20-35-56-84 and on. But these are something too. They’re the tetrahedronal numbers. They’re the number of things you need to make a tetrahedron. Try it out with a couple of balls. Oranges if you’re bored at the grocer’s. Four, ten, twenty, these make a nice stack. The fourth diagonal is a bunch of numbers I never paid attention to before. 1-5-15-35-70-126-210 and so on. This is — well. We just did tetrahedrons, the triangular arrangement of three-dimensional balls. Before that we did triangles, the triangular arrangement of two-dimensional discs. Do you want to put in a guess what these “pentatope numbers” are about? Sure, but you hardly need to. If we’ve got a bunch of four-dimensional hyperspheres and want to stack them in a neat triangular pile we need one, or five, or fifteen, or so on to make the pile come out neat. You can guess what might be in the fifth diagonal. I don’t want to think too hard about making triangular heaps of five-dimensional hyperspheres.

There’s more stuff lurking in here, waiting to be decoded. Add the numbers of, say, row four up and you get two raised to the third power. Add the numbers of row ten up and you get two raised to the ninth power. You see the pattern. Add everything in, say, the top five rows together and you get the fifth Mersenne number, two raised to the fifth power (32) minus one (31, when we’re done). Add everything in the top ten rows together and you get the tenth Mersenne number, two raised to the tenth power (1024) minus one (1023).

Or add together things on “shallow diagonals”. Start from a ‘1’ on the outer edge. I’m going to suppose you started on the left edge, but remember symmetry; it’ll be fine if you go from the right instead. Add to that ‘1’ the number you get by moving one cell to the right and going up-and-right. And then again, go one cell to the right and then one cell up-and-right. And again and again, until you run out of cells. You get the Fibonacci sequence, 1-1-2-3-5-8-13-21-and so on.

We can even make an astounding picture from this. Take the cells of Yang Hui’s triangle. Color them in. One shade if the cell has an odd number, another if the cell has an even number. It will create a pattern we know as the Sierpiński Triangle. (Wacław Sierpiński is proving to be the surprise special guest star in many of this A To Z sequence’s essays.) That’s the fractal of a triangle subdivided into four triangles with the center one knocked out, and the remaining triangles them subdivided into four triangles with the center knocked out, and on and on.

By now I imagine even my most skeptical readers agree this is an interesting, useful mathematical construct. Also that they’re wondering why I haven’t said the name “Blaise Pascal”. The Western mathematical tradition knows of this from Pascal’s work, particularly his 1653 Traité du triangle arithmétique. But mathematicians like to say their work is universal, and independent of the mere human beings who find it. Constructions like this triangle give support to this. Yang lived in China, in the 12th century. I imagine it possible Pascal had hard of his work or been influenced by it, by some chain, but I know of no evidence that he did.

And even if he had, there are other apparently independent inventions. The Avanti Indian astronomer-mathematician-astrologer Varāhamihira described the addition rule which makes the triangle work in commentaries written around the year 500. Omar Khayyám, who keeps appearing in the history of science and mathematics, wrote about the triangle in his 1070 Treatise on Demonstration of Problems of Algebra. Again so far as I am aware there’s not a direct link between any of these discoveries. They are things different people in different traditions found because the tools — arithmetic and aesthetically-pleasing orders of things — were ready for them.

Yang Hui wrote about his triangle in the 1261 book Xiangjie Jiuzhang Suanfa. In it he credits the use of the triangle (for finding roots) as invented around 1100 by mathematician Jia Xian. This reminds us that it is not merely mathematical discoveries that are found by many peoples at many times and places. So is Boyer’s Law, discovered by Hubert Kennedy.

## Reading the Comics, October 8, 2016: Split Week Edition Part 2

And now I can finish off last week’s comics. It was a busy week. The first few days of this week have been pretty busy too. Meanwhile, Dave Kingsbury has recently read a biography of Lewis Carroll, and been inspired to form a haiku/tanka project. You might enjoy.

Susan Camilleri Konar is a new cartoonist for the Six Chix collective. Her first strip to get mentioned around these parts is from the 5th. It’s a casual mention of the Fibonacci sequence, which is one of the few sequences that a normal audience would recognize as something going on forever. And yes, I noticed the spiral in the background. That’s one of the common visual representations of the Fibonacci sequence: it starts from the center. The rectangles inside have dimensions 1 by 2, then 2 by 3, then 3 by 5, then 5 by 8, and so on; the spiral connects vertices of these rectangles. It’s an attractive spiral and you can derive the overrated Golden Ratio from the dimensions of larger rectangles. This doesn’t make the Golden Ratio important or anything, but it is there.

Ryan North’s Dinosaur Comics for the 6th is part of a story about T-Rex looking for certain truth. Mathematics could hardly avoid coming up. And it does offer what look like universal truths: given the way deductive logic works, and some starting axioms, various things must follow. “1 + 1 = 2” is among them. But there are limits to how much that tells us. If we accept the rules of Monopoly, then owning four railroads means the rent for landing on one is a game-useful \$200. But if nobody around you cares about Monopoly, so what? And so it is with mathematics. Utahraptor and Dromiceiomimus point out that the mathematics we know is built on premises we have selected because we find them interesting or useful. We can’t know that the mathematics we’ve deduced has any particular relevance to reality. Indeed, it’s worse than North points out: How do we know whether an argument is valid? Because we believe that its conclusions follow from its premises according to our rules of deduction. We rely on our possibly deceptive senses to tell us what the argument even was. We rely on a mind possibly upset by an undigested bit of beef, a crumb of cheese, or a fragment of an underdone potato to tell us the rules are satisfied. Mathematics seems to offer us absolute truths, but it’s hard to see how we can get there.

Rick Stromoskis Soup to Nutz for the 6th has a mathematics cameo in a student-resisting-class-questions problem. But the teacher’s question is related to the figure that made my first fame around these parts.

Mark Anderson’s Andertoons for the 7th is the long-awaited Andertoon for last week. It is hard getting education in through all the overhead.

Bill Watterson’s Calvin and Hobbes rerun for the 7th is a basic joke about Calvin’s lousy student work. Fun enough. Calvin does show off one of those important skills mathematicians learn, though. He does do a sanity check. He may not know what 12 + 7 and 3 + 4 are, but he does notice that 12 + 7 has to be something larger than 3 + 4. That’s a starting point. It’s often helpful before starting work on a problem to have some idea of what you think the answer should be.

## Fibonacci’s Biased Scarf

Here is a neat bit of crochet work with a bunch of nice recreational-mathematics properties. The first is that the distance between yellow rows, or between blue rows, represents the start of the Fibonacci sequence of numbers. I’m not sure if the Fibonacci sequence is the most famous sequence of whole numbers but it’s certainly among the most famous, and it’s got interesting properties and historical context.

The second recreational-mathematics property is that the pattern is rotationally symmetric. Rotate it 180 degrees and you get back the original pattern, albeit with blue and yellow swapped. You can form a group out of the ways that it’s possible to rotate an object and get back something that looks like the original. Symmetry groups can be things of simple aesthetic beauty, describing scarf patterns and ways to tile floors and the like. They can also describe things of deep physical significance. Much of the ability of quantum chromodynamics to describe nuclear physics comes from these symmetry groups.

The logo at top of the page is of a trefoil knot, which I’d mentioned a couple weeks back. A trefoil knot isn’t perfectly described by its silhouette. Where the lines intersect you have to imagine the string (or whatever makes up the knot) passing twice, once above and once below itself. If you do that crossing-over and crossing-under consistently you get the trefoil knot, the simplest loop that isn’t an unknot, that can’t be shaken loose into a simple circle.

This scarf is totally biased. That’s not to say that it’s prejudiced, but that it was worked in the diagonal direction of the cloth.

My project was made from Julie Blagojevich’s free pattern Fibonacci’s Biased using Knit Picks Curio. The number of rows in each stripe is according to the numbers of the Fibonacci sequence up to 34. In other words, if you start at the blue side of the scarf and work your way right, the sequence of the number of yellow rows is 1, 1, 2, 3, 5, 8, 13, 21, 34. The sequence of the blue stripes are the same, but in the opposite direction. The effect is a rotationally symmetric scarf with few color changes at the edges and frequent color changes in the center. As I frequently tell my friends, math is beautiful.

If my geekiness hasn’t scared you away yet, here’s a random fun…

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## Fibonacci, a Comic Strip, and Venice

The comic strip Frazz, by Jef Mallett, touches another bit of mathematics humor. I imagine if I were better-organized I’d gather all the math comic strips I see over a whole week and report on them all at once, but, I’m still learning the rules of this blog, other than that anyone writing about mathematics has to bring up Fibonacci whether they want to or not.

The association that sequins brings up for me now, though, and has ever since a book I read about the United States’s war on the Barbary Coast pirates, is that the main coin of Venice for over 500 years of its existence as an independent republic was the sequin, giving me notions of financial transactions being all sparkly and prone to blowing away in a stiff breeze. It wasn’t that kind of sequin, of course or even any sort of particularly small coin. The Venetian sequin was a rather average-looking gold coin, weighing at least nominally three and a half grams, and the name was a mutation of “zecchino”, after the name of Venice’s mint. But, apparently, the practice of sewing coins like this into women’s clothing or accessories lead to the attaching of small, shiny objects into clothing or accessories, and so gave us sequins after all.

A listing on a coin collectors site tells me the Venetian sequin was about two centimeters in diameter, which isn’t ridiculously tiny at least. I’m not sure if that is a reliable guide to the size, although since it’s trying to sell me rare coins, probably it’s not too far off. Unfortunately most of the top couple pages of Google hits on “Venetian sequin coin size” brings up copies of Wikipedia’s report, which fails to mention physical size. An Ottoman sequin at the British Museum’s web site lists its diameter as 2.4 centimeters, but its weight at four and a third grams.