If We Only Had Twelve Fingers

Standard Kilogram Mass, one of 40 made in 1884 which were exact copies of the international prototype kilogram kept at the Bureau International des Poids et Mesures in Sèvres, France

Sitting around tonight arguing with my friend Gwyan about the Metric System, I found myself embroiled in a very interesting discussion about the nature of measurement and the extent to which people will follow rationalism.

The thing is, despite the fact that it's based on our own ten fingers, I don't like the metric system. I don't like a system that requires decimal calculations and which won't easily divide by anything other than 5 or 2. It is not ultimately logical for people who make clothes out of four basic panels (and have to size those panels up and down), and in my opinion anything that requires an infinitely repeating decimal to represent a third of the measuring unit is crazy. It just doesn't make my life better. The system was made up by a bunch of rationalists who got carried away with creating a completely new system that people in different countries would accept -- for the sole reason that they wanted something new. It figures that it caught on -- only something this untidy and bizarre would.

Well, mostly. The British didn't accept the metric system for many, many years, despite the Victorian institution of universal education -- probably because the system had originated in France. But that's a whole 'nother story. Curiously, though, the idea originated with an Englishman, John Wilkins, first secretary of the Royal Society of London, in 1668. The idea didn't catch on, and the English went right on with their intricate monetary system and their 20-ounce pints.


But then "in 1670, Gabriel Mouton, a French abbot and scientist, proposed a decimal system of measurement based on the circumference of the Earth... His ideas attracted interest at the time, and were supported by both Jean Picard and Christiaan Huygens in 1673." [wiki]

Well, that explains a lot. In the days of Reason and Enlightenment, systems which tidied up numbers and arranged them in clean lines and shapes were all the rage. Metrics are a perfect example:

"1000 litres = 1 cubic metre ≈ 1 tonne of water; 1 litre = 1 cubic decimetre ≈ 1 kilogram of water; 1 millilitre = 1 cubic centimetre ≈ 1 gram of water; and 1 microlitre = 1 cubic millimetre ≈ 1 milligram of water."

This would appeal enormously to a culture of gleeful intellectualism, the same one that came up with Napier's Bones and calculus.


It was dear, rational, idealistic France who went for the wholehearted changeover, of course:

"The inconsistency problem was not one of different units but one of differing sized units. Instead of simply standardising the size of the existing units, the leaders of the French revolutionary Assemblée Constituante decided that a completely new system should be adopted. It was felt that no country would accept standardising on the units of another country, but that there would be less resistance if a completely new system made change compulsory for all countries." [wiki]

In other words, they threw out measurements that had been working for individual people for hundreds of years or more, because of an ideal. Not a bad thing, maybe, and in talking with Gwyan, I was hard-pressed to describe my aversion to base-10 systems of measurement. I don't have a problem with base-10 monetary systems; money is, after all, pretty much about numbers, and our numeric system is base-10, so it follows. It's pretty straightforward that any being with ten digits is going to have a base-10 number system. And the beauty of the metric system is that if the units you're working with start to need dividing, you can simply slide down into the next unit level and viola! You're working with whole numbers again. It's a different way of thinking: you're not working so much with pieces and parts, but rather with a sort of layered mesh of wholes, through which you can move as needed. Which is fine for distance or weight, but not so good for discreet objects like eggs or minutes.

And that 1/3 measure, that sticks in my throat. You can go on sliding downward in unit size forever and never get to the bottom of the number; it will always be an estimate, a rounding-up or -down. And it bothers me, as someone who used to work in the garment industry, that dividing things in fourths involves such an awkward number as 2.5, or even 25. Those are not friendly numbers (*see below); they don't show up in the kind of kitchen that has a cast-iron pot at the fire and herbs hanging from the ceiling. These numbers don't believe in us and our four-cornered world; so I don't believe in them, either (so there).

The madness of post-Revolutionary France bears me out on this. They redesigned everything to be about tens: the 10-hour clock (as opposed to 12-hour); their new calendar had 12 months but with 10-day weeks; and of course, money, length, weight, volume and so on. The breadth of it was staggering: they were redesigning the universe to fit itself to our hands -- our five-fingered, flower-like hands.

(Image by Sue Ford)

True, there is something beautiful and otherworldly about the number 5. It exists in nature, but it doesn't fit into everyday symmetry the way the simple triangle can. Drawing a pentagram accurately is a tricky proposition. We don't think in fives: when we count pennies, most people make groups of twos and threes. It is beyond and above the natural grooves of our minds, and this may be why the pentagram (and pentagon) has always had such magical significance**.

But really -- should we be redesigning our whole cultural definition of space and mass into fives? They may be beautiful, but they are absolutely not practical, at least not in any world that I inhabit.

(Image thanks to The Steampunk Home)

Interestingly, Gwyan pointed out that the metric system is more useful in bureaucracies, mass-production, and science, where the numbers need to be able to go very large or very small. This is a wonderful point, because I think what I object to about the conversion is that it is designed to benefit those industries -- not individual people, moving through their individual lives. This is precisely why the calendar is still divided into twelves, and why the 10-hour clock simply failed; why dozens and grosses are still used in bakeries and eggs in many places. People like to be able to divide time and goods many different ways, not simply into two possible factors, and fractions thereof. The Romans had a unit called an uncia, which is the basis of our words for "inch" and "ounce"; it was part of a fractional system based on twelfths.

Interestingly, there are several languages who use duodecimal number systems (otherwise known as base-12). I'm not referring to Elvish here (apparently it's one example); in Nigeria, there are several, as well as a few obscure Nepalese and Indian languages.

Another place, at least in the US, that is unlikely to change very soon is in the kitchen. Cups, ounces, and teaspoons were arrived at through usage, through what worked easily with the tools at hand.

I have to say, it's not that I dont like tens; they work just fine in a mathematical context -- for counting things, it's certainly a good idea to have your counting system match your number of fingers. It's more that for everyday use you sometimes simply can't beat the number twelve. Even those of you who write in saying you're fine with the metric system still use a 12-hour clock and a 12-month year; would you prefer it differently? And for those of you, who like me, simply like the number 12, there are "dozenal" societies in the US and the UK (they forsake the word duodecimal because it means ten plus two, which they feel is beside the point). Perhaps I'll join. After all, what a fabulous number: dividable by 2, 3, 4, and 6. Definitely a keeper.



*Note: when I say friendly numbers here, I am referring to a different property than that of the friendly numbers of number theory; nor are they amicable numbers or sociable numbers, some of which have been around since Pythagorean times.

**Pentacles, on the other hand, do not originally have an association with the number five. I didn't know that until the moment of this writing.

Mr. Bowditch: Brilliance and Populism

I was looking through something the other day and I came across a reference to someone named Bowditch. Immediately, my mind was flooded with images from the life of Nathaniel Bowditch, a mathematical and navigational savant in the late 1700's and early 1800's, whom I read about at age 12 or so in a wonderful biographical novel for middle-graders called Carry On, Mr. Bowditch.

Bowditch was the son of a cooper in Salem, Massachusetts who left school at the age of ten to work for his father. At twelve he was indentured for nine years as a bookkeeping apprentice to a ship chandler. They were kind to him there and he was given access to the library; being immersed all day in numbers, and having a sharp brain, he became interested in some of the more complex ideas surrounding math, spending his days learning all about what it takes to outfit a ship and his nights studying.

"In 1787, aged fourteen, Bowditch began to study algebra and two years later he taught himself calculus. He also taught himself Latin in 1790 and French in 1792 so he was able to read mathematical works such as Isaac Newton's Philosophiae Naturalis Principia Mathematica. At seventeen, he wrote a letter to a Harvard University professor pointing out an error in the Principia... Serendipity aided Bowditch's self-directed study in as much as he found himself able to use the eminent Irish chemist Richard Kirwan's library: a privateer from Salem had intercepted the ship carrying the library between Ireland and England and brought the library back to Salem in June 1791." [wiki]

When he was twenty-two, Mr.Bowditch went to sea as a ship's clerk and captain's writer. Working from the error-prone navigational books of the day was so frustrating to him that by his fifth voyage, of which he was master and part owner of the ship, he "decided to write his own book, and to 'put down in the book nothing I can't teach the crew.' On that trip, it is said that every man of the crew of 12, including the ship's cook, became competent to take and calculate lunar observations and to plot the correct position of the ship." So in 1802 The American Practical Navigator, was published. It was such an accurate and useful book that it immediately revolutionized navigation, and is still carried onboard every commissioned U.S. Naval vessel today. The first edition of Bowditch's American Practical Navigator, which became the western hemisphere shipping industry standard for the next century and a half, and in 1866, the United States Hydrographic Office purchased the copyright. Since then the book, with appropriate revisions, has been in continuous publication, and to this day it is simply known as Bowditch.


Reading about him when I was a young person, I was struck mostly by the variety and extraordinariness of his experiences, and the fascinating idea that math had such concrete applications. It had never occurred to me to think of math the way Mr. Bowditch did, as a conceptual thing and a way of looking at the world. I never thought of myself as someone who was good at math, even though later in life I became adept at pattern-making (making 3D objects from 2D patterns) and had a keen mechanical understanding of math-related concepts, based more on intuition than education, which allowed me to do well in all sorts of arenas.

Nowadays, of course, children learn the beginnings of algebra in third and fourth grade. The more conceptual elements of math begin much earlier, and along with the usual times tables school children are taught to see math as a schema, a place to play around with numbers; the basic elements of math's abstractions are set in place in more malleable brains. Add to this the basic tenets of binary and hexidecimal systems, and you get some seriously young minds working on much more abstract levels than of old. ...At least, for those minds who are actually getting an education.

Which brings me to another point: this Bowditch person was pretty much self-taught. Remember, too, that calculus as a systemic study had only been fully developed a hundred years before, so when Mr. Bowditch found an error in Newton's masterpiece only three years into his study of higher math, it showed considerable brainpower. This is something I have only come to appreciate as an adult: both in terms of Isaac Newton and in terms of understanding Bowditch's learning rate. These, and the fact that such a brilliant man should go on to become an insurance actuary, a job which he kept until his death at the age of sixty-three, strike me strongly now that I have some perspective on them.

But what strikes me particularly, and what endears me most to the man, is his deep-seated belief that every man on a ship can and should be able to navigate, despite the mores of the day which would have it that men on a ship are expendable, and only the navigator has the knowledge (and only he should have the knowledge) to find the ship on a map - to navigate. His stubbornness on this point is legendary, and his willingness to prove it comes not only from a man who taught himself to be who he was, but from a time and a place: Massachussetts, in post-Revolutionary America. Best of all, he put his belief to work in writing, too - which is why his book is still relevant today, and why we have such people as Dee Caffari out there on the seas.

Well done, Mr. Bowditch.

A Rule of Thumb

"Fingerprints have been found on ancient Babylonian clay tablets, seals, and pottery. They have also been found on the walls of Egyptian tombs and on Minoan, Greek, and Chinese pottery — as well as on bricks and tiles in Babylon and Rome. ...on some pottery, fingerprints were impressed so deeply that they were likely intended to serve as the equivalent of a brand label." [wiki]

There is something eternally fascinating about the ridges and whorls on our hands and feet, those unrepeatable patterns which cover most of what is termed our "volar skin", that is, skin of the palms of the hands or the soles of the feet. When I was a kid I spent hours staring at the swirls and lines, looking at where they ended - and wondering why they were there. "Designs" I called them, when I was young.

I even went through a period, when I learned about fingerprinting and the idea that everyone has completely different fingerprints, where I made everyone around me (mostly adults) squash their fingers onto my ink-pad and leave their mark on the paper which I carried around for the purpose. Of course, it wasn't washable ink, so there seemed to be an inordinate number of long-suffering, black-fingered folks around my household.

The other night I got to talking with friends about fingerprints. How do they work? Why do we have them? The conversation didn't go very far, but it did make me decide to go look it up. Forty websites later, I am still no expert, but I continue to be fascinated.


(Koala fingerprint, above, versus human, below)

For example, did you know that koalas are one of the few mammals besides primates who have fingerprints, and in fact even with an electron microscope, it is difficult to tell koala prints apart from human prints? There's a mystery story in there somewhere, like The Murders in the Rue Morgue only (hopefully) more believable (anyone met a murderous orangutan lately?). Fishers are also said to have fingerprints, which seems to me very strange: if fishers do, why not stoats? Weasels? And so on?


Spider monkeys, whose prehensile tail-tips are so sensitive and flexible that they can pick a dime up off a floor, also have prints on the bare spot at the end of their tails. Since the tails are used not only as a sort of third arm when swinging in the trees (as a safeguard from falling), but often supports the entire weight of their bodies while they feed, this would make sense: fingerprints, and other places with "friction ridges" - the volar regions - generally tend to occur where one needs to grip something. This can mean gripping an object to keep from dropping it, or (as in the case of trees) to keep it from dropping you, or simply to keep your feet steady on the rocks so you don't fall off a cliff.

But how does it work? One source I was perusing posited that there could be a Van der Waals force element, like gecko's feet. The person cited the fact that our fingertips can feel the grittiness of a powder down to about 150 microns, and then it just didn't feel gritty anymore; since Van der Waals' forces tend to show up more when something is 150 microns or smaller, he conjectured a connection.


Other sources, however, didn't support this idea, even if it appealed to me. The general belief among my local pundits was that friction ridges weren't deep enough, enclosing enough or wet enough for either suction or for cohesion; and their structure wasn't complex enough for Van der Waals. The consensus was almost entirely on friction. Given that the flesh in these dermal ridges (to use another term) are notoriously squashy (thus making crime scene fingerprints - known as "latent prints" seriously difficult to decipher), the friction thing holds up as an answer. Just as tires made of squishy gel are more likely to stick to the road than ones made of hard plastic, so do the flexible, moist areas on our hands and feet provide an excellent surface to grip with. Thus does the fingerprint contribute to our development as tool-users.


Dermal ridges develop in the womb, and are pretty much developed by seventeen weeks. The patterns on our fingers are influenced by our time in the womb: subtle stresses and tensions affect how they grow, creating uniqueness through a combination of genetics and in utero experience (as can be seen by genetically identical twins, who don't have identical fingerprints). Once the fingerprints are set, they cannot be altered easily:

"Should the top layer of skin suffer any injury, the ridges grow back after healing in the exact pattern they had before. Therefore, superficial cuts or abrasions alter fingerprint characteristics only temporarily. If the injury reaches deep into the dermis and destroys the dermal papillae, then growth of new epidermal cells is impaired and a permanent scar is created."
[New South Wales Police Department]


The way the ridges develop, oddly, depends on the arrangement of the sweat glands, rising to pores which, in the volar regions, protrude in papillae (nipple-like structures) above the baseline of the skin surface. As these grow, they also grow connections to each other in rows - and this is how the lines and whorls of the fingerprint are created.

It also explains why fingerprints - the kind the police use for identification - are often made up of what appear to be rows of dots, rather than nice smooth lines:

"Such pore holes are critical to the production of latent prints since sweat reaches the surface of the hand and efficiently coats the tops of the fingerprint ridges with sweat. Sweat glands serve as small chemical reservoirs and contain a variety of water-soluble chemical compounds, produced or stored by the body."


In other words, we leave a chemical trace when we touch things, as rows of little oily mineral sweat-dots.

For those of you who have ever worried about the old hair-on-the-palm story, you can relax: both sebaceous glands and hair follicles appear in the dermal layer of other skin surfaces but don't in friction skin. Probably for good reason. How useful would it be to have painful pimples on the palms of your hands if your best escape from predators was to swing up into a tree?

Johann Christoph Andreas Mayer recognized in 1788 that although friction ridge patterns could appear similar, they never seemed to repeat themselves. Using fingerprints' unique patterns as an identification system, however came in much later, starting with the movement to the cities in the Industrial Revolution, when people began leaving their ancestral homes, where every face was familiar, and moving into more populous environments, where they were more difficult to identify and it was harder to find out their history.

"...felons quickly learned to lie about their names, and the soaring rate of urban crime forced police to search for a more exacting way to determine and keep track of identities. The first such system was devised in 1883 by a Parisian police clerk named Alphonse Bertillon. His method, called anthropometry, relied on an elaborate set of anatomical measurements -- such as head size, length of the left middle finger, face height -- and features like scars and hair and eye color to distinguish one person from another. Anthropometry proved useful, but fingerprinting, which was then coming into use in Britain, held more promise...


Francis Galton

"In 1880, Dr. Henry Faulds published the first comments, in the scientific journal Nature, on the use of fingerprints to solve crimes. Soon afterward, Charles Darwin's misanthropic cousin, Sir Francis Galton, an anthropologist and the founder of eugenics, designed a system of numbering the ridges on the tips of fingers -- now known as Galton points -- which is still in use throughout the world. (Ultimately, though, he saw fingerprints as a way to classify people by race.)"
-- [Michael Specter, from a fascinating article on the fallibility of fingerprints in the New Yorker]


Bertillon's method was actually quite popular in France long after fingerprints had become popular everywhere else (a member of the Bonnot Gang actually sent his fingerprints to the French police because he knew they only had his physical measurements on record). This popularity after his long struggle for the legitimization of his system meant that Bertillon was able to go on to implement such innovations as mug shots, systematized crime scene photography, ways to preserve footprints and ballistics, and the dynamometer, used to determine the degree of force used in breaking and entering.

Now, after a nearly hundred and fifty years of fingerprint analysis being considered unquestionably right, despite any evidence against it in trials across the world, a few cases have brought the practice into the limelight. Much of fingerprint analysis hasn't changed since it was first created, and its status as a "science" is coming into question, since scientific method, not to mention actual studies of the practice to see how accurate it is, seem to be missing from the process.


Some people are, actually, born without fingerprints. A genetic disorder due to defects in the protein Keratin 14 lead to two different diseases causing embryos not to form friction ridges. It makes it difficult to do certain things, like turn pages or deal cards. Most of all, it makes it difficult to get certain kinds of jobs - such as school teacher, nurse, and so on. Not to mention working for the government in either law enforcement or classified work.


In the old days, safecrackers used to sand the ends of their fingers to make them more sensitive and to make their fingerprints less identifiable; but that seems to be going out of fashion in contemporary times. Nowadays, you are more likely to affect your whorls by picking up a tiny virus-based skin tumor called a plantar wart (veruca), which deforms the skin striae as it grows, making the ridges go around it. When the wart finally goes away, your striae never look quite the same...

So, the next time you are lying on the couch with a loved one's feet in your lap, have a look, and marvel at the fanciful shapes and swirling minutae of their toes. Think about how long they have been on our feet, probably millions of years, and how even though we wear shoes, our bodies still create these wonderful artworks. They really are amazing.




Links:

A simple timeline on the history of fingerprints

Michele Triplett's Fingerprint Dictionary: Every term you could possibly want to know about fingerprint analysis and police procedure.

A little YouTube of the beginning of my favorite story about safecrackers, Butch Minds The Baby

Website about Sir Francis Galton, above

Photoshop brushes which give you fingerprint effects over at DeviantArt

Semaphore as Information Network

Claude Chappe had tried it out with his brothers and believed - no, knew it could work. He had a vision of quick national communication - much faster than a messenger could ride, much more communicative than a signal fire.

Messengers and signal fires had been around for thousands of years, and had served armies and governments relatively well. With the re-invention of the telescope in the early 1600s (and it subsequent popularity for naval and astronomical observation), technology had changed, allowing for more freedom of the seas and the land. By the late 1700s, the atmosphere of Europe was rife with invention, and people began to look to every technology and how it could be used for new (or old) applications.


Claude Chappe, who came from a well-to-do family and who had been an abbe with a secure income, had always been interested in physics, particularly optics. His uncle Abbé Jean Baptiste Chappe d'Auteroche (1722-1769), who was also in the clergy, as well as being a famous cartographer and astronomer, had traveled to Siberia in 1761 to observe the passage of Venus in front of the sun and published a well-known book about it, Voyage en Siberie. This same uncle had died of yellow fever in California while there to observe another celestial event. A eulogy was held in Paris to honor him, and his westerly observations were published posthumously.

Claude was well on his way to following in his uncle's footsteps, but the French Revolution intervened, and he and all his brothers were left unemployed. Despite the atmosphere of paranoia, where people were being beheaded for slight deviations in behavior, Claude decided to pursue an idea that had been put about, both as a reasonable scheme and as various harebrained ones: the idea of a telegraph, where signals would be passed along a line of stations, each one manned by someone trained to read the signal and reproduce it for the next station. After a few experiments, Claude came to the correct conclusion that the telegraph system should be optical - using telescopes to read the signals - because the distance between stations could then be lengthened quite a bit.


Here is an account of someone else who had experienced good results with an optical telegraph:

"One of the more practical proposals came from De Courrejolles, a captain in the French navy.[Note 12] In February 1783, De Courrejolles was engaged in battle with the English fleet, at what is described as the Turkish or Ionic Isles...He found himself surrounded by an English squadron commanded by Admiral Hood. De Courrejolles had a simple optical telegraph erected at a mountain top on the coast of one of the islands, and used it to monitor the enemy's movements. Every change in position was reported by the telegraph. Using this information De Courrejolles was able to overrun a squadron commanded by the then Captain (later Admiral) Nelson, and force the English fleet to retreat. Inspired by this success, De Courrejolles submitted a proposal to the French Minister of War to have the army adopt optical telegraphs for signaling purposes. Though De Courrejolles was unsuccessful at that time, he may well have paved the way for Chappe."

At a symposium in Sweden on the optical telegraph in 2004, a history of the Chappe network was presented in beautiful, researched detail. Unfortunately, this seems to have been taken down, but I found it via the Wayback Machine, and you can read it for yourself, if you like. Here's a quote about Claude casting about for methods to make his dream succeed:

"Abraham Chappe later wrote that Claude performed many experiments to find a good alternative, including the use of electrical signals traveling through conducting wires. He records that an optical method was only chosen. . . after having tried, unsuccessfully, electricity, various acoustical methods, the use of smoke produced by different types of combustible materials, etc. The idea to use an electrical signal had to be abandoned when no adequate insulators could be found for the wires."

So close! There were numbers of people thinking about electricity for telegraphic communication in those days, including one man in Spain who tried electrical sparks to illuminate tin-foil letters; but none of them were quite able to make it happen.

Chappe experimented and eventually adopted a design using weighted arms, which swiveled to create a large array of shapes, in effect a semaphore. Ignace is noted as saying, "Some time later [we] established with certainty that elongated objects were better visible than the sliding panels adopted before." It's interesting to note that semaphore was not used in the way we know it now, i.e., a flag-waving activity used to communicate between ships or between ships and shore, until the early 1800s.

By 1793, despite the beheading of Louis XVI and the beginning of the reign of terror, and despite the destruction of two of his signal towers by mobs who thought he was communicating with Royalist forces, Claude and his brothers had set up a telegraph line which ran between two locations near Paris, approximately 26 km apart. Having several allies within the new government, they received permission to test the line. The messages took approximately 10 minutes to transmit, an unheard-of speed at that time - and government people were there to see it happen.


This caused such excitement that within two weeks a decision had been made to establish a national telegraph system, and Claude Chappe was named Ingénieur Télégraphe (Telegraph Engineer), working for the government. Money was appropriated for the construction of a line of fifteen stations from Paris to Lille, at the frontier with the Austrian Empire; this line, when it was complete, could transmit a message in a little over half an hour, a key tool in the war between France and the Empire, as it meant the Capital could keep up on events as they happened.

If you look at this point in time, France was in a tricky position:

"France was surrounded by the allied forces of England, The Netherlands, Prussia, Austria, and Spain. The cities of Marseilles and Lyon were in revolt, and the English Fleet held Toulon. In this situation the only advantage France held was the lack of cooperation between the allied forces due to their inadequate lines of communications."[wiki]


As a result, with the success of the Lille line, optical telegraph lines were built over the entirety of France over the next twenty years or so. Napoleon loved the system, having his own portable station built which he carried with him on campaign. He also poured money into building more of the network. It wasn't cheap, because each station had to be manned by a highly-trained person, who observed the signal from other towers and knew how to pass it on. But the French system of fast communication was one of the key ingredients in France's success during the Napoleonic War, and so they hung onto it as long as they could. Claude Chappe himself remained in his position as the head of the system for over 30 years, until there was an administration change.

The Optical Telegraph system at its height covered most of the borders

By that time, however, people were finally starting to take the electric telegraph more seriously, and by 1844 America had begun work on a system of electric telegraphs which ultimately outmoded the semaphore system.

Even today you can find the towers with semaphore arms scattered all across France, sometimes with their arms drooping or missing. Some of the stations are still in working order, and you can go see them operated. It's one of those examples of a liminal moment, a place between two eras. I love things like that, things that changed the face of, say, communication while using only peoples' eyes and ears for the new technology rather than looking so far as circuits and fuses; and yet, ultimately, these systems become abandoned relatively quickly because, ultimately, someone is bound to work out the circuits and fuses...leaving behind artifacts and traces of something we couldn't possibly imagine for ourselves.



Links:

A site all about semaphores

A History of Information Highways and Byways, from NYU

Wikipedia's interesting page on the Chasqui, a network of Inca messengers who made for fast communication all over the Inca Empire.

Source for my images.

The Wonder of the Golden Proportions

Ever hear of the Golden Mean? Neither had I, until I was supposed to teach a graphic design course, and started (you know me) to do research on what, exactly, I should be teaching. It's one thing to be able to design things, and quite another to have to teach it to others.

A lot of what I found, gestalt theory and the principles of visual weight, and so on, were really interesting; but the Golden Mean was what really caught my fancy.

(NB: Math following. Don't be scared, all will be clear [and un-mathlike] in the end, I promise. I hope)

The Golden Mean, also known as the Golden Ratio, was developed as a proportional measurement by the ancient Greeks, as a way of making the most pleasing artworks. It was felt to be semi-divine, in that it seemed to show up in Nature as well. The ratio, an irrational number, began as 1.6180339887... and continued onward, pretty much forever. It is found by working out the following algebraic equation:

Essentially, if you take a line that is 1 long, and go from there, you will find that the above equation will work out to that same irrational number, which, when used as a proportional device, allows you to produce varying lengths of lines that are smaller and larger. But I'm not going to explain how, because though I really, really love math, I don't do well with equations, which are difficult for me: at least, to express the near-mystical magic that shows up in numbers.

So: now you have a bunch of varying line lengths. So what?

Well, let's see: take one of these lines and make a square out of it (putting four of them at right angles to each other, remember? I sometimes blank on these little leaps of logic). Then, starting at the center of one side, measure to one corner and draw an arc downward:


Aha! Now we start to have something. If the length of the square is 1, then the length of the rectangle (shown as the Greek letter phi here) is, of course, 1.6180339887... well, you get the point: the Greeks were smart. We call this shape the Golden Rectangle, and you can find it everywhere in Greek and Renaissance art (and elsewhere! Stage height proportions, window shapes, chair backs, believe me, they're everywhere. They are quite pleasing to the eye).

(If you really want to know the equation looks like this:)
(but don't ask me to explain that part).

Golden rectangles in the proportions of the Parthenon

Okay, onwards. I know this is looking like a lot of math, but bear with me here. Now we come to this guy named Fibonacci, born c.1170, who is considered "one of the most talented mathematicians of the Middle Ages" [wiki]. This is the man responsible for the introduction of the Hindu-Arabic numeral system when all of Europe was doing math in Roman numerals (I strongly urge you to read his link, above: it's fascinating). He used what is now known as the Fibonacci sequence - actually a pre-6th-century Indian concept - as an example in his famous and apparently brilliant book about math, Liber Abaci, or Book of Calculation, which is why we have it now. In any case, the Fibonacci Sequence, as it is now known, leads to all kinds of interesting events.

Try this. Draw a square, measuring one unit across (make it a small unit, like a centimeter, or perhaps the distance between binder-paper lines - otherwise you will need big paper). Now draw another square exactly the same right up alongside it (so they are sharing a side). Now, say you take the line along the top of both squares and use that to draw a bigger square, which of course is two units on a side - right? Okay, now moving clockwise (or counter-clockwise, like the picture below) around this construction, draw another square along the side where the edge of the big square and one small square align. Keep going clockwise and keep drawing bigger and bigger squares. The length of each consecutive square should make a sequence, like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... (by now you've probably run out of paper). Viola! Your own personal Fibonacci sequence, right there in your own home! And...lookie there. It sure looks kind of like...is it? Pretty close to a Golden Rectangle, isn't it?


Okay, okay, you say. That's kind of neat. But aren't we just wanking with numbers?

Well, perhaps. But check this out. You can draw an arc, starting with a point at the middle of the two first squares where they touch the third square. The arc goes from corner to corner of each square, so that the two squares together make a semicircle. Then, by expanding the arc and drawing one in each square, working around the structure, you can build a lovely spiral:

This Fibonacci spiral does not have the two original squares visible

This spiral is one that closely mimics the Golden Spiral, based on the Golden Mean, above. The main difference is that the Golden Mean goes in both directions, both up and down, whereas the Fibonacci spiral only goes upwards from 1 (though you could take it down if you were into math, I'm sure). Both are considered logarithmic spirals, which are found everywhere in nature. Jakob Bernouli, a mathematician from a great family of brilliant people, called the logarithmic spiral spira mirabilis, or "the Miraculous Spiral," so called because the size increases but its shape is unaltered with each successive curve. This kind of spiral shows up in shells, in hurricanes, in the shape of a cat's claw or a wave; galaxies and flowers all work with logarithmic spirals. The Fibonacci sequence can be found many places as well, such as in the ancestry patterns of bees, the branching of trees, the whorls of a sunflower and the fruitlets of a pineapple.

Technically, though, the Fibonacci spiral has a slight wobble; it is not perfect, so not really a proper logarithmic spiral.

Which brings me to something which I find absolutely wonderful: if you chart that wobble on a graph, it begins to look as if it is ocillating around something, some specific number. Guess which number?

You guessed it. Now tell me there's no mystery in numbers.



(For a nice, step by step, even better mathematical explanation of this, check out this site. It certainly got me going.)

Tycho Brahe, Bibliodyssey, and other Astrological Landmarks

Thanks to BoingBoing, who were featuring something interesting, as usual, I have discovered Bibliodyssey, which, in case you didn't happen to read that particular BoingBoing post, is a blog devoted to "Books -- Illustrations -- Science -- History -- Visual Materia Obscura -- Eclectic Bookart" and is full of amazing Wonder-ful images and sources for that kind of Mobius-think which I associate with my favorite time period.


Looking through the site, which has a lovely side-bar with visual links to previous posts, I came across a reference to Tycho Brahe's Astronomiæ Instauratæ Mechanica, a book originally published in 1598 in an effort to secure more funding after a royal death cut off Brahe's sponsorship. The illustrations show what seem to be enormous, house-sized azimuth quadrants and Christmas-tree sized sextants.

Apparently, Brahe came up with the idea for this long-term project: to make a reliable chart of the heavens, using observations from a fixed point over time. He accomplished this, with the aid of royal patronage, by building a huge observatory on the island of Hven (between Denmark and Sweden). Please note that this was long before the first telescopes, which is why he did his measuring with large, accurate versions of those instruments that ships' captains had been using for many years. Bibliodyssey mentions that this was the "first" of his observatories, and is, unfortunately, long-lost.

This reminds me of the Jantar Mantar Observatory [wiki], which I saw when I was in India, at the palace in Jaipur. It was built in 1734 by Sawai Jai Singh, the first Maharaja of Jaipur, who "...succeeded to the throne of Amber in 1700 at the age of thirteen. Abandoning that capital, he founded the city of Jaipur in 1727. A soldier, ruler, and scholar with a lifelong interest in mathematics and astronomy, Jai Singh built observatories in Delhi, Jaipur, Ujjain, Mathura and Benares. Jai Singh was conversant with contemporary European astronomy through his contacts with the Portugese Viceroy in Goa. He supplied corrections to the astronomical tables of de la Hire, and published his own tables in 1723."
- Quote loaned from Michael D. Gunther's Old Stones website, which has very detailed descriptions of the observatory's workings


Each of the eighteen observation instruments, which are mostly built out of masonry and are simply enormous, are meticulously constructed, with markings for the positions of the known heavens, and truly ingenious ways for people to get inside or to stand far enough above to be able to measure things as accurately as possible. Standing within one of the sundials, for example, manages to take you out of human scale far enough that you begin to get an inkling of the enormity of the heavens.

I can only imagine what that first observatory of Brahe's must have been like. The instruments look impressive, though without people in there to scale them, it's hard to tell if it's quite as stunning as Jai Singh's contribution. Sigh.

Astrolabes, old and new

Detail of the earliest known Persian astrolabe, from the MHS in Oxford (see below)

I'm posting this early because I'll be traveling next week and don't know if I'll be able to access the Internet. Suffice to say, I'm thinking of indulging a lifelong fascination with astrolabes, buying one and (gasp) learning how to use it. I've always found them to be deeply mysterious and quite beautiful.

Astrolabes were first used in the Islamic world, starting around 700 AD. The Europeans didn't pick them up until about 1000 AD, and went on using them until the mid-17th century (they were used in Islamic areas until the mid-19th century, being available, traditional and quite accurate for voyages in smaller seas). They are generally made of brass or copper, sometimes with gilding. Curiously, manufacturers of brass astrolabes in 17th-century India were two centuries more advanced than their European peers (see this study).

Thanks to Brill publishers for the lovely photo

There is something marvelous about the idea that you can take a large medallion, hang it, sight the altitude of various heavenly bodies, and then know where on earth you are. It anchors one in the universe, and in such a beautiful, accurate, mathematical, and low-tech way.

While looking at all the wonderful pictures of astrolabes I found (and by the way, the Museum of the History of Science at Oxford University, which looks like an amazing resource for Cabinet materials, has a whole bundle of astrolabes for your perusal) I came across Max Chen, whose oilycog site is full of flash things he's made out of bicycle parts and other bits. Among other steampunk-related items (including a very bike-messenger version of an orrery) I found this astrolabe, which I think is really wonderful.





The front side has a map of San Francisco, but only the bicycle lanes are shown (no Bay Bridge crossing). The rete rotates freely, and on the perimeter is a little prayer, which reads: "Oh great god of all bicycles,/Grant me safe passage,/Through the cars that kill, potholes that maim,/And glass that deflates my tires and my spirits."

Even though it doesn't exactly orient one toward the stars, it is a sort of microcosmic astrolabe, orienting the user to his or her small world of the City as seen from the Heavens. It carries the same sort of mystique that many of the ancient astrolabes probably had for their owners.




If you're feeling in a DIY mood, I highly recommend the book Latitude Hooks and Azimuth Rings: How to Build and Use 18 Traditional Navigational Tools, by Dennis Fisher. This book has excellent explanations and diagrams, and from it you can learn to make astrolabes, quadrants, seagoing sundials, latitude hooks (which are Pacific Islanders' way of finding latitude), and nocturnals (star clocks), among other things. It really is a treasure trove of wonderful instruments that people have used for hundreds of years - and sometimes millenia.

Lastly, here are some other purveyors of more traditional newly-minted astrolabes, in case you are interested in owning one:

- Jim Fanjoy makes a nice copper medieval-style astrolabe, based on the descriptions by Chaucer and others.
- The Brass Compass makes four models of brass astrolabe, including a desk model, for a reasonable price.
- Brian Greig, in Australia, makes hand-made replica astrolabes.