Typing Monkeys

More observations stimulated by John Barrows new book (see my recent blog) Everybody has heard the suggestion that a million (or some other number) of monkeys typing continuously for many millenia would eventually produce a) Shakespeare, b) all of known science, c) the bible, d) all of the above). It began with Jonathon Swift and Gulliver's Travels, 1872, according to Professor Barrow. In the tale "a mythical professor of the Grand Academy of Lagado who aims to generate a catalogue of all scientific knowledge by having his students continuously generate random strings of letters..." (I think, see emphasis in the excerpt below, that it was random strings of words).. Anyway, according to the good Professor Barrow, the story was embellished in different forms until French Mathematician Emile Borel{there is a street and a square named for him in the 17th District in Paris} suggested that random typing monkeys could duplicate the French national library. A few years later(1929), Arthur Eddington Anglicised that to "books in the British Museum." By 1972, Arthur Koestler writing in The Case of the Midwife Toad, New York, 1972, page 30, refers to Monkeys typing Shakespeare as "proverbial":"Neo-Darwinism does indeed carry the nineteenth-century brand of materialism to its extreme limits--to the proverbial monkey at the typewriter, hitting by pure chance on the proper keys to produce a Shakespeare sonnet." Ok, so eventually someone had to put this to a more scientific test, and they did. "A website entitled The Monkey Shakespeare Simulator, launched on July 1, 2003, contained a Java applet that simulates a large population of monkeys typing randomly, with the stated intention of seeing how long it takes the virtual monkeys to produce a complete Shakespearean play from beginning to end. For example, it produced this partial line from Henry IV, Part 2, reporting that it took "2,737,850 million billion billion billion monkey-years" to reach 24 matching characters:"RUMOUR. Open your ears; 9r5j5&?OWTY Z0d... " Even more impressive, to me, is the fact that 'in another part of that book, Swift tells of how the astronomers on the flying island of Laputia had: "discovered two lesser stars, or satellites, which revolve around Mars, whereof the innermost is distant from the center of the primary exactly three of his diameters, and the outermost five: the former revolves in the space of ten hours, and the latter in twenty-one and a half". Swift wrote this in 1726, but it was not until 1877 that Asaph Hall discovered the two moons of Mars.'... I just did a little checking on the orbit and periods he "predicted?" and the actual periods are about 7 hours for Phobos, and about 30 for Deimos... and their distance from the planet were about 9 x 10³km and 23.5 x 10³km. Mars has a diameter of 6.794 x 10³km so they are closer to 1.5 and 3 radii away it seems, but wow, for 100 years before the actual discovery??? Don't you wonder what made him use Mars instead of Venus or ??? Wait, maybe authors typing randomly can describe the true nature of the universe (with some limits of error)... ------------------------------------------------------------------------------------------ Ok best guess,  Swift had heard of a misinterpretation of Johannes Kepler. He had interpreted the anagram that Galileo Galilei sent to him in 1609 to tell him about the discovery of the phases of Venus as the discovery of two moons of Mars. 

 If you have your copy, here is what I found in Chapter Five of Gulliver's Travels. The whole thing is available at the Guttenburg Project. "The first professor I saw, was in a very large room, with forty pupils about him. After salutation, observing me to look earnestly upon a frame, which took up the greatest part of both the length and breadth of the room, he said, "Perhaps I might wonder to see him employed in a project for improving speculative knowledge, by practical and mechanical operations. But the world would soon be sensible of its usefulness; and he flattered himself, that a more noble, exalted thought never sprang in any other man's head. Every one knew how laborious the usual method is of attaining to arts and sciences; whereas, by his contrivance, the most ignorant person, at a reasonable charge, and with a little bodily labour, might write books in philosophy, poetry, politics, laws, mathematics, and theology, without the least assistance from genius or study." He then led me to the frame, about the sides, whereof all his pupils stood in ranks. It was twenty feet square, placed in the middle of the room. The superfices was composed of several bits of wood, about the bigness of a die, but some larger than others. They were all linked together by slender wires. These bits of wood were covered, on every square, with paper pasted on them; and on these papers were written all the words of their language, in their several moods, tenses, and declensions; but without any order. The professor then desired me "to observe; for he was going to set his engine at work." The pupils, at his command, took each of them hold of an iron handle, whereof there were forty fixed round the edges of the frame; and giving them a sudden turn, the whole disposition of the words was entirely changed. He then commanded six-and-thirty of the lads, to read the several lines softly, as they appeared upon the frame; and where they found three or four words together that might make part of a sentence, they dictated to the four remaining boys, who were scribes. This work was repeated three or four times, and at every turn, the engine was so contrived, that the words shifted into new places, as the square bits of wood moved upside down. Six hours a day the young students were employed in this labour; and the professor showed me several volumes in large folio, already collected, of broken sentences, which he intended to piece together, and out of those rich materials, to give the world a complete body of all arts and sciences; which, however, might be still improved, and much expedited, if the public would raise a fund for making and employing five hundred such frames in Lagado, and oblige the managers to contribute in common their several collections. He assured me "that this invention had employed all his thoughts from his youth; that he had emptied the whole vocabulary into his frame, and made the strictest computation of the general proportion there is in books between the numbers of particles, nouns, and verbs, and other parts of speech."

Logic Diagrams, A Brief History

I just read a nice blog over at the NUMBER WARRIOR by Jason Dyer about using logic diagrams for graphic organizers to point out relationships between math objects. It is a nice post and I wouldn't (couldn't) improve on it, so check it out. BUT..... (isn't there always a but..) In the article, which was titled Carroll Diagrams, somewhere along the way he pointed out that,"I’ve used something like this before in geometry for sorting triangle types, but I never knew there was a name for it..." [emphasis added].

Well that set off the math historian in me.... Jason is a clever guy who knows lots of math and somehow, when we let kids that clever grow up and teach math and they haven't been introduced to (at least a mini-) history of logic diagrams, something HAS to be done.. so some notes about the history of logic diagrams, as I understand them at this moment in time.

When I grew up these type of logic diagrams were always known as Venn Diagrams, after the Cambridge mathematician John Venn(1834-1923). Venn was a lecturer at Cambridge and worked mainly in logic and probability theory. He used diagrams of circles to represent the unions and intersections of subsets of a Universal set in non-overlapping regions. You can find more about his life at this page from the Electronic Journal of Combinatorics.
It appears that the first person to call these types of diagrams "Venn diagrams" was Clarence Irving in his work, A Survey of Symbolic Logic in 1918.

Then, in a note at the Euler Project web site, maintained by Prof. Ed Sandifer, I found a note suggesting that the diagrams are actually the creation of Euler. "Letters to a German Princess is likely to be the source of much of what people attribute to Euler. For example, I know that what we call Venn diagrams first appear in there. (Venn himself first called them "Eulerian Circles", but then managed to get them called Venn Diagrams later on.)"

"WAIT!" You scream, Venn, Euler, but the title was Carroll Diagrams,... Ok, I'm getting there.... you see, Lewis Carroll, who was in his other reality the Oxford math lecturer Charles Dodgeson, also did some nice work on Logic Diagrams. He approached set diagrams with rectangles. The image at the top shows an example of how Carroll's diagram might look for three sets(above the middle, right or left of middle, and inside or outside the inner rectangle).


Carroll probably was most influential in his use of a finite set for the Universal set, as Venn often simply used the infinite plane outside the boundaries drawn to indicate the set of things not belonging to any group. Carroll's book, The Game of Logic can be found free on the web at the Guttenburg Project. Perhaps one of the great logic statements of all time occurs in the beginning of Carroll's book when he writes, "Besides the nine Counters, it also requires one Player, AT LEAST. I am not aware of any Game that can be played with LESS than this number: while there are several that require MORE: take Cricket, for instance, which requires twenty-two. How much easier it is, when you want to play a Game, to find ONE Player than twenty-two." (Ok, you already knew he was a clever guy.)


So why do I (and many others) persist in calling them Venn Diagrams.. In his book Cogwheels of the Mind, The Story of Venn Diagrams, Professor Anthony Edwards of Cambridge explains that the Venn Diagram were much broader in scope than Euler's, and in a comparing Venn's work to previous, and sometimes similar, work he states, "Venn's own contribution, which fully justifies our attaching his name to the general diagram was the first to see that the diagram could and should be generalized to any number of sets..." Professor Edwards is a Fellow of Gonville and Caius college as was Venn, and played a part in the design of the commemorative glass shown above, which is at the college. The Glass is part of a set of six that are all commemorative of math and science people.




Professor Edwards is an accomplished mathematician, statistician, geneticist in his own right, as well as being the last graduate student of the great R. A. Fisher. He came up with a method of extending set diagrams to any indefinite size by drawing them on a sphere, and sterographically projecting them back onto a plane to create the cogwheels of the title of his book. See his book.

Like Venn before him, Professor Edwards is a Fellow of Gonville and Caius (pronouced "keys" for us Americans), at Cambridge. He is not only very brilliant, but a nice guy to boot. He showed my wife and me around the Great Hall at G&C and let me see the inside view of the stained glass tribute to Venn in the hall there. He also gave me directions to Venn's grave site at the Trumpington Parish Extension cemetery. His grave was so covered with vines that I would have never found it except my very psychic wife stops by a clump of brambles and says, "I think this is it.." Sure enough, after clearing away the vines, we managed to expose the grave site, which includes Venn, his wife, his son, and his daughter-in-law.
Venn's grave and memorials can be found at the "Find a Grave" website

For more information about Venn Diagrams check this Survey of Venn Diagrams from the Dept of Computer Science at the University of Victoria.

I took some of my math students down to the graveside a few years ago thinking it would be nice to plant some flowers so they would always remember they honored a mathematician. We planted three colors of Tulips near his grave so that they would come up in a set of Venn Rings... and the grave that hadn't been mowed for years before I tore back the brambles to expose it, was mowed the next spring.. so I saw three tiny circles of green shoots cut very close to the ground.. oh well, it was a nice drive out in the spring.

SO that's what I know, and if you want to add more, send me a comment, I would love to add your information.

Don't Write the Law of Sines Upside Down, Please!

Mostly I'm a teach and let teach kind of guy... you do what you think is important, and let me do the same... but sometimes when I see people teach the law of sines... I wonder... DO THEY KNOW?????
The ideas behind the law of sines, like those of the law of cosines, predate the word sine by over a thousand years. Theorems in Euclid on lengths of chords are essentially the same ideas we now call the law of sines. The law of sines for plane triangles was known to Ptolemy and by the tenth century Abu'l Wefa had clearly expounded the spherical law of sines (in 2014 Thony Christie sent a note telling me that "Glen van Brummelen in his "Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry" says the spherical law of sines was discovered either by Abū al-Wafā or Abu Nasr Mansur .  It seems that the term "law of sines" was applied sometime near 1850, but I am unsure of the origin of the phrase (and if you have a reference, please advise).



A simple proof of the law of sines begins with a triangle, ABC, inscribed in a circle with radius R. A diameter is drawn with one endpoint at A terminating at D and the right triangle ADC is created. Using the right triangle definitions of Sine, we see that sin (ADC)=AC/AD.

Because Angles ABC and ADC are both inscribed angles cutting the same arc, they have equal measures, and therefore equal sines. By substitution then we get sin(B)=AC/AD and since AD is a diameter equal to 2R , we may also write sin(B) =AC/2R . Now if we adopt the modern convention of calling the side AC opposite angle B, side b, we can rewrite this as sin(B)= b/2R. With one last algebraic manipulation we exchange the positions of sin(B) and 2R to get 2R= b/sin(B) [Thanks to Joshua Zucker who reminded me that the radius of the circumcircle is usually capitalized, with r used for the radius of the incircle]. Since the choice of angle B was arbitrary, we could show that the same holds for each side and opposite angle pair, producing the typical high school textbook theorem below. 


(as per Joshua Zucker's note below, that r should probably be R since that is more commonly used for the radius of the Circumcircle, with r used for the incircle. I leave it rather than having to scrabble together a new graphic)
Addendum: [ I still get emails and comments that seem not to realize that by writing the sides on top you can reduce the entire thing to a single geometric relationship.  For any triangle, the ratio of a side to the sine of the opposite angle is always equal to the diameter of the circle which circumscribes the triangle.  Perhaps I should have added this earlier.]

I am frequently amazed to see this theorem presented in math texts without the "=2R" which seems to give it visual or geometric life. It is especially curious since the property dates back to Ptolemy. I get even more frustrated when it is presented with the Angles on top, thus destroying the geometric meaning. I can't think of a good reason for doing that, but if you consciously do it the other way for some reason, I would love to hear it.

As a footnote, in spherical triangles it is customary to work with a sphere of unit radius, thus allowing the sides to be expressed in radian or angle measure as well as the angles. Since all great circles have length 360 degrees, we may express the length of a side by the fraction of a complete great circle it occupies. With this convention, the spherical law of sines states that in a spherical triangle with sides a, b, and c and angles A, B, and C, it is true that

\( \frac{sin a}{sin A} = \frac{sin b}{sin B} = \frac{sin c}{sin C}  = \frac{sin a sin b sin c}{6 Vol(OABC)} \)



That is the ratio of  the sin of any side to the sin of its opposite angle  the product of the sines of the sides over six times the volume of the tetrahedron formed by the center of the sphere and the points A, B, and C.

According to Ubiratàn D'Ambrosio and Helaine Selin, the spherical law of sines was discovered in the 10th century. It is variously attributed to al-Khujandi, Abul Wafa Bozjani, Nasir al-Din al-Tusi and Abu Nasr Mansur. 
Ptolemy knew the formula  for the planer law of sines and something like the angle addition formula but he expressed them in terms of chords of arcs, not sines of angle.  The half chords, or sines, were introduced by the Hindu mathematician Aryabhata around 500.
The spherical law of sines was first presented in the west by Johann Muller, also known as Regiomontus,in his De Triangulis Omnimodis in 1464. This was the first book devoted wholly to trigonometry (a word not then invented). David E. Smith suggests that the theorem was Muller's invention. The word trigonometry, by the way, seems to have been the creation of Bartholomaus Pitiscus, who used it in the title of a book, Trigonometriae sive de dimensions triangulorum libri cinque in 1595. Among other things the book includes a demonstration of the law of sines and the law of cosines. I find it highly unusual that the first use of a word would be in the title of a book.

As a second footnote, it may be of interest to teachers and students that the use of the unit circle was "unknown much before 1800". I found that out in an article on "Benjamin Banneker's Trigonometry Puzzle" by Florence Fasanelli, Graham Jagger, and Bea Lumpkin that appeared in the MAA online magazine Convergence. Unfortunatly the magazine is no longer free on-line. Older trig tables gave the measurements for the sine, tangent and secant on a circle of very large radius (van Schooten used 10,000,000) rather than on a circle of radius 1, as we do today. Thus, the sin 90°, also called the “total sine” was given as 10,000,000, and the sine of 45° was 707,107 and not 0.707107, as we would use today. Anyone using these tables would use rules of proportion to make any necessary conversions.

Next I hope to talk about Descartes Rule of Signs.. (which has almost completely disappeared from my current textbooks).