Notes on Etymology and History of Math Terms - Tangrams

 Tangram is a name of a Chinese puzzle of seven pieces that became popular in England around the middle of the 19th century. It seems to have been brought back to England by Sailors returning from Hong Kong. The origin of the name is not definite. One theory is that it comes from the Cantonese word for chin. A second is that it is related to a mispronunciation of a Chinese term that the sailors used for the ladies of the evening from whom they learned the game. [Concubines on the floating brothels of Canton, Hong Kong, and many other ports belonged to an ethnic group called the Tanka whose ancestors came from the interior of the country to become fishermen and pearl divers. They were considered as non-chinese by the govenments of China until 1731. They were unique among Chinese women in refusing to have their feet bound. ] A third suggestion is that it is from the archaic Chinese root for the number seven, which still persists in the Tanabata festival on July seventh in Japan which celebrates the reunion of the weaver (vega) and the herdsman (altair). Whatever the origin of the name, the use of the seven shapes as a game in China were supposed to date back to the origin of the Chou dynasty over one thousand years before the common era. The Chinese name is Ch'i ch'iao t'u which translates, so I am told, as "ingenious plan of seven".

Puzzle books with shape challenges were common in Tangram books.

It appears however, that the game and the name are both much more modern than believed. From the MathPuzzle.com website, I found that " The Tangram was invented between 1796 and 1802 in China by Yang-cho-chu-shih. He published the book Ch'i ch'iao t'u (Pictures using seven clever pieces). The first European publication of Tangrams was in 1817. The word Tangram itself was coined by Dr. Thomas Hill in 1848 for his book Geometrical Puzzles for the Young. He became the president of Harvard in 1862, and also invented the game Halma.

Jeff Miller's Earliest Use of Some Words in Mathematics contains the following.

"According to various dictionaries, the word may be derived from a Chinese word tang, or it may be derived from the obsolete English word trangam, meaning a trinket or a gimcrack. Merriam-Webster says the word is of unknown origin.

Trangam is found in a 1658 dictionary.

On June 1, 1809, the American Citzien reported, "Vast numbers of those 'tangrams and gimcracks' are piled up in the office, of every shape and size, making it a great toy shop. [Joel S. Berson]

A classified advertisement in the Franklin Gazette of Feb. 24, 1818, offers "Chinese Tangrams," which were probably puzzles [Bill Mullins]."

The Shoemaker's Knife Cuts Beautiful Math Across the Centuries

The term "arbelos" means shoemaker's knife in Greek, and an example is shown at the top of the blog.   The term is also applied to the shaded area in the figure below which resembles the blade of a knife used by cobblers. 

 

The height of the line segment HA is the geometric mean of the segments r and 1-r. The area of the arbelos (blue) is equal to the area of a circle with diameter AH.  

Archimedes himself is believed to have been the first mathematician to study the mathematical properties of this figure. 

One of Archimedes famous results is shown here. When the two circles drawn on each side of AH and tangent to it and the inner and outer circle,   he showed in his Book of Lemmas (proposition 5) that no matter how the larger diameter semi-circle was divided to produce the two smaller ones, the area of the two smaller circles were equal to each other. The circles are known as the Archimedean circles, Archimedean twins, and other similar names.

Then it got quiet for awhile... a long while.
But in 1954 a Los Angeles dentist (you read that right) named Leon Bankoff found a triplet for the two twins (A Mere Coincidence, Los Angeles Mathematics Newsletter, Nov. 1954).  
Often called the Bankoff triplet circle, it can be found by drawing a third circle tangent to all three semi-circles of the arbelos. Then the triplet emerges from the common points of tangency of this new circle.

Bankoff was not just any dentist, Along with his interest in dentistry were the piano and the guitar. He was fluent in Esperanto, created artistic sculptures, and was interested in the progressive development of computer technology. Above all, he was a specialist in the mathematical world and highly respected as an expert in the field of flat geometry. Since the 1940s, he lectured and published many articles as a co-author. Bankoff collaborated with Paul Erdős in a mathematics paper and therefore has an Erdős number 1.  
After 2000 years, the dam had broken: In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to Scientific American's "Mathematical Games" editor Martin Gardner. The manuscript was forwarded to Leon Bankoff. Bankoff gave a copy of the manuscript to Professor Clayton Dodge of the University of Maine in 1996. The two were planning to write an article about the Arbelos, in which the Schoch circles would be included; however, Bankoff died the year after.
Schoch's paper can be found here with images of his dozen additions to the Archimedean circle clan.
Then, in 1998, Peter Y. Woo of Biola University, published Schoch's findings on his website. By generalizing two of Schoch's circles, Woo discovered an infinite family of Archimedean circles named the Woo circles in 1999.
And today, well you can see an Online Catalogue of Archimedean circles maintained by Floor van Lamoen, who has a few geometric objects named after himself as well. .

Updating the History of the Pigeon Hole Theorem

 The Pigeon Hole Principle......The basic idea behind this mathematical principle is what students would call common sense; if there are n objects to be placed in m receptacles (with m less than n), at least two of the items must go into the same container. While the idea is common sense, in the hands of a capable mathematician it can be made to do uncommon things. The late Alexander Bogomolny used the principle to argue that there must be at least two persons in New York City with the same number of hairs on their head. This "counting hairs" approach dates back to the earliest version of the principal I have ever seen.

The same axiom is often named in honor of Dirichlet who used it in solving Pell's equation. The pigeon seems to be a recent addition, as Jeff Miller's web site on the first use of some math words gives, "Pigeon-hole principle occurs in English in Paul Erdös and R. Rado, A partition calculus in set theory, Bull. Am. Math. Soc. 62 (Sept. 1956)" (although they credit Dedekind for the principle). In a recent discussion on a history group Julio Cabillon added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portugese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish
we use also:"the principle of the drawers of Dirichlet"
that is 'Zasada szufladkowa Dirichleta' ". I received a note that said, "Dirichlet first wrote about it in Recherches sur les formes quadratiques à coefficients et à indéterminées complexes (J. reine u. angew. Math. (24 (1842) 291 371) = Math. Werke, (1889 1897), which was reprinted by Chelsea, 1969, vol. I, pp. 533-618. On pp. 579-580, he uses the principle."

He doesn't give it a name. In later works he called it the "Schubfach Prinzip" [which I am told means "drawer principle" in German]

The idea has been around much longer than Dirichlet, however, as I found out in June of 2009 when Dave Renfro sent me word that the idea pops up in the unexpected (at least by me) work, "Portraits of the seventeenth century, historic and literary", by Charles Augustin Sainte-Beuve. During his description of Mme. de Longuevillle, who was Ann-Genevieve De Bourbon, and lived from 1619 to 1679 he tells the following story:
"I asked M. Nicole (See below for description of M. Nicole) one day what was the character of Mme. de Longueville's mind; he told me she had a very keen and very delicate mind in knowledge of the character of individuals, but that it was very small, very weak, very limited on matters of science and reasoning, and on all speculative matters in which there was no question of sentiment ' For example,' added he, ' I told her one day that I could bet and prove that there were in Paris at least two inhabitants who had the same number of hairs upon their head, though I could not point out who were those two persons. She said i could not be certain of it until I had counted the hairs of the two persons. Here is my demonstration/ I said to her: M lay it down as a fact that the best-fiimbhed (not sure what this word was supposed to be, ..Plumbed??) head does not possess more than 200,000 hairs, and the most scantily furnished head b that which has only 1 hair. If, now you suppose that 200,000 heads all have a different number of hairs, they must each have one of the numbers of hairs which are between 1 and 200,000; for if we suppose that there were 2 among these 200,000 who had the same number of hairs, I win my bet But suppose these 200,000 inhabitants all have a different number of hairs, if I bring in a single other inhabitant who has hairs and has no more than 200,000 of them, it necessarily follows that this number of hairs, whatever it b, will be found between 1 and 200,000, and, consequently, b equal in number of hairs to one of the 200,000 heads. Now, as instead of one inhabitant more than 200,000, there are, in all, nearly 800,000 inhabitants in Paris, you see plainly that there must be many heads equal in number of hairs, although I have not counted them.' Mme. de Longuevillle still could not understand that demonstration could be made of the equality in number of hairs, and she always maintained that the only way to prove it was to count them. "

The M. Nicole who demonstrated the principal was Pierre Nicole, (1625 -1695), one of the most distinguished of the French Jansenist writers, sometimes compared more favorably than Pascal for his writings on the moral reasoning of the Port Royal Jansenists. It may be that he had picked up the principal from Antoine Arnauld, another Port Royal Jansenist who was an influential mathematician and logician. Here is a segment from his bio at the St. Andrews Math History site.
-------------------------
He published Port-Royal Grammar in 1660 which was strongly influenced by Descartes' Regulae. In Port-Royal Grammar Arnauld argued that mental processes and grammar are virtually the same thing. Since mental processes are carried out by all human beings, he argued for a universal grammar. Modern linguistic theorists consider this work as the beginnings of the modern approach their subject. Arnauld's next work was Port-Royal Logic which was another book of major importance. It was also strongly influenced by Descartes' Regulae and also gave a first hand account of Pascal's Méthode. This work presented a theory of ideas which remained important in philosophy courses until comparatively recent times. In 1667 Arnauld published New Elements of Geometry. This work was based on Euclid's Elements and was intended to give a new approach to teaching geometry rather than new geometrical theorems."
He was a correspondent of Gottfried Wilhelm Leibniz, and of course Pascal, who wrote the Pascal "Provincial Letters" in support of Arnauld. I enjoyed the quote about him from the Wikipedia bio: "His inexhaustible energy is best expressed by his famous reply to Nicole, who complained of feeling tired. 'Tired!' echoed Arnauld, 'when you have all eternity to rest in?"

I have not been able to find any thing in Arnauld's personal writing at this time to confirm that he was aware of or used the Pigeon-hole Principle. I have also seen a comment that there is a book by Henry (or Henrik) van Etten (pseudonym of Jean Leurechon, who coined the term thermometer) , circa 1624, which uses the method for problems involving "if there are more pages than words on any page" and various other illustrations. The writer suggests that the problem is in the French version but not the English translation. Would love to hear from someone who can confirm, and perhaps send a digital image.

A Bird in the Hand is worth... a Pigeon Hole Principle

Sometimes problems that seem very hard, can be very easy if they are viewed in the right way, and one of those easy ways to make some hard problems manageable is the Pigeon-Hole Principle. Over the last few weeks seems like lots of problems involving this idea have shown up, so I thought I would bring it to you.
The basic idea is so easy any sixth grader would agree; if you have two boxes, and you are going to put three balls in the boxes, then at least one box will get more than one ball..... "well, Duh!" they answer... and yet... it seems easier to apply than it might be. Now that you know the secret, try these two problems. I'll post the answer down lower on the page where you must not look until you take a few minutes to ponder the problems.
Here is the first from a recent blog I read: "39 people are attending a large, formal dinner, which must of course occur at a single, circular table. The guests, after milling about for a while, sit down to eat. It is then pointed out to them that there are name cards labeling assigned seats, and not a single one has sat in the seat assigned to them. Prove that there is some way to rotate the table so that at least two people are in the correct seats."
This one seems tougher, but really isn't, it just requires a different way of thinking. "Suppose you pick six unique integers from 1 to 1000. Prove that at least two of them must have a difference that is a multiple of five.
Before I give you the answers, I will throw in a little cultural information that may amuse and entertain you. The same axiom is often named in honor of Dirichlet who used it in solving Pell's equation. In a discussion on a history group a few years ago Julio Cabillon added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portuguese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish we use also:"the principle of the drawers of Dirichlet" that is 'Zasada szufladkowa Dirichleta' ". You just can't have TOO many names for a really useful idea.
Ok, The Proofs... for number one... Suppose you handed each person a number that was how many seats they needed to move to the right to find their assigned seat. Since no one is at the right seat, the number can not be zero or thirty-nine. SO each of the people has a number between 1 and 38...wait, there are 39 people...two of them (at least) must be the same distance away from their assigned seats.... admit it…..that’s pretty cool. (I have a slight question about whether this actually proves the solution of "rotating the table" to put two in their correct seats.  Suppose we know that persons A and B are in each others seats and 5 seats apart.  Rotating the table five seats in either direction would only put one of them in their correct seat.  Maybe all we proved is that there are , at least, two people who are the same distance from their seat.  If we had specified how far away in a clockwise direction they are from their correct seat, we would have a solution to the rotation of the table problem.)
For number two it is sort of the same idea, but you have to think about how much each number would have for a remainder if you divided them by five. The only possible choices are 0, 1, 2, 3, or 4... , five different remainders, but there are six numbers, so two of them have the same remainder...and two numbers that have the same remainder on division by five, are a multiple of five apart.... think of 1,6, 11, etc for remainders of one. If you want to read more about how remainders can play a part in solving problems, see my blog on "casting out sevens"  

And for some history about this beautiful problem solving idea, see this.