Automorphic Numbers and some history notes

 As far back as the Babylonians, and maybe much before, someone looked at the symbol for five, and the symbol for 25 and noticed that both ended with the same symbol for five.  Even 5 x 25 ended in the symbol tor two x sixty and the same 5 symbol.  In Roman numerals V times V = XXV, and V times XXV  =  CXXV.  Even XXV times XXV , written as DCXXV repeats the original factor in its ending.  6^2 = 36 also appears easily in Roman numerals, VI x VI = XXXVI, and 76^2 as well.

It's hard to imagine that number crunchers like Diophantus didn't recognize some of these repetitive patterns, and if there is earlier mention, would one of those more knowledgeable historians give me a note.  

For at least 200 years, the one digit autonomic numbers, 0, 1, 5, and 6 have been called circular numbers. The earliest mention of the term I can find is An Introduction to the knowledge and variety of Numbers, John Smith (Schoolmaster of Norwich), 1809.  On page 104 he writes, "The numbers 5 and 6 are called circular numbers ; because , like the circle , terminating where it begins , these numbers , multiplied by themselves ever so often , always end in the same number : 5 by 5 make 25 , and that product  multiplied by 5 makes 125, So 6 by 6 makes 36, and 6 times this product make 216, etc."  This term and usage is still preserved in some recreational math books into the 21st Century. 

The expanse of this idea to numbers that repeated the last two, or three, or more digits took on the term automorphic numbers, (formed on oneself).  The first use of the term automorphic in mathematics, seems to be by Arthur Cayley in terms of functions, "... invariant with respect to a group of linear transformations of a certain kind leaving a certain function invariant.  This meaning was superseded by Felix Klein's choice of "a function f(x) is automorphic with respect to a group G, then f(Tₙ(x)) = f(x) for every element Tₙ of G."

The term automorphic number now means an n digit number that when squared has the number appear as the last n digits of the square.  thus 25^2 =625 and 376^2 = 141376.  Many more examples below.

The first instances of the use of autonomic numbers I can find is from the early 1940's.

Mathematical Recreations by Maurice Kraitchik, 1942

He explains the term in a footnote :

He explains that prime numbers will not work except for the trivial n=0 or 1, and illustrates numbers in base 6 and 10, He explains that prime numbers will not work as bases except for the trivial n=0 or 1, and he illustrates numbers in base 6 , And lists numbers that are automorphic in base 6, "which end in 4, 44, or 344 and 3, 13, 213, and so on are automorphic."  

Then he does the same with base 10.

He closes by providing "the last digits may be 3740081787109376 or 6259918212890625."  He nowhere mentions other composite numbers whose bases work, 12, 14, 15.

Early in the same year, a problem in The American Mathematical Monthly, Vol. 49, No. 2 (Feb., 1942), pp. 120-121 (2 pages)  used the term. 

Since this was new to me, I tested it with n=2, 90625^4.  It came out as, 67451572418212890625; the last 11 digits were 18212890625 which is itself automorphic , producing a larger number that ends in these same 11 digits.

The Previous reference is from a question in 1941 that show what are now called trimorphic numbers, by H. S. M. Coxeter, 

These numbers are the 13 six-digit trimorphic numbers,  109375, 109376, 218751, 281249, 390625, 499999, 500001, 609375, 718751, 781249, 890624, 890625, 999999. Two of these are the six digit autonomic numbers 109376 and 890625. Tri-morphic are n-digit numbers whose cubes preserve the original in their last n digits. All automorphic numbers are automatically tri-morphic, but there are others which are not automorphic.

The automorphic numbers in base 10 have two sets (ignoring the trivial 0, 1). The fives are mostly self describing, 5^2 =25, 25^2 = 625, 625^2 presents a glitch, since it produces 390625, there is not a four digit automorphic number in this series, but we extend to the next digit to get 90625 which when squared produces 8212890625, and we get the six digit automorph, 890625.

The six series is a little more complex, 6^2=36, but 36 is not automorphic, instead we use the tens compliment of 3, 7 as the lead digit, and 76 is the two digit automorph. 75^2 = 5776, but 776 is not a 3 digit automorph, but the tens compliment of 7 yields 376 which is the three digit automorph.

there is an interesting combination for the numbers in base ten. Notice that 6+5 = 10+1, and 25 + 76 = 100+1. 625 + 376 = 1000+1, and 90625 + 9376= 10000+1.

There are also a-automorphic numbers for which ax^2 preserves the n-digits of x, so 2-automorphic would be 2x^2, and contain numbers like 8, 88, 688, 4688, 54688...

The 3-automorphic numbers I have found end in 2, 5, and 7. (still exploring these). 2, 92, 792, ... ; 5, 75, 875,..; 7, 67, 667, 6667...;

I think any of these would be great explorations for even middle school students....or old retired guys like me, so enjoy!

A Brief History of Blackboards and Slates

 

My career in education began with the use of chalk and a blackboard, transitioned into a room with only dry erase marker boards, and finished in a class with only an electronic smart board. Except for the few times I was inconvenience by a projector bulb picking a bad moment to go bad (and if a chalk board had been available, I would have been grateful) I loved the improvements that each transition brought to my ability to more effectively convey my ideas.
But for the period from 1800 to 2000 few things were as ubiquitous in a mathematics classroom as the blackboard. Today modern "white boards" may have taken their place in many institutions, or even an electronic version called a smart board; but board work still seems to be a part of the current classroom procedure. In a recent talk, Keith Devlin began by saying, "I step back from the (now largely metaphorical) blackboard and .. "
Whatever the present state of its demise, the classic chalkboard was so common a classroom presence that it was part of a frequently repeated gag sequence on the popular Simpsons cartoon series.(Not sure if the use shown above could be termed "educational")

It appears that the blackboard first came into American education around 1800. The National Museum of American History website on colonial education says,:

"Mathematics teachers with ties to England and France introduced blackboards into the United States around 1800. By the 1840s, these erasable surfaces were used for teaching a wide range of subjects in elementary schools, colleges, and academies. The Massachusetts educator William A. Alcott visited over 20,000 schoolhouses. “A blackboard, in every school house," he wrote, "is as indispensably necessary as a stove or fireplace."

James Pillan, a Scottish teacher and education reformer is often cited as the "inventor" of the blackboard, but this seems to be a misunderstanding based on a letter from Pillan which appeared in Jeremy Bentham's Chrestomathia (1815). It was entitled Successful application of the new system to language-learning, and dated 1814; it mentions the use of chalk and blackboard in teaching geography. But Pillan only began teaching in 1810, almost a decade after the board made its way to America, and as we shall see, literally hundreds of years too late to "invent" the chalkboard.  He may, however, be the inventor of colored chalk. He is reported to have had a recipe with ground chalk, dyes and porridge.

Blackboards and slates were seemingly used well before any of the previous examples in musical study. In Composers at Work, author Jessie Ann Owens devotes several pages to the existence of several types of slate and wood "cartella" which were used to write out musical ideas. She describes the discoveries of these with five or ten line staves dating to the 16th century. Much larger wall size examples seem to have been used but have only been confirmed by iconography. The book includes an image from a woodcut by Hieronymus Holtzel of Nuremberg in 1501.

In America they seem to have very quickly become and essential part of daily school life. [From a web page of Prof. Rickey]

Perhaps no one method has so influenced the quality of the instruction of the cadets as the blackboard recitations. Major Thayer (Superintendent from 1817) insisted on this form, although old records show that it was introduced at West Point by Mr. George Baron, a civilian teacher, who in the autumn of 1801 gave to Cadet Swift "a specimen of his mode of teaching at the blackboard." Today it is the prominent feature in Academic instruction. [Quoted from Richardson 1917, p. 25] There is indication that the blackboard was used in a few schools in the US before it was used at USMA. See Charnel Anderson, Technology in American Education, 1650-1900, published by the
US Dept of Health, Education, and Welfare 1961(I have read this document and he credits Frenchman Claude Crozet with introducing the blackboard to the USMA and that he built and painted one to teach his classes.  It may well be that after Baron left under a cloud in 1802, the method was not used by other teachers there until Crozet arrived in 1817. 

Thayer had visited the Ecole Polytechnique in France to study their methods and was heavily influence by the "French" method when he became superintendent, even to the point of extending instruction in French so that the students could better master the French texts in advanced math.. I can't find an early example of the use of chalk or slate in France, but they seem to have been very much a part of the educational process by the time Galois threw an eraser at his examiner in July of 1829.

Galois was not the only one who reacted negatively to some of the innovations in education connected to the blackboard. At Yale, there were two "rebellions" in which students refused to accept some changes in testing practice. Here is a paragraph from Stories in Stone: Travels Through Urban Geology By David B. Williams.

This "rebellion" occurred in 1830, with 43 rebels expelled, including Andrew Calhoun, the son of John C Calhoun, and Alfred Stillé, who eventually did get a degree from Yale and another from  U of Pennsylvania before he became a somewhat famous doctor, and one of the first to distinguish Typhus from Typhoid fever. His rebellious side wasn't limited to college however, as he refused to accept germ theory and laboratory medicine. There had been a similar event in 1825 at Yale, but those students recanted and were readmitted.

One of the earliest mentions of blackboards I have found has nothing to do with education, however. It seems that a custom developed in London's financial district in the later part of the 19th century to list the names of debtors on a blackboard to shame them into paying, and it seems to have persisted for a long time. Here is a description of the practice from Chronicles and Characters of the Stock Exchange
By John Francis, Daniel Defoe; printed in 1850.

From Wikipedia I learned that the Oxford English Dictionary provides a citation from 1739, to write "with Chalk on a black-Board". I know it is common in England for Pubs to advertise with a blackboard outside their doors on the sidewalk, but have no idea how far back this idea originated.
 
Prior to the use of blackboards students learned their early lessons from an object called a hornbook. Here is a description of one from the Blackwell Museum webpage at Northern Illinois University

Paper was pretty expensive once and hornbooks were made so children could learn to read without using a lot of paper. A hornbook was usually a small, wooden paddle with just one sheet of paper glued to it. But because that paper was so expensive, parents and teachers wanted to protect it. So they covered the paper with a very thin piece of cow's horn. The piece of cow's horn was so thin, you could see right through it. That's why these odd books were called "hornbooks."

Hornbooks seem to have been totally imported from England into the American Colonies, and almost all had a cross on the upper left, with the Lord's Prayer at bottom.  The American Revolution seemed to have almost completely eliminated the import of Hornbooks in rejection of all things English at the time.  The education conversion to the blackboard seems to have finished the hornbooks very quickly afterward judging from this quote from the OED about Hornbooks, (a1842 HONE in A. W. Tuer Hist. Horn-Bk. I. i. 7) " A large wholesale dealer in..school requisites recollects that the last order he received for Horn-books came from the country, about the year 1799. From that time the demand wholly ceased..In the course of sixty years, he and his predecessors in business had executed orders for several millions of Horn-books".
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Early blackboards were usually made of wood, (but some may have been made of paper mache') and painted with many coats as true slate boards were very expensive. Schools purchased large pots of "slate paint" for regular repainting of the boards. The Earliest quotes from the OED date to 1823.

1823 PILLANS Contrib. Cause Educ.    A large black board served my purpose. On it I wrote in chalk. 1835 Musical Libr. Supp., Aug. 77 The assistant wrote down the words..on a blackboard. 1846 Rep. Inspect. Schools I. 147 The uses of the black board are not yet fully developed.

However under "slates" I found other  earlier uses. In "1698 FRYER Acc. E. India & P. 112 A Board plastered over, which with Cotton they wipe out, when full, as we do from Slates or Table-Books" which indicates that boards covered with Plaster or other materials were used to write upon much earlier than the earliest use of "blackboards" in classrooms.

Another early use of slates is given in David E. Smith's Rara arithmetica of a book printed in 1483 in Padua of the arithmetic of Prosdocimo containing a mention of the use of a slate. This led Smith to conclude that at this time the merchants would actually erase and replace numbers (as was originally done by the Hindu mathematicians working in their sand trays) in division rather than showing the cross-outs that distinguish the galley method of division after it was adopted to use on paper.

The very earliest claim for slates I have found is of use in the 11th century. A work called Alberuni's India (Tarikh Al-Hind), "They use black tablets for the children in the schools, and write upon them along the long side, not the broadside, writing with a white material from the left to the right."

Chalkboards became so important for teaching that teachers in the 19th century sometimes went to extremes to create one. In Glen Allen, Virginia; a school is named for Elizabeth Holladay, a pioneer teacher who started the first public school in the Glen Allen area of Henrico County at her home in 1886. On a note about the history of the school it says she had, "Black oilcloth tacked to another part of the shipping crate served as a blackboard." 

The slate was used even after paper became a relatively commonplace item. Many school histories report the use of slates into the 20th Century. This use may have been significant. The Binney & Smith company, better known to many for their creation of the Crayola Crayon, began the production of slate pencils, for writing on slate, in the year 1900. As an aside, they also won a Gold Medal at the St. Louis Fair.

A local museum in Roxbury, New Zealand has Binney and Smith Slates and Pencils displayed in a school started in 1872.  I am searching for records of cost of slates, lead pencils, paper and such in 1800-1920 period, and any help is appreciated.  These slate pencils were solid cylinders of lead or stone.

An add for an "Andrews Quiet Drawing Slate'" from 1870's was priced at 40 cents.

The same museum also has a display of wooden slate pencils, which look like regular lead pencils, but with lead filing.  

Slate pencils prior to 1800 were known as Dutch Pencils in England, but increased slate mining in Wales around 1800 led to more domestic production, and use of slates, and slate pencils in England.   In the journal Australian Historical Archaeology, (2005) Peter Davies reports that in the excavation of a site called Henry Mill that was only operational from 1904 until around 1930 they found 30 slate pencils, remnants of four slates, and a single graphite pencil core. 

The National Museum of American has a  pastel of “School Boy with Slate.” from 1822.

In "Slates Away!": Penmanship in Queensland, Australia, John Elkins, who started primary school in 1945, writes that he used slates commonly until around the third year of school.

I think in Prep 1 that we had some paper to write on with pencils, but my memory of the routine use of slates is much more vivid. Each slate was framed in wood and one side was inscribed with lines to guide the limits for the upper and lower extremities of letters. The slate "pencils" were made of some pale gray mineral softer than slate which had been milled into cylinders some one-eighth of an inch in diameter and inserted into metal holders so that about an inch protruded.
Each student was equipped with a small tobacco tin in which was kept a damp sponge or cloth to erase the marks. Sharpening slate pencils was a regular task. We rubbed them on any suitable brick or concrete surface in the school yard. Teachers also kept a good supply of spares, all writing materials and books being provided by the school. It is possible that the retention of slates stemmed from the political imperative that public education should be free.

Slates were advertised in newspapers in the US as early as 1737. Slates, as indicated above, show up as commonplace in quotes from the OED as early as 1698. It seems they may have been used for some artistic or educational purposes as early as the end of the 15th Century. In the famous painting of Luca Pacioli,
Ritratto di Frà Luca Pacioli, Pacioli is shown drawing on a slate to copy an example from Euclid in the open book before him. The closed book, which has the dodecahedron upon it, is supposedly Pacioli's Somma di aritmetica which was written in 1494.
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In the Dec 2003 issue of Paradigm, the Journal of the Textbook Colloquium, is an article by Nigel Hall titled, "The role of the slate in Lancasterian schools as evidenced by their manuals and handbooks". A couple of snips from the article appear below:

The Oxford English Dictionary gives as its first citation for slate being used as a writing tool a quotation from Chaucer’s Treatise on the Astrolabe written about 1391. Whether usage began around this time or had begun much earlier is unknown, although as a technology it shared many characteristics with the wax tablet, used extensively from before the time of the Greeks until the 1600s in Europe, and even surviving in some usages until the early twentieth century (Lalou, 1989). Knowledge of the use of slate for writing after Chaucer is limited until one reaches the second half of the eighteenth century. The mathematician Digges (1591) refers to writing on slates and in the new colony of America an inventory (Plymouth Colony Archive, n.d.) made on 24 October 1633 of the possessions of the recently deceased Godbert and Zarah, noted among many items, ‘A writing table of slate’ (table here being a tablet of slate).

Hall goes on to suggest that, in fact, the use of slates may not have been very common in England until the end of the 18th Century because reading (beginning with hornbooks) was much more commonly taught than writing. He credits Lancaster for the promotion of slates for writing and math, but suggests that the slate was a principal element in the "monotorial system" in which more advanced students taught the lower group. An illustration showing the use of slates (now broken) and the student monitor below is taken from the article. [See the full article here]

The blackboard was extended to some specialty uses as well. A "Slated Globe" was advertised in The New York Teacher, and the American Educational Monthly, Volume 6 in 1869 for use in spherical geometry and geography classes. A four inch diameter globe sold for $1.50

I also recently found this image on a Wikipedia article about Benjamin Pierce. He seems to be standing beside a stand with a spherical blackboard resting on it, but can not be sure that is what it was.

In an 1899 article for the proceedings of the Society for the Promotion of Engineering Education, Professor Arthur E Haynes of the University of Minnesota had an article for, "The Mounting and Use of a Spherical Blackboard, which included this image.

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Recently, J F Ptak posted an article on his Science Books blog from Scientific American, (Sep 13, 1890) about a pen-tip eraser for slate pens meant to be wetted to erase the marks on a slate by the pen.  The article described the invention with credit to the inventor, Mrs Emma C. Hudson

Regular Polygons Inscribed in a Similar Regular Polygon

A long time back, I wrote about some geometric options for a problem I had found on Greg Ross' Futility Closet. Shortly afterward I got a note from a blogger at "Five Triangles" who mentioned he had posted a very similar problem (above) about a year earlier.

What I especially enjoyed about his presentation of the problem was the obvious invitation to generalize the idea to regular polygons of more sides.
Each of the figures is simplified by the visual approach of rotating the inner polygon until it has its vertices at the midpoints of the sides of the larger. From there it is almost trivial that for the triangles, the smaller is 1/4 the larger (the logo of the five-triangles web site shows this clearly), and the smaller square is 1/2 the larger.

When you go to five sides the quick visual solutions disappear, but a generalization should offer itself to a clever trig student. If we assume the sides of the larger n-gon are each of unit length, then the area of the two polygons should be in the same ratio as the square of the side of the smaller polygon..... ( some were confused by this, the area of two similar polygons are in the ratio of the square of their corresponding side, but since we et the larger at one unit, its square is 1 square unit, and so the ratio of the two area is the square of the inner edge length over one.)....and a clever trig student looking at all those triangles (such as the blue FGB) formed between the two polygons should know a quick rule for finding the square of the side lengths of the inner polygon.... the beautiful extension of the Pythagorean theorem they know as the law of cosines.

Since each leg on the outside of the triangle is \(\frac{1}{2}\) unit, and the angle is \(\frac{\pi(n-2)}{n}\) it should be easy to determine that the square of the sides of the inner polygon is \(\frac {1}{2})^2 + (\frac {1}{2})^2 - 2 *(\frac {1}{2}*\frac {1}{2} * \cos(\frac{\pi(n-2)}{n})\) Or more simply, \(\frac {1}{2}(1-\cos(\frac{\pi(n-2)}{n}))\)
For values from n= 3 to 12 I came up with the following with the support of Wolframalpha:

Only the triangle, square, and hexagons produced rational roots, in convenient consecutive quarters for easy remembering.  The decimal approximations clearly support the intuitive idea that the limit should approach one as n grows larger without bound. By the time you get to the hectogon, the ratio is ,9990.  Fans of the "golden ratio will appreciate its appearance in the pentagon even if slightly camouflaged.

N      Ratio

3 ..... 0.25

4 ..... 0.5

5 ..... 0.654508

6 ..... 0.75

7 ..... 0.811745

8 ..... 0.853553

9 ..... 0.883022 

10 .... 0.904508

11 .... 0.920627

12 .... 0.933013

An interesting exploration, I think, for good trig students to explore.  Enjoy

A Unique approach for Odd Order Magic Squares

Lo Shu Magic Square

I have been interested in Math History and Recreation Math for a really long time,  (yes, I'm that old), so when I came across a new approach on twitter that I had never seen, I was a little surprised.  When I read that it was about 400 years old, I was even more surprised (and no, I'm not THAT old).

I've written about Magic Squares over the years, from the earliest known 3x3 supposedly found on the back of a turtle in Chinese Mythology, called the Lo Shu Square ( literally: Luo (River) Book/Scroll)  and about the magic square on the Passion Facade on the Sagrada Familia Cathedral in Barcelona and then about  a magic square relationship to Matrices I just learned this year (2018) from John D. Cook's blog.

I usually not surprised in finding out new relationships in magic squares, but part of what surprised me this time, was that it was a method created by Claude Gaspard Bachet de Méziriac, Who I've read a lot about, and written a little about, and was aware that he worked with recreational math and number theory. He published a Latin translation of the Greek text of Diophantus’s Arithmetica in 1621. This is the translation that Fermat made his famous margin note that became the famous Fermat's Last Theorem. He asked the first ferrying problem: Three jealous husbands and their wives wish to cross a river in a boat that will only hold two persons, in such a manner as to never leave a woman in the company of a man unless her husband is present.

So, anyway, if I'm not the only person in the world who never saw it before, here is a really unique method of constructing nXn magic Squares when n is odd. which I found on a animated tweet created in Geogebra by Jason-Automaths@palajsn, and thanks to Vincent Pantaloni for sharing.

You start by constructing a Diamond stack of squares with 1 square in the first row, three squares in the second, etc. until you get to 2n-1 squares for the desired nXn desired, then descending back down to one. Here is the example for the 7x7 square.

Then you start at the 1'st diagonal down the right side and write the numbers in order, 1 to 7. Skip to the third diagonal and do the next 7 digits. Continue in like fashion and you get something like this.

Now here is the slickest little move imaginable, you take the pyramid of six numbers above the the top row of seven squares, and move it down until the number one is just below the center square (25 in this case)...

Make a similar translation of the pyramids on the other three sides across to the similar position on the other side of the center square, and you have a magic square.

The first known magic square was the Lo Shu, shown above. If you learned the quick method I did for odd squares, you start at the bottom center with one (apparently the Chinese put North on their calendars at the bottom, and that was an influence on the future evolution of magic squares). Then you just number up and right (or down and right) on the diagonal (as if the edges were connected right and left, top and bottom like a torus). Each time you come to a multiple of n, you drop down one and continue. Notice this works the same way, except that the diagonals go down and right, and at ever multiple of n, you drop down two rows, instead of one.