Tuesday, 31 October 2023

Soul Cakes, Halloween Began in Britain??

Well, you get old and eventually you learn stuff. I got a note from Charles Wells who said, " The *name* Halloween came from the British. It is the eve of All Saints Day (Hallowmas, November 1) which is celebrated all over Catholic Europe, not just in Britain. November 2 is All Souls Day, meaning the day for sinners as well as saints. That is the Day of the Dead in Mexico."
Later I added, "Thanks, Charles,but given that the Scottish, the English, the Welsh, and the Irish ALL seem to object to being included as "British", I will simply confirm what you have said with a quote from the Online Etymology Dictionary, "c.1745, Scottish shortening of Allhallow-even "Eve of All Saints, last night of October" (1556), the last night of the year in the old Celtic calendar, where it was Old Year's Night, a night for witches. Another pagan holiday given a cursory baptism and sent on its way. Hallowmas "All-saints" is first attested 1389."

 
I had just seen a BBC show that morning in 2009 and Sting had just released the Soul Cakes song, don't ask why, some things must be left unexplained.
 

 Here is the story of soul cakes and halloween as told by Wikipedia: "A Soul cake is a small round cake which is traditionally made for All Souls' Day to celebrate the dead. The cakes, often simply referred to as souls, were given out to soulers (mainly consisting of children and the poor) who would go from door to door on Hallowmas (new word to me, obviously the eve of All Souls Day) singing and saying prayers for the dead. Each cake eaten would represent a soul being freed from Purgatory. The practice of giving and eating soul cakes is often seen as the origin of modern Trick or Treating." "The tradition of giving Soul Cakes originated in Britain during the Middle Ages, although similar practices for the souls of the dead were found as far south as Italy." "The cakes were usually filled with allspice, nutmeg, cinnamon, or other sweet spices, raisins or currants, and later were topped with the mark of a cross. They were traditionally set out with glasses of wine on All Hallows Eve, and on All Saints Day children would go "souling" by calling out: Soul, Soul, a soul cake! 
 I pray thee, good missus, a soul cake! 
 One for Peter, two for Paul, 
 three for Him what made us all! 
 Soul Cake, soul cake, please good missus, a soul cake.
 An apple, a pear, a plum, or a cherry,
 anything good thing to make us all merry. 
 One for Peter, one for Paul, 
& three for Him who made us all. ...

lyrics from A Soalin', a holiday song written and performed by Peter, Paul and Mary (1963)."

See, nothing scary here. Have a Happy Halloween.

Monday, 30 October 2023

Notes on the History of Graph Paper

Direct repost from 2011 to preserve notes.If you have corrections, comments, or additions, please send me a note, comment, telegram.....



 



Just re-ordered graph paper for next year for my department. We don't use nearly as much these days as five or ten years ago... calculators have made them much less common in schools. It reminded me that I hadn't actually put anything here about my notes on the history of graph paper, so for those who are interested...

Graph paper, a math class staple, was developed between 1890 and 1910. During this period the number of high school students in the U.S. quadrupled, and by 1920, according to E.L Thorndike, one of every three teenagers in America “enters High School”, compared to one in ten in 1890. The population of “high school age” people had also grown so that the total number of people entering HS was six times as great as only three decades before. Research mathematicians and educators took an active interest in improving high school education. E. H. Moore, a distinguished mathematician at the University of Chicago, served on mathematics education panels and wrote at length on the advantages of teaching students to graph curves using paper with “squared lines.” When the University of Chicago opened in 1892 E.H. Moore was the acting head of the mathematics department. “Moore was born in Marietta, Ohio, in 1862, and graduated from Woodward High School in Cincinnati. “(from Milestones in (Ohio) Mathematics, by David E. Kullman) Moore was President of the American Mathematical Society in 1902. The Fourth Yearbook of the NCTM, Significant Changes and Trends in the Teaching of Mathematics Throughout the World Since 1910, published in 1929, has on page 159, “The graph, of great and growing importance, began to receive the attention of mathematics teachers during the first decade of the present century (20th)” . Later on page 160 they continue, “The graph appeared somewhat prior to 108, and although used to excess for a time, has held its position about as long and as successfully as any proposed reform. Owing to the prominence of the statistical graph, and the increased interest in educational statistics, graphic work is assured a permanent place in our courses in mathematics.” [emphasis added]
Hall and Stevens “A school Arithmetic”, printed in 1919, has a chapter on graphing on “squared paper”.

Using google's n-gram viewer I arrived at the conclusion that from 1880 until appx 1925 the term square paper was the most popular with coordinate paper close behind.  The earliest mention of graph paper in relation to math education was in Advanced Algebra by Joseph Victor Collins.  I found a use in 1890 of logarithmic graph paper (defined as "Ordinate scales printed on logarithmic graph paper" in a report " Geological Survey Water-supply Paper - Issues 1890-1894").  The term may have been more used in engineering fields before it was adapted in mathematics.  However the shift occurred, by 1930 "Graph paper" was the most common term, and by 1940 it was more common than the other two terms combined, and by 1960 it reached eight times the usage of either of the others.  *PB 

John Bibby has written (August,2012) to advise me that John Perry, who was at the time President of the Institution of Electrical Engineers, has a section on "Use of Squared Paper" in an article in Nature in 1900 (The teaching of mathematics, Nature Aug 1900 pp.317-320.) They wanted $19 to see the article, so I take John at his word. I did find another similar endorsement of "squared paper" by Perry in "Englands Neglect of Science" published in 1900 also. On page 18 after several lamentations about trained engineers who had no ability/understanding of the mathematics applying to their field, he writes: "I tell you, gentlemen, that there is only one remedy for this sort of thing. Just as the antiquated method of studying arithmetic has been given up, so the antiquated method of studying other parts of mathematics must be given up. The practical engineer needs to use squared paper." 

The actual first commercially published “coordinate paper” is usually attributed to Dr. Buxton of England in 1795 (if you know more about this man, let me know). The earliest record I know of the use of coordinate paper in published research was in 1800. Luke Howard (who is remembered for creating the names of clouds.. cumulus, nimbus, and such) included a graph of barometric variations. [On a periodical variation of the barometer, apparently due to the influence of the sun and moon on the atmosphere. Philosophical Magazine, 7 :355-363. ]
[The above was gathered from a numbur of authoritive sources including a Smithsonian site, but on a recent visit to Monticello, the home of my longtime favorite American Prisident, Thomas Jefferson, I discovered it was in error. I found a use by Jefferson in his use of the paper for architectural drawings earlier than any of these dates. Here is the information from the Moticello web site.]
Prior to 1784, when Jefferson arrived in France, most if not all of his drawings were made in ink. In Paris, Jefferson began to use pencil for drawing, and adopted the use of coordinate, or graph, paper. He treasured the coordinate paper that he brought back to the United States with him and used it sparingly over the course of many years. He gave a few sheets to his good friend David Rittenhouse, the astronomer and inventor:

"I send for your acceptance some sheets of drawing-paper, which being laid off in squares representing feet or what you please, saves the necessity of using the rule and dividers in all rectangular draughts and those whose angles have their sines and cosines in the proportion of any integral numbers. Using a black lead pencil the lines are very visible, and easily effaced with Indian rubber to be used for any other draught." {Jefferson to David Rittenhouse, March 19, 1791}
A few precious sheets of the paper survive today.

 

The increased use of graphs and graph paper around the turn of the century is supported by a Preface to the “New Edition” of Algebra for Beginners by Hall and Knight. The book, which was reprinted yearly between the original edition and 1904 had no graphs appearing anywhere. When the “New Edition” appeared in 1906 it had an appendix on “Easy Graphs”, and the cover had been changed to include the subhead, “Including Easy Graphs”. The preface includes a strong statement that “the squared paper should be of good quality and accurately ruled to inches and tenths of an inch. Experience shews that anything on a smaller scale (such as ‘millimeter’ paper) is practically worthless in the hands of beginners.” He finishes with the admonition that, “The growing fashion of introducing graphs into all kinds of elementary work, where they are not wanted, and where they serve no purpose – either in illustration of guiding principles or in curtailing calculation – cannot be too strongly deprecated. (H. S. Hall, 1906)” The appendix continued to be the only place where graphs appeared as late as the 1928 edition. The term “graph paper seems not to have caught on quickly. I have a Hall (the same H S Hall as before) and Stevens, A school Arithmetic, printed in 1919 that has a chapter on graphing on “squared paper”. Even later is a 1937 D. C. Heath text, Analytic Geometry by W. A. Wilson and J. A. Tracey, that uses the phrase “coordinate paper” (page 223, topic 153). Even in 1919 Practical mathematics for Home Study by Claude Irwin Palmer introduced a section on “Area Found by the Use of Squared Paper” and then defined “paper accurately ruled into small squares” (pg 183). It may be that the term squared paper hung on much longer in England than in the US. I have a 1961 copy of Public School Arithmetic (“Thirty-sixth impression, First published in 1910) by Baker and Bourne published in London that still uses the term “squared paper” but uses graphs extensively.

I recently found one other earlier use of coordinate grid on paper. The Metropolitan Museum of Art owns a pattern book dated to around 1596 in which each page bears a grid printed with a woodblock. The owner has used these grids to create block pictures in black and white and in color.


Of course "graph paper" could not have preceded the term "graph" for a curve of a function relationship, and many teachers and students might be surprised to know that it was not until 1886 when George Chrystal wrote in his Algebra I, "This curve we may call the graph of the function." The actual first known use of the term "graph" for a mathematical object actually predates this event by only eight years and occurred in a discrete math topic.   J. J. Sylvester published a note in February 1878 using 'graph' to denote a set of points connected by lines to represent chemical connections. In that note "Chemistry and Algebra", Sylvester
wrote: "Every invariant and covariant thus becomes expressible by a graph precisely identical with a Kekulean diagram or chemicograph" .


A more or less famous Kekule structure is the benzine shown at right.
(August Kekule von Stradonitz was one of the founders of structural organic chemistry, and is remembered for his dreams of the structure of benzene as a snake swallowing its own tail.)

This short note in Nature was more a notice of the more complete paper he had written in American Journal of Mathematics, Pure and Applied, would appeared the same month,  "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, — with three appendices,"  The term "graph" first appears in this paper on page 65.  The images he uses appear below. 





Graph has come to have multiple meanings in mathematics, but for most students it relates to the graph of functions on the coordinate axes.  The origin is from the Greek graphon, to write, perhaps with earlier references to carving or scratching. Jeff Miller's web site suggests that the use of graph as a verb may have first been introduced as late as 1898. 
In a post to a history newsgroup, Karen Dee Michalowicz commented on the history of graphing:
It is interesting to note that the coordinate geometry that Descartes introduced in the 1600's did not appear in textbooks in the context of graphing equations until much later.  In fact, I find it appearing in the mid 1800's in my old college texts in analytic geometry.  It isn't until the first decade of the 20th century that graphing appears in standard high school algebra texts. [This matches rise of  graph paper in the same periods].  Graphing is most often found in books by Wentworth.  Even so, the texts written in the 20th century, perhaps until the 1960's, did not all have graphing.  Taking Algebra I in the middle 1950's, I did not learn to graph until I took Algebra II

Math historian Bea Lumpkin has written about the early graphs by the Egyptians in what was an early use of what painters call the grid method:
In my article ... I suggest, "It is possible that the concept of coordinates grew out of the Egyptian use of square grids to copy or enlarge artwork, square by square.  It needs just one short, important step from the use of square grids to the location of points by coordinates.  
In the same posting she comments on the finding of graphs in Egyptian finds dating back to 2700 BC: 
"An architect's diagram of great importance has lately been found by the Department of Antiquities at Saqqara.  It is a limestone flake, apparently complete, measuring about 5 x 7 x 2 inches, inscribed on one face in red ink, and probably belongs to the III rd dynasty"  Here is the reason that Clark and Engelbach attached great importance to the diagram.  It shows a curve with vertical line segments labeled with coordinates that give the height of points on the curve that are equally spaced horizontally.  The vertical coordinates are given in cubits, palms and fingers.  The horizontal spacing, the authors write "... most probably that is to be understood as one cubit, an implied unit elsewhere."  To clinch their analysis, Clarke and Engelbach observe:  "This ostrakon was found near the remains of a solid saddle-backed construction, the top of which, as far as could be ascertained from its half-destroyed condition, closely approximated tot he curve obtained from the data on the ostrakon. 


This certainly lays claim to the oldest line graph I have ever heard.  

Sunday, 29 October 2023

Barycenter....History and Etymology of Math terms

 Barycenter The word barycenter is another term for the center of gravity or centroid. The Greek root is barus which generally refers to weighty or heavy. The more ancient Indo-European root seems to have come from a word like "gwerus" and has relatives in our words for gravity and grave.

The term "centroid" is of recent coinage (1814). It is used as a substitute for the older terms "center of gravity" and "center of mass" when the purely geometrical aspects of that point are to be emphasized. The term is peculiar to the English language; 

Another word derived from the same root is baryon, the name for a family of particles that are heavier (more massive) than mesons. The word barometer also comes from the same root and is so named because, in a sense, it measures how heavy the air is. Another related word still in current use is baritone, which literally means heavy voiced. The science names for the chemical barium and the ore from which we obtain it, barite, also called "heavy spar", are both from the same root.
The History of Math web site at St. Andrews University in Scotland credits the creation of barycenters to August Möbius (1790-1868):

In 1827 Möbius published Der barycentrische Calcul, a geometrical book which studies transformations of lines and conics. The novel feature of this work is the introduction of barycentric coordinates. Given any triangle ABC then if weights a, b and c are placed at A, B and C respectively then a point P, the center of gravity, is determined. Möbius showed that every point P in the plane is determined by the homogeneous coordinates [a,b,c], the weights required to be placed at A, B and C to give the center of gravity at P. The importance here is that Möbius was considering directed quantities, an early appearance of vectors.


Saturday, 28 October 2023

Versine, Haversine... History and Etymology of Math Terms

  Versine

The versine of an angle, A, is an almost extinct expression for the quantity 1-cos(A). Up to the 1600's this was probably the second most common trigonometric value used. The Latin word versed relates to turning, and the "versed sine" was, in essence, the sine turned 90 degrees.

In 1835, James Inman introduced the term haversine to describe a value of 1/2 of the versine, "half-versine". The haversine was an important formula in spherical geometry and navigation, since it gave a simple way to find the approximate distance between two points on the earth using the Longitude and Longitudes. If we consider two points on the unit sphere, with positions given as (lat1, long1) and (lat2, long2) in radians, then the distance between them is given by  where dLat and dLong are the differences in the latitiudes and longitudes. Tables for Navigation contained both Hav(x) and its inverse invHav(x) and the logs of these values to assist in prosthaphaeresis . To find the distances on the earth, the answer would be multiplied by the radius of the earth. According to Jeff Miller's web site, the word first appeared in the third edition of Navigation and Nautical Astronomy for the use of British Seamen.

The mathematical terms converse and inverse are both from the same root. Many other words come less directly from this root. A plow turns dirt up and over and creates a furrow, a straight line of dirt along the ground. Things laid out along a straight line were sometimes said to resemble the furrow and called verses, and thus words in a line of a poem became a verse. To reverse is to turn back, and the obverse side is the side you see when you turn something over, and your vertebra are the joints that allow you to turn.

Friday, 27 October 2023

Problems from the Land down Under

 

Looking through the Gazette of the Australian Mathematical Society, and found their puzzle corner (July 2009 , so the exponents in the first problem are explained) ... really nice problems. I think I have this one, but I didn't prove it.... 


  Digital deduction The numbers 2^2009 and 5^2009 are written on a piece of paper in decimal notation. How many digits are on this piece of paper? And this one has me puzzled (which is why they call them puzzles, I guess).. 

  Piles of stones There are 25 stones sitting in a pile next to a blackboard. You are allowed to take a pile and divide it into two smaller piles of size a and b, but then you must write the number a×b on the blackboard. You continue to do this until you are left with 25 piles, each with one stone. What is the maximum possible sum of the numbers written on the blackboard? Anyone know how to a) prove the first, or b) solve the second... 

Do let me know....mostly down to chewing my pencil tips now.... 


 Spoiler (I think) x x x x x x x 
OK, for number one I went back to that old Polya-ism, "If there is a problem you can't solve, find a smaller problem you can solve."  Instead of 2010 I put in 1.  Well 2^1 has one digit and 5^1 has 1 digit so the answer is 2.  Repeating this with more numbers it seemed the solution was always n+1 digits for any exponent n.  
Sue VanHattum gave a nice approach using base ten logarithms, 
digits in 2^n = ceiling(log(2^n))
digits in 5^n = ceiling(log(5^n))
adding gives n+1, so we have n+1 digits.

OK, I think the total for the 25 stones will always be 300... I tried it about three different ways and they all came out the same... hmmmm... In fact, if we look at some smaller numbers for a guide, it seems that for any n, the sum of the products by this process will lead to \( \binom{n}{2}\)... now why is that? Anyone, Anyone??? Bueller?
Well I was right on that one, it seems, but the real understanding came when master problem solver, Joshua Zucker, explained, "The second problem I have seen many times in books as a strong induction exercise, but ... WHY does it come out the triangular numbers?

Well, the triangular numbers are the solution to the handshake problem.
When all the pebbles are in one pile, let them all shake hands.
At each splitting step, the number of points you score is equal to the number of handshakes you destroy.
At the end, you have all the pebbles in their own individual pile, so there are no more handshakes possible - they have all been destroyed.
Hence the score is equal to the initial number of handshakes.


For example, if you start with five stones/handshakes, there are 10 handshakes(edges) connecting the five points.



 If you break away a group of two, (say V1 and V2) you break the connection between each of these two in one group and the three in the second group, or six handshakes, leaving four edges (handshakes) One between the set of two, and three in the triangle of V3, V4, V5.  
------------------------

Just for a kick, I picked out a couple of newer ones for you to try.  Enjoy and share your solutions:

For the geometry lovers, try this one.

In a regular nonagon, prove that the length difference between the longest diagonal and the shortest diagonal is equal to the side length. In other words, prove c−b = a in the diagram below.
*Australian Mathematical Soc. Gazette
And here is one for that I think is an excellent problem for younger students to intuit a wonderful mix of problems solving ideas.  "Let S be a set of 10 distinct positive integers no more than 100. Prove that S contains two disjoint non-empty subsets which have the same sum."

I will come back in awhile and address possible approaches to each, (If I can solve them).

Thursday, 26 October 2023

#26 Repunit and Residual... History and Etymology of Math Terms

 Repunit The term repunit, for a number made up of all unit digits, was created by Albert Beiler in the 1960's as a contraction of repeated unit. It still does not appear in most dictionaries, and is therefore difficult to trace... more to come, I hope.

One of the most famous puzzle masters of the early 1900's was Henry Ernest Dudeney. In his puzzle book,The Canterbury Puzzles, of 1907 he poses a problem about Repunit factoring, without using the term repunit. The problem is posed thus:

"It used to be told at St Edmondsbury," said Father Peter on one occasion, "that many years ago they were so overrun with mice that the good abbot gave orders that all the cats from the country round should be obtained to exterminate the vermin. A record was kept, and at the end of the year it was found that every cat had killed an equal number of mice, and the total was 1,111,111 mice. How many cats do you suppose there were?"
Later the problem provides that there is more than one cat, and each cat kills more than one mouse, in fact each cat killed more mice than there were cats.

I will begin to discuss the answer shortly, so those who wish to solve the problem first might stop reading and take a few moments to work on a solution.
Dudeney points out that for the solution to be unique, there must be only two factors of 1,111,111. In truth, the factors of 1,111,111 are 239 and 4649, so there must have been 239 cats who each caught 4649 mice.

In the solutions in the second edition in 1919, Dudeney discusses the general problem of finding solutions of repunits (again without using the term) and gives some interesting tables and remarks which I have tried to recopy accurately

Lucas, in his L'Arithmetique Amusante, gives a number of curious tables which he obtained from an arithmetical treatise, called the Talkhys, by Ibn Albanna, an arabian mathematician and astronomer of the first half of the thirteenth century. In the Paris National Library are several manuscriptes dealing with the Talkhys, and a commentary by Alkalacadi, who died in 1486. Among the tables given by Lucas is one giving all the factors of numbers of the above form (repunits) up to n=18 (eighteen ones in a row). It seems almost inconceivable that Arabians of that date could find the factors where n=17 as given in my introduction [On page 18 of the introduction he gives the factors of 11,111,111,111,111,111 as 2,071,723 and 5,363,222,357.]. But I read Lucas as stating that they are given in Talkhys, though an eminent mathematician reads him differently, and suggest to me that they were discovered by Lucas himself. This can, of course, be settled by an examination of the Talkhys, but this has not been possible during the war.(WWI)

The difficulty lies wholly with those cases where n is a prime number. If n=2, we get the prime 11. The factors when n=3, 5, 11, and 13 are respectively (3x37), (41 x 271), (21,649x513,239), and (53 x 79 x 265,371,653)[Dudeney used a raised dot for multiplication which I have replaced with an x for my convenience]. I have given in these pages the factors where n=7 and n=17. The factors where n=19, 23, and 37 are unknown, if there are any.* [emphasis added]

In a footnote He points out that "Mr. Oscar Hoppe, of New York, informs me that, after reading my statement in the introduction [where he says n=19 is prime], he was led to investigate the case of n=19, and after long and tedious work he succeeded in probing the number to be a prime. He submitted his proof to the London Mathematical Society, and a specially appointed committee of that body accepted the proof as final and conclusive. He refers me to the Proceedings of the Society for 14th February, 1918.

Dudeney also points out three "curious series of factors" that he thought would "doubtless interest the reader." They are shown here: n=2 --> 11
n=6 --> 11 x 111 x 91
n=10 --> 11 x 11,111 x 9091
n=14 --> 11 x 1,111,111 x 909,091
Can you guess n=18?

Or the same numbers can be written as
n=6 --> 111 x 1,001
n=10 --> 11,111 x 100,001
n=14--> 1,111,111 x 1,000,000,001


For the cases where n is a multiple of 4, we get
n=4 --> 11 x 101
n=8 --> 11 x 101 x 10,001
n=12--> 11 x 101 x 100,010,001
n=16--> 11 x 101 x 1,000,100,010,001


Residual Sit back, stay right there, and I will tell you the origin of residual. Wait! I just did. The common re prefix means back and sid is from the Latin sedere which means to sit, so the literal meaning of residual is one who sits back or, more appropriately, stays seated. In statistics we use it in the same sense as residue, that which remains (stays seated) when something else is taken away; what remains from the observed amount when the predicted amount is removed. Another closely related word is residence. Other words drawn from the sedere root include sedentary [one who sits around a lot], sediment [stuff that settles], and sedative [something that keeps you from moving around].

Wednesday, 25 October 2023

Almost Integers, almost incredible

 



I first came into teaching just as computers and programmable calculators did. It was  me, Apple and Texas Instruments side by side against mathematical illiteracy. One of the things I loved doing in the classroom , and still enjoy out of it, were  expressions that were "nearly" an integer, now called "almost integers" in recreational math.

I think my first introduction came a half-dozen or so years before I first became a teacher. It was the famous article by Martin Gardner in April of 1975 when he posted his "Six sensational discoveries that somehow or another have escaped public attention" in his column in Scientific American. One of his six things was his claim that eπ√163 = 262,537,412,640,768,744, a perfect integer. He credited this discovery to Ramanujan, and it has come to be known as Ramanujan's constant, but the nearness to an integer was noted in 1859 by the mathematician Charles Hermite. The number actually ends in ...743.999,999,999,999,25..., but in the age of calculators only given ten to 12 digits, it was enough to be confusing and difficult to check. You can use much smaller numbers to get similar near integers, eπ√43 comes within about 1/5000 of an integer, and my TI-84 silver Edition gives 884736744, an exact integer as the result.

The Golden Ratio, \( \phi =\frac{\sqrt{5}+1}{2} \) will produce almost integers for any power above about ten. \( \phi^{10} =122.991... \) while \( \phi^{18} =5777.9998.. \). Successive powers tend to successively overreach and undershoot an integer. It seems that the twenty-fourth power is sufficient to fool my TI which gives \( \phi^{24} =103,682 \). (I just checked this last one on Desmos online graphing calculator and it gives 103681.99999.  Science marches on.  

The numbers 134903170 and 196418 form a pair that introduce lots of almost integers. 134903170/196418 = 5777.999826..., these are the 45th and 27th Fibonacci numbers, but you get nearer to an integer each time you double the index of the two Fibonacci numbers. Wikipedia tells me that Fib(45*8) /Fib(27*8) will produce a string of about 30 nines after the decimal point.

An unusual one that is more of a mathematical coincidence, and not explained, is the fact that \( e^{\pi} - \pi = 19.99909997... \) The constant \( e^{\pi} \) is called Gelfond's Constant, after Aleksandr Gelfond, who proved that the number was transcendental.

For Trig Students, Sin (11) is an almost integer, (almost -1) and you can totally fool the TI-84 with \( Sin(2017*\sqrt[5]{2})\) which I'm using for my special day number next year on 2/5/2017 . The Sin(11) is nice to introduce when working with half-angle identities since \( sin^2(11)=1/2(1-cos^2(22)\).

\( 163 ( \pi - e ) \) is almost an integer, and powers of pi and e can be fun almost integers, and \( \pi^4 + \pi^5 - e^6 \) is almost zero. Students might search for such an expression that will fool their calculator.

And for Geometry class, the image at the top of the blog shows a triangle with all integer sides and segments except segment d.  Ed Pegg Jr is credited with finding this gem, and showing that the segment marked d is 7.0000000857....

Know of others, ....? SHARE!

Tuesday, 24 October 2023

Which Platonic Solid is Most-Spherical? (and The Archimedian Solids)

 


If you inscribe a regular polygon into a given circle, the larger the number of sides, the larger the area of the polygon. I guess I always thought that the same would apply to Platonic solids inscribed in a sphere..... It doesn't. I noticed this as I was looking though "The Penny cyclopædia of the Society for the Diffusion of Useful Knowledge
By Society for the Diffusion of Useful Knowledge" (1841). As I browsed the book, I came across the table below:


The table gives features of the Platonic solids when inscribed in a one-unit sphere. At first I thought they must have made a mistake, but not so. The Dodecahedron fills almost 10% more of a sphere (about 66%) than the icosahedron (about 60%). So the Dodecahedron is closer to the sphere than the others.

Interestingly, if you look at the radii of the inscribing spheres, it is clear that solids which are duals are tangent to the same internal sphere.

But if you look at the table of volumes when the solids are inscribed with a sphere inside tangent to each face:


When you put the Platonic solids around a sphere, the one smallest, and thus closest to the sphere is the icosahedron.

This leads to the paradox that when platonic solids are inscribed with a sphere, the icosahedron is closest to the circle in volume (thus most spherical?) but when they are circumscribed by a sphere, the dodecahedron is the closest to the volume of a sphere (and thus most spherical?)... hmmm

Here is a table of the same values when the surface area (superficies) is one square unit.Notice that for a given surface area, the icosahedron has the largest volume, so it is the most efficient "packaging" of the solids (thus more spherical?).
I guess that makes it 2-1 for the icosahedron, so I wasn't completely wrong all along.

POSTSCRIPT:::  Allen Knutson's comments on the likely cause of this reversal of "closeness" to the sphere:


I think it's about points of contact. On the inside, the dodecahedron touches the sphere at the most points (20), and on the outside, the icosahedron touches the sphere at the most points (again 20).
Indeed: my recipe would suggest that inside, the 8-vertex cube is bigger than the 6-vertex octahedron, and outside, the 8-face octahedron is smaller than the 6-face cube. Both are borne out by your tables. Thank you, Allen

Some other pertinent comments too good to ignore:
Anonymous said...

I think "most-spherical" would need to be better defined.

Not that I want to do it (and not that I am certain that I could), but perhaps summing the squares of the distances from each point on the surface of the solid to the closest point on the sphere would be the way to go (like a sort of physical variance).

But as I say, I'm not sure I could do this (maybe I can kill some time today, I'm visiting a puzzle-friend), and I'm not sure that it makes a lot of sense.

Oh, oh, and inscribe or circumscribe or find the best match in-between?

(it's play with this, read on-line, or grade. Choices.)

Jonathan

(Love When anonymous signs his comment)


Mary O'Keeffe said...

What a fascinating post! Here is a nice way to think about your first result in terms of the empty space.

The icosahedron leaves ~20% of the space in its circumscribing sphere empty. The dodecahedron leaves ~12% of the space in its circumscribing sphere empty.

That means that if you start with a solid sphere (let's imagine it's something easy to carve, like soap!) and carefully cut away the 20 portions needed to turn it into an icosahedron, each of the 20 pieces you trimmed off will have volume of about 1% of the sphere.

Similarly, if you do the same thing to carve a dodecahedron out of a sphere, each of the 12 pieces you trimmed off will also have volume of about 1% of the sphere.

From now on, whenever I look at either polyhedron (icosa or dodeca) I will always think of it a little differently--because I will think of those ~1% extensions on each face needed to round it out to a sphere.


I really liked this idea, and want to take time to find the amount of volume reduced by cutting each face into a sphere.

It seems that the cube is third best, just ahead of the octahedron.   The cube volume in a unit sphere is 1.5396.. and the volume of a unit sphere is 4/3 Pi, or 4.1888, cutting off 2.64919 cubic units (more than half the sphere, about 63%) .  That means to sculpt a cube

 from a unit sphere we cut off an average of just over 10% for each face. (I did that quickly,

 do check)

It might be a wonderful challenge to imagine a slicing approach to take out just the right fraction of the volume

 to be removed to cut the same amount when revealing each face.  I think I see some of them, but some are.... "difficult"?


----------------------------------------------------------------------------------------------------------------------------

Later I tried working on the Archimedian Solids.

Measuring Sphereocity of The Archimedian Solids

mathworld.wolfram.com


A few years ago I wrote a blog about which of the Platonic solids (above) was most spherical. I compared which ones had the most volume inscribed into a unit sphere, and the which had the smallest volume when circumscribed about a unit sphere. Surprisingly I got different answers to the two methods, and lots of good mathematical comments about why this might be so. Then a  while ago (July 2015) I posted it again. As part of a tongue-in-cheek exchange with Adam Spencer ‏@adambspencer I challenged him to find the roundest of the thirteen Archimedean Solids. For those who are not familiar with the distinction, both the Platonic and Archimedean solids are made of up faces that are regular polygons, and both have the property that the view at each vertex is identical to every other, but where the Platonic solids consist of only a single type of regular polygon, (for example the tetrahedron is made up of four equilateral triangles), the Archimedean solids may have more than one (in the cubeoctahedron each vertex is surrounded by two squares and two equilateral triangles).

Then only a day or so later, I got to wondering about the actual answer and started doing some research. Along the way I found a couple of papers on the topic, one of which was published earlier in the same month I began my search for the Platonic Solids Sphere-ness. What was great was that they re-exposed me to a formula for comparing according to George Polya from his Mathematics and Plausible Reasoning: Patterns of plausible inference. I sheepishly admit that I had read this book ( it's on my bookcase nos) several times years ago, but somehow this didn't pop up when I was thinking of the Platonic solids.

The problem of comparing the ratio of the numerical values of the surface area to the volume is that the answer changes over size. In a sphere, for instance, if the radius is one, then the volume is 4 pi/3, and the surface area is 4pi, so the V= 1/3 SA. Now increase the radius to three units and the volume is 36 pi, and the surface area is also 36 pi, Now V = SA, and if we keep making the radius bigger, the volume becomes larger than the surface area, in fact the ratio of Volume to surface area can be reduced to \( \frac{V}{SA} = \frac{r}{3} \). Now that kind of thing happens with all the solids, the larger the volume gets, the larger the ratio of V to SA gets. It is one of those things that amazes students (and well it should) that for any solid, there is some scalar multiplication which will transform it into a solid with Volume = Surface Area.  In this sense, every solid is isoperimetric (same measure)

So Polya found a way to neutralize this growth. He created an "Isoperimetric Quotient" that served to null out this scalar alteration. By setting the IQ = \( \frac{36 \pi V^2}{S^3} \) With this weapon he was able to compare, for instance, the "roundness" of the Platonic Solids. Try this with any sphere and you always get one. Try it with anything else, you always get less than one.

The IQ of the Platonic Solids follows the number of faces, with the tetrahedron at the bottom with an IQ of about .3 and the icosahedron at the top with an IQ of about .8288.

SO what about the Archimedian Solids? Well here they are
TruncatedTetrahedron...... 0.4534
TruncatedOctahedron ....... 0.749
TruncatedCube ............. 0.6056
TruncatedDodecahedron ..... 0.7893
TruncatedIcosahedron ...... 0.9027
Cuboctahedron ............. 0.7412
Icosidodecahedron ......... 0.8601
SnubCube .................. 0.8955
SnubDodecahedron .......... 0.94066
Rhombicuboctahedron ....... 0.8669
TruncatedCuboctahedron .... 0.8186
Rhombicosidodecahedron .... 0.9357
TruncatedIcosidodecahedron. 0.9053



So the roundness winner is the snubdodecahedron, with a pentagon and four equilateral triangles around each vertex. That is four 60o angles and one of 108o for a total of 348o.  Students might check if any other of the Archimedean solids can top that.  Is that "flatness" at vertices somehow related to "roundness"? I have to admit I first thought it might be the truncated icosahedron.  Students may have heard of this one more than others.  A molecule of C-60, or a “Buckyball”, consists of 60 carbon atoms arranged at the vertices of a truncated icosahedron.  It's roundness is a feature of many of its applications, but it only comes in fourth. 

Monday, 23 October 2023

Alien Division by Fractions

   From the archives in 2008, strange things happen when you're giving a down day and showing a video....



The day before Christmas break one of my seminar students brought in the old (1951) video of "The The Day the Earth Stood Still". I worked at my desk as they watched, and about thirty minutes in they called my attention to ask if the math on the blackboard was "real". The Alien in the movie, Klatu (Michael Rennie), in the company of a young boy who lived in the house where he was renting a room, had entered the home of a professor who was supposedly knowledgeable about Astro Physics. I did not recognize any physics I knew from the brief shot of what looked like differential equations of no particular relation, but that could be my limited physics more than the actual images.
I returned to work, but in a few minutes in another scene, Klatu is helping Bobby with his homework and the only line you hear is "All you have to remember is first find the common denominator, and then divide." My head pops up... what were they doing? "Common denominators" leads to thoughts of fractions, but almost no one teaches finding common denominators as a prelude to dividing fractions (which is sort of a shame because it makes division of fractions work like multiplication...the way kids think it should.) It works in fact, if you do not find the common denominator first, but sometimes the answer is as confusing as the problem.
When you multiply fractions, as every fifth grader learns, you just multiply top times top and bottom times bottom... 2/3 x 5/7 = 10/21. The fact that division works the same way is often missed, or misunderstood because it so often leads to nothing simpler... 2/3 divided by 5/7 is indeed (2 divided by 5) over (3 divided by 7) but that seems not to give the classic simple fraction we seek. For some fractions, it will work out fine... if 4/27 is divided by 2/3, the answer is (four divided by two ) over (27 divided by 3) = 2/9 and that is the answer you get by the method you memorized (but never understood, most likely) in the fifth grade.
But what if we follow the advice of the alien Klatu. If we convert 2/3 and 5/7 to fractions with a common denominator, we get 14/21 and 15/21, and if we divide top by top and bottom by bottom we get 14/15) over 1, which is just 14/15... job done...


I can imagine including some visuals and suggestive images to help it make sense... It is after all, just a reversal of the multiplication process. If we say "3 dogs times 5 = 15 dogs" then by division we should have the equivalent expressions that "15 dogs divided by 5 = 3 dogs." and just as naturally "15 dogs divided by 3 dogs = 5" .  

This may be one of the most under used and under understood parts of elementary math instruction. If teacher's taught multiplication and division with units; feet ounces, miles; before they began units on fractions, this natural connection will carry over.  Students who have learned (I've been in England too long, I just had to edit "learnt") that "eighths" and "fifths" are just units like "dogs" and "kittens" should then understand that 5 eighths divided by three eighths is just as clearly 5/3.  

Sunday, 22 October 2023

Combinations, Choose .... Notes on History and Etymology of Math Terms

 Choose

The creation of the expression "N choose R" for a term of a binomial coefficient is credited to Richard Guy sometime around 1950. Although the symbol for the number of ways of selecting a group of r distinct items from a group of n distinct items still varies, the most consistent usage today is . The symbol is read as "N choose r". The number and symbol is also called the Combination symbol, and is sometimes read as "The combinations of n things taken r at a time." Other symbols still used include and the less common . It is also very common to use a Capital C between the values of n and r with the values subscripted nCr, and many calculators still use a notation such as nCr with the numbers on the same line level with the C. Mathematica uses "Binomial[n,r]", and at one time I know that the TEX formatting language used (n\choose r}.

Most students first see the binomial coefficients as elements in the array (mis)named Pascal's Triangle


More information about computing binomial coefficients can be found in the page on combinations and permutations. The use of the term Binomial Coefficients comes from the fact that the numbers are the same as the coefficients of each term when a binomial, such as (x+y) is raised to a power. For example the expansion of (x+y)4 gives the five terms in the fifth row of Pascal's triangle.

In Early July of 2004 I received a note from Matthew Hubbard, the curator of Pascal's Triangle From Top to Bottom. In it he informed me that I had failed to include, in particular, the contribution of India to the study of the arithmetic triangle. A quick visit to his web site led me to :

The idea of taking "six tastes one at a time, two at a time, three at a time, etc." was written down correctly in India 300 years before the birth of Christ in a book called the Bhagabati Sutra, a text from the Jainist religion; this gives the subcontinent of India the distinction of being the earliest civilization to have an understanding of the binomial coefficients in their combinatorial form "n choose k" in a text that survives to this day.

The site contains much additional material about Indian study of the triangle, and other information that makes it well worth a visit.

Matthew also called me to task for my suggestion above that the use of "Pascal's Triangle" was somehow inappropriate. He wrote in justification of the term, " One of the reasons I wrote is the idea of misnomy in mathematics; you put the word (mis)named in front of Pascal's Triangle. While it is certainly true that many, many people had studied the binomial coefficients prior to Pascal, his work is honored because it was read by people who came after him, most notably Monmort and deMoivre, who credited Pascal's Treatise in their works several decades later. Moreover, it is worth reading, as Pascal finds many identities in the triangle that no one before him had written down.
It's too late to get the world to call it Pingala's Triangle, and I fully appreciate the desire of civilizations to honor their own, but I think if anybody's name is going to be linked to this famous array, Blaise Pascal is as good a candidate as any and significantly better than most."

My thanks to Matt for the additional material on the Indian contribution, and for helping to insure a balance of credit where credit is due, and certainly Pascal is due much credit for his exposure of many aspects of the triangle, by whatever name it is called.


Saturday, 21 October 2023

Viete on Pythagorean Triples

 




Viete is too little known to American High School Students (and might I say teachers???) Most would enjoy his method of taking any two Pythagorean Triples, and producing two more with a common hypotenuse. Pretty brilliant guy.


Reading The Analytic Art by Francois Viete, or at least the T R Witmer translation, and came across an interesting way of combining the legs of any two Pythagorean triples to create two others. Viete calls the two methods synaeresis and diaeresis, which seem to be language terms Viete appropriated. Synaeresis is cramming two vowel sounds together to make one... like the way people in New Orleans say "Nor"leans. I think the official term is diphthong, but check with an English major for confirmation. The actual Greek roots mean “a joining or bringing together" or something similar Diaeresis is stretching one vowel out into two....and you can find your own example...
To illustrate Viete's approach, we can take two simple right triangles, say a 3-4-5 and a 5-12-13 as examples. Viete's method would produce two triangles whose hypotenuses( hypotenii?) were both 5x13 = 65 units. Viete distinguished between the legs calling them base and the perpendicular, so in the 3-4-5 triangle the base is 3 and the perpendicular is 4. It doesn't matter which is called what name, of course except that it reverses the outcomes of the two methods. The Synaeresic method would be to add the products of each base with the perpendicular of the other triangle; 3x12+ 4x5 = 56. This would give one leg of the new triangle. To find the other leg take the difference of the products of the two bases from the two perpendiculars; 4x12 - 3x5 = 33. This completes a triple of 33-56-65.

The second method, simply reverses the signs of conjunction. Subtract the two perpendicular x base products and add the two products of a common part. The crossed terms gives 3x12-4x5 = 16 for one leg, while the products of like parts gives 4x12+3x5=63 for the other, completing a 16-63-65 right triangle.

There is a complexity about this simple method that bothered me for awhile before it hit me.  More on that later.

Ok, it's later, and I wrote more here