## Sunday, 30 August 2009

### The Sagrada Familia, Magic Squares, and the Hitch-Hiker's Guide to the Universe

I have known for a long time that there was a magic square, above [click to expand to full picture], on the Passion facade at Gaudi's famous Cathedral in Barcelona, the Sagrada Familia. But I recently got some additional information so here is the related part of the story from my MathWords web page:

Each row and column add up to 33, the supposed age of Christ at his death. In fact there are supposed to be exactly 33 different four number groupings that add up to 33; can you find them all?

In October of 2004 Ms. Sheila Knight sent me a note indicating that she and her Year 7 students at The Grays School Media Arts College had found a total of 42 four number groupings. She added, "Perhaps Gaudi was a fan of “The Hitch-hiker’s Guide to the Galaxy” where 42 is “the answer to life , the universe and everything” – this works when you consider it is on a church!". She was kind enough to send a PDF file of her results.

In Dec of 2006, Ahto Truu sent me an email in which he expanded her list to 54 different sums. He also reasoned that probably Gaudi had no knowledge of "Hitch-hiker’s Guide". Here is his post, clipped a bit:

"In fact, there are 48 of them if we don't allow the two 10's or the two 14's to be used at once, and 54 if we do.

In the list below, all the possible sums are listed in a systematic order, with the entries not on Ms. Knight's list marked:

1 3 14 15

1 4 13 15

1 5 13 14

1 6 11 15

1 7 10 15

1 7 11 14

1 8 9 15

*1 8 10 14

1 8 11 13

1 9 10 13

2 3 13 15

2 4 13 14

2 5 11 15

2 6 10 15

2 6 11 14

2 7 9 15

2 7 10 14

2 7 11 13

2 8 9 14

2 8 10 13

3 4 11 15

3 5 10 15

3 5 11 14

3 6 9 15

3 6 10 14

3 6 11 13

3 7 8 15

3 7 9 14

3 7 10 13

3 8 9 13

3 9 10 11

4 5 9 15

4 5 10 14

4 5 11 13

4 6 8 15

4 6 9 14

4 6 10 13

*4 7 8 14

*4 7 9 13

*4 8 10 11

5 6 7 15

5 6 8 14

5 6 9 13

5 7 8 13

*5 7 10 11

5 8 9 11

6 7 9 11

*6 8 9 10

*1 4 14 14

*2 3 14 14

*2 10 10 11

*4 9 10 10

*5 8 10 10

*6 7 10 10

Also, I find it quite unlikely that Antoni Gaudi could have been a fan of "The Hitch-Hikers Guide to the Galaxy", as he died in 1926, more than 50 years before the Guide appeared (the first episode of the radio show was aired in 1978, and the first book was published in 1979 :) "

A couple of years later, in 2009, I received a note from Christopher Mata that pointed out that the wall may have been influenced by the Hitch-Hikers Guide after all...

"I'd just like to note that the Passion façade was sculpted by Josep Maria Subirachs when he started work on the Sagrada Familia temple in 1987, not by Gaudí, and by then the hitchhiker's guide to the galaxy was in full circulation. Not that it is of any relevance to the matter, but for historical accuracy it isn't so much Gaudí's magic square as it is Subirach's, and was acquired bu rotating Dürer's Melancholia square and subtracting 1 from 15, 11, 10 and 16."

More information about the design and its transformation from the Durer Square can be found at this web page

I'm not sure if there is any symbolism intended in the location of the square, but it is located on the wall behind the sculpture depicting Judas betrayal of Christ with a kiss. You can see a photo of the setting here.

As a footnote about the number 42 and the "hitch-hiker's guide"; if you enter, "life , the universe and everything" in Wolfram-alpha, this is the response:

## Wednesday, 26 August 2009

### 80's Music and Math

Just read a link about the song below on Dave Richeson's "Division by Zero" blog. First, listen and try to figure out what is the cool math in the song...

Ok, Dave's web site is in response to a previous post about the Twin Primes Conjecture, and here is what he found when he checked it out on Wolfram Alpha

"Type 8675309 into WolframAlpha and it says: “Jenny’s phone number.” (Har, har, har, we all know that song.)

At the top of the page it says: Assuming “8675309″ is a phrase | Use as a number instead.

Click on that link and you’ll see the following interesting facts about 8675309.

1. 8675309 is a prime.

2. 8675309 is a twin prime (8675311 is also prime).

3. 8675309 is the hypotenuse of a (primitive) Pythagorean triple: 8675309

That’s so cool! Who knew?!?!"

Well, Not me, Dave, but I'm willing to help spread the word..

by the way, What is the smallest number that would meet all three conditions above, that is it is a twin prime and it is also the hypotenuse of a primitive Pythagorean triple.... Ok, way to easy ... what is the NEXT smallest?

The obvious question, is it really a phone number???

Yep, and here is the straight scoop from the folks at Snopes

To add a little statistical flavor..."In a project he called Jenny, are you there?, Dan Stecz, a New Jersey man, called all the 867-5309s in every area code within North America and found that nearly all the numbers were not in service. A handful of the numbers did refer to Jenny, however, and some even played bits of the song on their answering machine greeting."[wikipedia]

In 2015 I came across this while searching Google, ""Dexter" writes that if you use Vonage as your ..." Have not verified that, but

Hey, Dave, Now THAT's Cool

Ok, Dave's web site is in response to a previous post about the Twin Primes Conjecture, and here is what he found when he checked it out on Wolfram Alpha

"Type 8675309 into WolframAlpha and it says: “Jenny’s phone number.” (Har, har, har, we all know that song.)

At the top of the page it says: Assuming “8675309″ is a phrase | Use as a number instead.

Click on that link and you’ll see the following interesting facts about 8675309.

1. 8675309 is a prime.

2. 8675309 is a twin prime (8675311 is also prime).

3. 8675309 is the hypotenuse of a (primitive) Pythagorean triple: 8675309

^{2}= 2460260^{2}+8319141^{2}.That’s so cool! Who knew?!?!"

Well, Not me, Dave, but I'm willing to help spread the word..

by the way, What is the smallest number that would meet all three conditions above, that is it is a twin prime and it is also the hypotenuse of a primitive Pythagorean triple.... Ok, way to easy ... what is the NEXT smallest?

The obvious question, is it really a phone number???

Yep, and here is the straight scoop from the folks at Snopes

To add a little statistical flavor..."In a project he called Jenny, are you there?, Dan Stecz, a New Jersey man, called all the 867-5309s in every area code within North America and found that nearly all the numbers were not in service. A handful of the numbers did refer to Jenny, however, and some even played bits of the song on their answering machine greeting."[wikipedia]

In 2015 I came across this while searching Google, ""Dexter" writes that if you use Vonage as your

*phone*company, when making a call to any "867-5309"*number*, they play the chorus of "*Jenny*" before completingHey, Dave, Now THAT's Cool

## Tuesday, 25 August 2009

### More About One Letter Math Words

I questioned Craig Conley for an oversight in missing the letter m for slope in my last blog, but I have to give him credit for catching an oversight of mine.

I write a lot about the history of math and math words and symbols, and I thought I pretty well knew the History of &pi, so I was a little surprised to see a note that he had that said that the first person to use a single symbol for the ratio of the circumference to diameter of a circle was J. Christoph Sturm, who used the letter e for the value 3.14159.. Sturm, was considered the leading experimental physicist in Germany of his time.

Cajori writes that "perhaps the earliest use of a single letter to represent the ratio of the length of a circle to its diameter" occurs in 1689 in Mathesis enucleata by J. Christoph Sturm, who used e for 3.14159....

Cajori cites a note by A. Krazer in Euleri opera omnia as a reference for the above.

The first known use of the symbol π for its present purposes was in 1706 by William Jones, an English mathematician, although it was the use of the symbol by Euler that brought it its permanency. Euler, of course, is the one who popularized the use of e for 2.71828..., the base of the natural logarithms.

I had earlier mistakenly thought that Euler was the discoverer of the value, but in fact the number was published in Edward Wright's English translation of Napier's work on logarithms in 1618, almost 100 years before Euler's birth. The number represented by e is approximately 2.718281828459045... Euler actually computed the number to eight more decimal places. This was done in 1727, and would seem almost impossible accuracy for anyone else, but of Euler it was said, "Euler calculates as other men breathe."

There has been lots of speculation about why he might have chosen e, since it is almost sure that is was not for his initial. My favorite supposition is that e is the first letter in the German for one,

I write a lot about the history of math and math words and symbols, and I thought I pretty well knew the History of &pi, so I was a little surprised to see a note that he had that said that the first person to use a single symbol for the ratio of the circumference to diameter of a circle was J. Christoph Sturm, who used the letter e for the value 3.14159.. Sturm, was considered the leading experimental physicist in Germany of his time.

Cajori writes that "perhaps the earliest use of a single letter to represent the ratio of the length of a circle to its diameter" occurs in 1689 in Mathesis enucleata by J. Christoph Sturm, who used e for 3.14159....

*si diameter alicuius circuli ponatur a, circumferentiam appellari posse ea (quaecumque enim inter eas fuerit ratio, illius nomen potest designari littera e).*Cajori cites a note by A. Krazer in Euleri opera omnia as a reference for the above.

The first known use of the symbol π for its present purposes was in 1706 by William Jones, an English mathematician, although it was the use of the symbol by Euler that brought it its permanency. Euler, of course, is the one who popularized the use of e for 2.71828..., the base of the natural logarithms.

I had earlier mistakenly thought that Euler was the discoverer of the value, but in fact the number was published in Edward Wright's English translation of Napier's work on logarithms in 1618, almost 100 years before Euler's birth. The number represented by e is approximately 2.718281828459045... Euler actually computed the number to eight more decimal places. This was done in 1727, and would seem almost impossible accuracy for anyone else, but of Euler it was said, "Euler calculates as other men breathe."

There has been lots of speculation about why he might have chosen e, since it is almost sure that is was not for his initial. My favorite supposition is that e is the first letter in the German for one,

*eins*as he was studying the value of x for which the area under the hyperbolic curve y= 1/x from x=1 to this value would equal one. Well, that's my story and I'm sticking to it.## Monday, 24 August 2009

### A is for Vector?.

Found a copy of “One Letter Words” by Craig Conley in a cheap resale shop the day before I flew back to England, so I bought it and read it on the plane…Some interesting references to mathematical symbols and constants got me thinking…somebody should put together a mathematical book of one letter words… I offer the title above for free… it comes from the quote in the book that said , noun (Mathematics) A vector… and then justifies with a quote from Marie Vitulli’s “A Brief History of Linear Algebra and Matrix Theory”… The use of a single letter A to represent a matrix was crucial to the development of matrix algebra.

Ok, did he miss the point? Or did I… I assume Vitulli meant the use of a single letter (such as A) to represent a vector was a huge leap… but; I think the give-away was calling it a noun… variables like this are more like pronouns in my mind… using x in place of the number “four” or n for “some number not named”… correct me if you disagree…

Sadly, after all that, he left out the one that is known to almost every Algebra I student, “M is for slope”. Interestingly, that one has led to more mis-history in classrooms than any other topic, with the possible exception of the life of that Great American-Indian Mathematician, Chief Soh Cah Toa.

Here is what I have found about the slope, as it appears at my MathWords page; Slope is derived from the Latin root slupan for slip. The relation seems to be to the level or ground slipping away as you go forward. The root is also the progenitor of sleeve (the arm slips into it) and, by dropping the s in front we get lubricate and lubricious (a word describing a person who is "slick", or even "slimy").

Many variations of where the idea of M for slope originated seem to be mostly myth. One of the most common is that the letter was used by Descarte because it was the first letter of some French word or another that related. In a recent post to the AP Stats discussion list, Hector Hirigoyen shared the following story:

I was told by Mary Dolciani herself, that the SMSG group "decided "to use y=mx+b because of the French (Descartes, I presume)-"montant"; I found it strange because the "logical word" would be "pente"(which is slope (and the standard term in Spanish is pendiente, which matches this). However, several years ago, while visiting a French high school, I noticed the teacher used y=sx+b. I inquired, and she said because of the "American" word "slope." I believe they are using ax+b for the most part these days.

. Here are several other clips from postings about the topic on a discussion group about math history.

In his "Earliest Uses of Symbols from Geometry" web page, ... Jeff Miller gathered the following information: Slope. The earliest known use of m for slope is an 1844 British text by Matthew O'Brien entitled _A Treatise on Plane Co-Ordinate Geometry_ [V. Frederick Rickey]. George Salmon (1819-1904), an Irish mathematician, used y = mx + b in his _A Treatise on Conic Sections_, which was published in several editions beginning in 1848. Salmon referred in several places to O'Brien's Conic Sections and it may be that he adopted O'Brien's notation.

According to Erland Gadde, in Swedish textbooks the equation is usually written as y = kx + m. He writes that the technical Swedish word for "slope" is "riktningskoefficient", which literally means "direction coefficient," and he supposes k comes from "koefficient."

According to Dick Klingens, in the Netherlands the equation is usually written as y = ax + b or px + q or mx + n. He writes that the Dutch word for "slope" is "richtingscoefficient", which also means "direction coefficient." In Austria k is used for the slope, and d for the y-intercept. In Uruguay the equation is usually written as y = ax + b or y = mx + n, and the "slope" is called "pendiente", coeficiente angular", or "parametro de direccion".

It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for "modulus of slope." One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is monter. However, there is no evidence to make any such connection. Descartes, who was French, did not use m. In _Mathematical Circles Revisited_ (1971) mathematics historian Howard W. Eves suggests "it just happened."

Jeff Miller's web site cited above now has updated the earliest use of m for slope.

The earliest known use of m for slope appears in Vincenzo Riccati?s memoir De methodo Hermanni ad locos geometricos resolvendos, which is chapter XII of the first part of his book Vincentii Riccati Opusculorum ad res Physica, & Mathematicas pertinentium (1757):

Propositio prima. Aequationes primi gradus construere. Ut Hermanni methodo utamor, danda est aequationi hujusmodi forma y = mx + n, quod semper fieri posse certum est. (p. 151)

The reference is to the Swiss mathematician Jacob Hermann (1678-1733). This use of m was found by Dr. Sandro Caparrini of the Department of Mathematics at the University of Torino.

*Early in the development the formula det(AB) = det(A)det(B) provided a connection between matrix algebra and determinants.*[Italicized sentence added by me]Ok, did he miss the point? Or did I… I assume Vitulli meant the use of a single letter (such as A) to represent a vector was a huge leap… but; I think the give-away was calling it a noun… variables like this are more like pronouns in my mind… using x in place of the number “four” or n for “some number not named”… correct me if you disagree…

Sadly, after all that, he left out the one that is known to almost every Algebra I student, “M is for slope”. Interestingly, that one has led to more mis-history in classrooms than any other topic, with the possible exception of the life of that Great American-Indian Mathematician, Chief Soh Cah Toa.

Here is what I have found about the slope, as it appears at my MathWords page; Slope is derived from the Latin root slupan for slip. The relation seems to be to the level or ground slipping away as you go forward. The root is also the progenitor of sleeve (the arm slips into it) and, by dropping the s in front we get lubricate and lubricious (a word describing a person who is "slick", or even "slimy").

Many variations of where the idea of M for slope originated seem to be mostly myth. One of the most common is that the letter was used by Descarte because it was the first letter of some French word or another that related. In a recent post to the AP Stats discussion list, Hector Hirigoyen shared the following story:

I was told by Mary Dolciani herself, that the SMSG group "decided "to use y=mx+b because of the French (Descartes, I presume)-"montant"; I found it strange because the "logical word" would be "pente"(which is slope (and the standard term in Spanish is pendiente, which matches this). However, several years ago, while visiting a French high school, I noticed the teacher used y=sx+b. I inquired, and she said because of the "American" word "slope." I believe they are using ax+b for the most part these days.

. Here are several other clips from postings about the topic on a discussion group about math history.

In his "Earliest Uses of Symbols from Geometry" web page, ... Jeff Miller gathered the following information: Slope. The earliest known use of m for slope is an 1844 British text by Matthew O'Brien entitled _A Treatise on Plane Co-Ordinate Geometry_ [V. Frederick Rickey]. George Salmon (1819-1904), an Irish mathematician, used y = mx + b in his _A Treatise on Conic Sections_, which was published in several editions beginning in 1848. Salmon referred in several places to O'Brien's Conic Sections and it may be that he adopted O'Brien's notation.

According to Erland Gadde, in Swedish textbooks the equation is usually written as y = kx + m. He writes that the technical Swedish word for "slope" is "riktningskoefficient", which literally means "direction coefficient," and he supposes k comes from "koefficient."

According to Dick Klingens, in the Netherlands the equation is usually written as y = ax + b or px + q or mx + n. He writes that the Dutch word for "slope" is "richtingscoefficient", which also means "direction coefficient." In Austria k is used for the slope, and d for the y-intercept. In Uruguay the equation is usually written as y = ax + b or y = mx + n, and the "slope" is called "pendiente", coeficiente angular", or "parametro de direccion".

It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for "modulus of slope." One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is monter. However, there is no evidence to make any such connection. Descartes, who was French, did not use m. In _Mathematical Circles Revisited_ (1971) mathematics historian Howard W. Eves suggests "it just happened."

Jeff Miller's web site cited above now has updated the earliest use of m for slope.

The earliest known use of m for slope appears in Vincenzo Riccati?s memoir De methodo Hermanni ad locos geometricos resolvendos, which is chapter XII of the first part of his book Vincentii Riccati Opusculorum ad res Physica, & Mathematicas pertinentium (1757):

Propositio prima. Aequationes primi gradus construere. Ut Hermanni methodo utamor, danda est aequationi hujusmodi forma y = mx + n, quod semper fieri posse certum est. (p. 151)

The reference is to the Swiss mathematician Jacob Hermann (1678-1733). This use of m was found by Dr. Sandro Caparrini of the Department of Mathematics at the University of Torino.

## Monday, 3 August 2009

### The Tower of Hanoi

Back awhile, in a blog about Fibonacci, I mentioned that Edouard Lucas had created the "Tower of Hanoi" game and received comments and mail from people who thought I must be mistaken because the game was "really old". Turns out, it really isn't, but just the creation of a master mathematical story teller. Here are some notes about the man, and the history of the Towers of Hanoi from my Math Words Etymology page.

Also, you can find a java applet to play the game at this site... and if you've never done it (where HAVE you been?) don't start with all 12 discs, that takes 4095 moves to solve (see below).

The Lucas sequence is similar to the Fibonacci sequence. The Lucas sequence is given by {1, 3, 4, 7, 11, 18, ...} . Each term is the sum of the two previous numbers, as in the Fibonacci sequence. Just as in the Fibonacci sequence, the limit of the ratio of consecutive terms is the Golden Ratio. The Lucas numbers can also be constructed from the Fibonacci numbers by the function Ln = Fn-1 + Fn+1, thus the fifth Lucas number, 11, is the sum of the fourth and sixth fibonacci numbers (3+8).

The sequence is named for Edouard Lucas, a French mathematician of the later half of the nineteenth century. He used his sequence and the Fibonacci sequence to develop techniques for testing for prime numbers. Lucas is also remembered for his unusual death, caused by a waiter dropping a plate which shattered sending a piece of plate into his neck. Lucas died several days later from a deadly inflamation of the skin and subcutaneous tissue caused by streptococcus. The disease, officially listed as erysipelas (from the Greek for "red skin") was more commonly known as "Saint Anthony's Fire".

Lucas was also the creator of a popular puzzle called The Tower of Hanoi in 1883. You can see the original box cover above. Note that the author on the box cover is Professor N. Claus de Siam, an anagram of Lucas d' Amiens (his home). The professors college, Li-Sou-Stian, is also an anagram for "Lycee Saint-Louis" where Lucas worked.

France was building an Empire in Indochina (the peninsula stretching from Burma to Viet Nam and Malaysia) and the "mysterious East" was a very fashionable topic. Lucas created a legend (some say he embellished an existing one, but I can find no earlier record of one) of monks working to move 64 gold disks from one of three diamond points to another after which the world would end. The solution for a tower of n disks taks 2n -1 moves, so the game often had less than the 64 disks of the legend. Solving the 64 disks at one move a second would require 18,446,744,073,709,551,615 seconds, which at 31,536,000 seconds a year would take 584 Billion years. (and you thought Monopoly took a long time to finish). The reference in his instructions to Buddhist monks in a temple in Bernares(Varanasi

**),**India seems, even now, to make people believe there was such an activity taking place. Varanasi is considered the holiest of the seven sacred cities (Sapta Puri) in Hinduism, and Jainism, and is important to Buddhism because it was in nearby Sarnath that Buddha gave his first teaching after attaining enlightenment, in which he taught the four noble truths and the teachings associated with it. There is a Buddhist temple there with many relics of the Buddha, but so far as I can find, no monks moving golden disks on needles.

Students/teachers interested in further explorations of the history and math of the famous game should visit the work of Paul K Stockmeyer who maintains the page with the cover illustration mentioned above, and his Papers and bibliography on the Tower of Hanoi problem.

Lucas developed several other mathematical games of his on, including the well known children's pastime of dots and boxes (which he called La Pipopipette), which on large boards is still essentially unsolved, I believe. He also (probably) invented a Mancala type game called Tchuka Ruma.

Lucas is also remembered for suffering an unusual death. At a 1banquet in 1891 a waiter dropped a dining plate and one of the pieces cut Lucas on the neck and cheek. Within a week he was dead from what was called the "Holy Fire" or St Anthony's Fire, a form of septicemia.

## Saturday, 1 August 2009

### Ours Not to Reason Why, Just Flip and Multiply

I still encounter math teachers, especially in the middle grades, who are not aware that there is a fractional division approach analogous to the multiplication method commonly used for fractions. In the same way that multiplication can be thought of as "multiply top times top, bottom times bottom", division can be done by using, "divide top by top, bottom by bottom". Of course this often leads to the result of a fraction divided by a fraction again, and is not of practical use without one additional step which may have once been more common than the current "reciprocals" approach. That additional step was simply to do what was done with addition and subtraction of fractions, find a common denominator. It is the preferred or "best" method in the "Text-book of arithmetic, for the use of teachers" by John Hunter, 1847; as shown in the clip from page 50.

This is not an isolated usage as David E. Smith pointed out in his "History of Mathematics: Vol. I, General Survey of the History of Elementary Mathematics; Vol. II, Special Topics of Elementary Mathematics (1923)

In "The elements of arithmetic in theory and practice", 1903 By John William Hopkins, and Patrick Healy Underwood we see the reciprocal method used, but a historical note points out that the method is very old, probably not what would happen if it was a common part of textbooks for a long period.

An even earlier reference is given by Smith, who credits it to "both the Hindus and the Arabs" of the early Middle Ages.

A translation of "Līlāvatī of Bhāskarācārya" which was written in 1150 contains the following:

Smith adds a Historical footnote to his description of the second common method, using the cross.

This is not an isolated usage as David E. Smith pointed out in his "History of Mathematics: Vol. I, General Survey of the History of Elementary Mathematics; Vol. II, Special Topics of Elementary Mathematics (1923)

In "The elements of arithmetic in theory and practice", 1903 By John William Hopkins, and Patrick Healy Underwood we see the reciprocal method used, but a historical note points out that the method is very old, probably not what would happen if it was a common part of textbooks for a long period.

An even earlier reference is given by Smith, who credits it to "both the Hindus and the Arabs" of the early Middle Ages.

A translation of "Līlāvatī of Bhāskarācārya" which was written in 1150 contains the following:

Smith adds a Historical footnote to his description of the second common method, using the cross.

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