14 March 2001 Bill Colson (Morgan Park HS, Math)
noted that tomorrow is p Day [3.14 --- get it?]. As an multicultural mnemonic to remembering the digits of p, he presented the following phrases, in which the number of letters in each word form the sequence

10 April 2001 Porter Johnson (IIT)
mentioned the golden rectangle ratio

( 1 + Ö5)/2 = 1.61803... 
GOLDEN RECTANGLE

This number arises out of the definition of a golden rectangle; specifically, that the ratio of its height h {short side) to breadth b (long side) is the same as the ratio of its breadth b to the sum of its height and breadth (h + b):

or

If we define the "golden ratio" x as the long side b divided by the short side h;  i.e.  x = b / h, this equation may be written as

This quadratic equation has two solutions, one positive and one negative. The positive solution is

This golden ratio can also be understood as the limit of ratios of successive number pairs in the Fibonacci Sequence:

In particular, note that 89 / 55 = 1.6181818 ... is fairly close to the limit.  The sequence is generated from the first two entries y₁ = 1 and y₂ = 2 by taking the sum of the two previous elements:

y_n+1 = y_n+ y_n-1 .

Let us assume that the ratio y_n+1/ y_n approaches a limiting value, x, at very large n; i.e. y_n+1/y_n ® x and y_n /y_n-1 ® x.

The iteration formula

y_n+1 = y_n+ y_n-1 .

is equivalent to

At very large n, the ratios may be replaced by their limiting values to obtain this equation for the limit:

Thus the golden mean is the limit of the Fibonacci Sequence, independently of the starting seeds y₁ and y₂. 

One may express any real number uniquely through its continued fraction expansion [http://archives.math.utk.edu/articles/atuyl/confrac/]:

where the  coefficients a; b, c, d, e, ¼ are positive integers.  If the number A is rational, the continued fraction expansion will terminate; otherwise it will go on forever.  We may identify the number with its continued fraction:  A = (a; b, c, d, e, ¼ ).  For the golden mean the continued fraction has the simplest form, in that the coefficients a; b, c, d, e, ¼ are all equal to 1. That is,

The golden mean is related to Penrose Tilings; see the website http://mathworld.wolfram.com/PenroseTiles.html.  By terminating this continued fraction after various steps we recover the ratios of Fibonacci numbers,

The continued fraction for e, the base of the Natural Logarithms, is relatively simple [see http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#othercfsE]

e = (2; 1, 2 ,1, 1, 4, 1, 1 ,6 ,1 ,1, 8, 1, 1, 10, 1, ... )

On the other hand, the continued fraction expansion of p is less elegant looking:
p =

(3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, 45, 1, 22, 1, 2, 2, 1, 4, 1, 2, ... )

The rational approximations are 3, 22/7, 333/106, 355/113 = 3.14159292, ... . The last approximation is rather accurate, because the next number in the continued fraction, 292, is rather large. *** History of Mathematics ***

07 May 2002: Hoi Huynh (Clemente HS) Division with Mirrors [*** but without smoke!]
Hoi taped together two big mirror tiles [30 cm ´ 30 cm] along the edges, and stood them on the table.  The tape acted like a hinge, so that the angle between the mirror faces could be changed as one wished.  By placing a ubiquitous cola can between the mirrors and varying the opening angle between the mirrors, we could produce various patterns of cola can reflections.  In particular, we noticed that at certain angles the reflected images of the cola can formed regular patterns.   Specifically, when the angle between the mirrors was 360º/N, these images lay at the vertices of a regular polygon with N sides.  The cases N = 3, 4, 5, 6, and 7 are shown below:

Hoi described what happened as the opening angle  q = 360º/N becomes small, as N becomes large.  In the limit as the opening angle q goes to zero, the number of images, N, becomes infinite. It must be so!  Incidentally, the choice of 360º for one revolution was made by the Babylonians; see http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_mathematics.html.  The website http://mathforum.org/library/drmath/sets/select/dm_circle360.html contains several explainations while another contains the following information:

"Why, when, where and by whom was it decreed that there should be 360 degrees of arc in a complete circle?

It was during the reign of Nebuchadnezzar (605-562 BC) in the Chaldean dynasty in Babylon that the circle was divided into 360 degrees. This was because the Chaldeans had calculated by observation and inference that a complete year numbered 360 days. The basis of angular measure for the mathematicians of Babylon was the angle at each of the corners of an equilateral triangle. They did not have decimal fractions and thus found it difficult to deal with remainders when doing division. So they agreed to divide the corner of an equilateral triangle into 60 degrees, because 60 could be divided by 2, 3, 4, 5 and 6 without remainder. Each degree was divided into 60 minutes and each minute into 60 seconds. If the angles at the corners of six equilateral triangles are placed together they form the angle formed by a complete circle. It is for this reason that there are six times 60 degrees of arc in the complete circle."

We then discussed the operation of reflecting mirrors for drivers of motor vehicles, which usually have an "anti-glare" setting, in addition to the regular one.  These mirrors are actually wedge-shaped, and most of the reflection occurs from the silvered "back surface" of the mirror.  However, a small amount of reflection occurs from the front surface, which is slightly tilted relative to the back surface.  When a bright light falls on the mirror, the driver may switch to the "anti-glare" front surface reflection, in order to avoid being distracted by the light.  Of course, the intensity of all images is reduced in this process, and the driver should switch back to the regular setting when the bright light is removed, to maintain visibility.

06 May 2003
Porter Johnson found some interesting entries on the Islam and Islamic History and the Middle East website:  (1) Arabic Numerals http://www.islamicity.org/mosque/ihame/Ref6.htm and (2) Arabic Writing and Calligraphy:  http://www.islamicity.org/mosque/ihame/Ref3.htm.

In addition, Ishaque Khan [IIT Chemistry] has called attention to the stamp Eid Greetings, which was released about 2 years ago by the US Postal Service.   The words EID MUBARAK [Blessed Festival:  Eid-al-Fitr at end of Ramadan or Eid al-Adha at the end of the hajj] are written in continuous cursive Arabic calligraphy. For details see the website http://www.usps.com/news/2001/philatelic/sr01_054.htm.

23 September 2003: Walter McDonald  [CPS Substitute and VA Hospital Technician]        Fibonacci Numbers
Walter began by writing some of the Fibonacci Numbers on the board:

f₂ = f₁ + f₀ = 1 + 0 = 1
f₃ = f₂ + f₁ = 1 + 1 = 2
f₄ = f₃ + f₂ = 2 + 1 = 3
f₅ = f₄ + f₃ = 3 + 2 = 5

Porter Johnson pointed out that the the number x, the limit of the ratio of adjacent Fibonacci numbers, is the "simplest" non-rational number --- all "1's" in its continued fraction representation.. For more information as well as additional references, see his comments at  the Math-Physics SMILE meeting of 10 April 2001: mp041001.htm.

Biographical information about Leonardo Pisano (sobriquet "Fibonacci") [1170-1250??] can be found at the ST Andrews University History of Mathematics website [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html], from which the following is excerpted:

"Fibonacci was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father's job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia. ... Fibonacci was taught mathematics in Bugia and travelled widely with his father, recognising ... the enormous advantages of the mathematical systems used in the countries they visited."

Fascinating topic.  Good job, Walter!

07 October 2003: Porter Johnson touted the book The Golden Ratio and Fibonacci Numbers {Springer Verlag 1997: ISBN 981-02-3264-D).  In this invaluable book, the basic mathematical properties of the golden ratio and its occurrence in the dimensions of two- and three-dimensional figures with fivefold symmetry are discussed.  In addition, the generation of the Fibonacci series and generalized Fibonacci series and their relation to the golden ratio are presented.

02 December 2003: Walter McDonald  [CPS Substitute Teacher; X-ray technician -- Veterans Administration Hospital, North Chicago]       Fibonacci Number Programs on HP 48G Programmable Calculator
Walter passed around information on two programs to calculate the Fibonacci numbers on the calculator, called FIB1 and FIB2.  The listings and other information from these programs (presumably) came from the HP 48G Series Advanced User's Reference Manual:  http://www.hpcalc.org/hp48/docs/books/aur.html. This is an extension of Walter's explanation of Fibonacci numbers in the Math-Phys SMILE meeting on 23 September 2003 mp092303.html].  As explained there, the sequence of  Fibonacci numbers, F(n), is generated by iteration, starting from the seeds F(0)=0 and F(1) = 1, using the recursive formula F(n) = F(n-1) + F(n-2).  In particular, F(6) = 8, whereas F(10) = 55 and F(13) = 233.  The first program, FIB1 (Recursive Version), calculated and stored each Fibonacci number in an array, and then went on to the next one.  It took about 25.17 econds to obtain F(13).  By contrast, the second program, FIB2 (Loop Version), stored only the last two Fibonacci numbers, and took just 0.0897 seconds to calculate F(13). This remarkable time difference was shown in a third program, FIBT (Comparing Program-Execution Time) which determines the execution time for each of the two programs.

Why are the execution times so dramatically different?  According to the handout, the Program FIB1 calculates intermediate values F(i) more than once, while Program FIB2 calculates each intermediate value F(i) only once.  Thus, the time required to calculate F(n) grows exponentially with n in FIB1, and linearly with n in FIB2.  FIB2 is obviously the "way to go", whereas FIB1 bogs down, even with a relatively fast CPU [central processing unit].

Porter Johnson pointed out that most currently available programmable computers are at least as sophisticated as the first commercially available mainframe computers, e.g. IBM 650. This means that (1) they can perform absolutely marvelous tasks for you if you know how to program them but (2) when a particular program crashes, you generally have to reboot the system before it will run again.  Note:  Programs that crash the computer will cause it to end up in computer limbo land, producing meaningless results!  In fact, no hand-held calculator is ever of any value, until and unless you know how to use it, and use it properly!  The more sophisticated the calculator, the easier it is to crash it and get nonsensical answers!

Very interesting, Walter!

  23 March 2004: Bill Colson [Morgan Park, Mathematics}         Euler's Disk
Bill showed us a carefully designed coin-shaped disk.  Known as Euler's Disk, it is about 1 cm thick and 8 cm in diameter, with one side mirror-like and the other side with a bright and colorful iridescent pattern.  He set it spinning on a smooth glass surface in the shape of a round, slightly concave mirror about 25 cm in diameter.  It seems like it would never stop, only gradually losing energy because of friction and vibration.  As time goes by, it begins to "hum" with a progressively higher pitch and louder sound.  Lee Slick reflects a laser beam off the disk and onto the ceiling.  We can follow the motion of the disk more easily by watching that reflection, a circle that gradually decreases in diameter as the disk loses energy. Eventually, it simply stops, abruptly -- at which point the small circle n the ceiling condenses to a small dot.  Amazing, but why does it happen?

For additional details see The Official Euler's Disk website:  http://www.eulersdisk.com/.  In particular, for information on the physics of this object see the page http://www.eulersdisk.com/physics.html, from which the following is excerpted:

"When Euler's Disk is spun, the disk contains both potential and kinetic energy. The potential energy is given to the disk when it is placed upright on its side. The kinetic energy is given to the disk when it is spun on the mirrored base. Euler's Disk would spoll (i.e., spin and roll) forever if it were not for friction and vibration. ... "

For insights gained by Richard Feynman concerning a plate wobbling on a table, see the following excerpt from his book Surely You're Joking, Mr Feynman:  http://www.wwnorton.com/catalog/spring97/surely.htm.

Euler's Disk was named in honor of the famous mathematician Leonhard Euler (1707-1873) [ http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html], who maintained a life-long interest in puzzles.  His puzzle concerning the crossing of the 7 bridges over the river Pregel in the city of Konigsberg [ http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Konigsberg.html] represented the beginning of modern topology.  Euler's beautiful equation e ^ip + 1 = 0 connects the five most important numbers in mathematics --- two of which were invented by Euler himself (i and e).  For additional details see Tour of the Calculus by David Berlinski [ISBN 0-679-74788-5].  

An anonymous student once said "infinity is where things happen that don't".

Thanks for these delightful math and physics insights, Bill!

04 May 2004: Don Kanner [Lane Tech HS, physics]           Funnel Multiplication
Don recently learned of a novel method of multiplication, which he illustrated with an example such as 4567 ´ 9876 = 45103692.  He wrote the numbers directly  below one another, and performed the following manipulations:

numbers arranged in                     4 | 5 | 6 | 7
four columns of two                 ´     9 | 8 | 7 | 6
                                     --------=---------
product of numbers in same column     |3 6|4 0|4 2|4 2|
sum of diag products; adjacent cols:     7 7|8 3|8 5|
sum of diag products; next col pairs      |8 3|8 6|
sum of diag products; 1st and last cols:    |8 7|
(now, add everything up)              --------------------
 (et, voilá!)                         |4 5|1 0|3 6|9 2|

For discussion of an ancient description of this algorithm for multiplication, see the Iowa State University Department of Mathematics webpage Math Night Module:  Multiple Methods of Multiplication [ http://www.math.iastate.edu/mathnight/activities/modules/multiply/aboutmod.shtml] from which the following has been excerpted:

"The history of mathematics in India is ancient. The Hindu Vedic tradition is an oral tradition of knowledge passed down in short verses, dating to before the invention of paper. The Vedas encompass a broad spectrum of knowledge, including the sutras (verses) pertaining to mathematics. In the early 20th century Swami Shri Bharati Krishna Tirthaji Maharaja claimed to have rediscovered a collection of 16 ancient mathematical sutras from the Vedas and published it in a book called Vedic Mathematics. Historians do not agree on whether or not these were truly part of the Vedic tradition. If these sutras date back to the Vedic era they were certainly part of an oral rather than a written tradition. However, they are a novel and useful approach to computation: they are flexible in application and easy to remember. They can often be applied in algebraic contexts as well as in simple arithmetic. 'Vertically and Crosswise' [sic: URDHVATIRYAGBHYAM] bridges the gap between arithmetic and algebra: the algorithm is very similar to the standard algorithm used in the US and also is similar to the "FOIL" (first, outer, inner, last) rule used for multiplying binomials in algebra."

Thanks, Don!

12 October 2004: Walter McDonald [CPS Substitute Teacher]           Perfect Numbers and Triangular Numbers 
Walter  passed around Playthink 528, taken from the reference[1000 PLAYTHINKS  Games of Science, Art, & Mathematics by Ivan Moscovich; Workman Publishing  [http://www.workman.com/] 2002;  ISBN: 0-7611-1826-8 ], from which the following information on PERFECT NUMBERS has been excerpted:

"A perfect number is the sum of all the factors that divide evenly into it -- including 1, but excluding the number itself. The first perfect number is 6, which is divisible by 3, 2, and 1, and is the sum of 1, 2, 3. So far; thirty-eight perfect numbers have been found. Can you work out what the second perfect number is?" 1 + 2 + 3 = 6

"If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect."
... or ...
If, for a positive integer n, the number 2ⁿ -1 is a prime, then 2^n-1 ´ (2ⁿ -1) is a perfect number.

As pointed out by Bill Shanks,  a triangular number is a number N that can be written as the sum of integers from 1 through p:

Fascinating stuff! Thanks, Walter.

26 October 2004: Don Kanner [Lane Tech HS, physics]           A Book Review 
Don  touted the book A History of Mathematical Notations by Florian Cajori [Dover Publication 1994], ISBN 0-486-67766-4.   Don cited as an example that Cajori described the origins of the signs + and  - to represent addition and subtraction, respectively.  They were developed in Germany during the last twenty years of the fifteenth century.  The + sign first occurs in print in a book by Johann Widman (1489).  [See also Widman's biography: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Widman.html.  It is considered by experts to be a misprint for the symbol vñ, representing vnnd or "und"-- "and" in modern German. The - sign first occurs in print in a book by H Brugsch, and it is given the name minnes.  For a biography of Florian Cajori see the website http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cajori.html.

Thanks, Don.

29 March 2005: Don Kanner [Lane Tech HS, physics]              Signs of the times + p a la mode
Don described the history of the multiplication symbol ´, as described in the references History of Mathematics and History of Mathematical Notation by Florian Cajori.  The original representation of multiplication of one and two digit decimals was written as follows

            3                        3   7                  7   8  
            |                        | X |                  | X |
            3                        3   7                  5   6 
          -----        ...         ---------       ...   ----------
            9                          4 9                    4 8 
                                     2 1                    4 2 
                                     2 1                    4 0
                                     9                    3 5
                                   ---------              --------- 
                                   1 3 6 9                4 3 6 8

              7 8  
               X
              5 6
            ------
               48
              42
              40
             35
            ------ 
             4368

                3     2     3 ° 5     15
               --- X --- =  -----  =  --
                7     5     7 ° 2     14

"The European Renaissance brought about in due course a whole new mathematical world. Among the first effects of this reawakening was the emergence of mathematical formulae for p. One of the earliest was that of Wallis (1616-1703)

2/p = (1_°3_°3_°5_°5_°7_° ...)/(2_°2_°4_°4_°6_°6_° ...)

and one of the best-known is

p/₄ = 1 - ¹/₃ + ¹/₅ - ¹/₇ + ....

This formula is sometimes attributed to Leibniz (1646-1716) but it seems to have been first discovered by James Gregory (1638- 1675)."

Don also mentioned the formula used by Francois Vièta (1540-1603), for which the following information appears on the ST Andrews website:

"Viète used Archimedes method with polygons of 6 ´ 2¹⁶ = 393216 sides to obtain 3.1415926535 < p < 3.1415926537. He is also famed as the first to find an infinite series for p."

Don is engaged in an investigation to see which of the expressions converges most rapidly to p. Place your bets now!

04 October 2005: Fred Schaal (Lane Tech HS, mathematics)             Dubito Ergo Sum
A student recently asked Fred this question:  "Who invented graph paper?" Fred 'Googled' this question; that is, he used the Google website:  http://www.google.com. The answer came back as René Descartes, the inventor of Cartesian coordinates; see the biographical entry on the ST Andrews University (UK) website: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html. See also Descartes' Geometric Solution to a Quadratic equation: http://www-groups.dcs.st-and.ac.uk/~history/Diagrams/DescartesSoln.gif. The most famous assertion of Descartes is "Cogito ergo sum" [I think; therefore I am], but he may actually have meant "I doubt; therefore I am".

Fred also asked the age of The Naughty Lady from Shady Lane:  http://www.theguitarguy.com/naughtyl.htm.  Aficionados of songs from the 50's will no doubt recall that, in the 1954 song, "She's only nine days old!". Fred has started to have a regular Google Session in his math classes, to address such issues.  Thanks, Fred