High School Math Physics SMILE Meeting
10 April 2001
Notes Prepared by Porter Johnson
Math/Physics

Earnest Garrison (Jones Academic Magnet HS)
showed some Discovery Kits on electromagnetism that he had obtained from the source:

Science Kit & Boreal Laboratories
P.O. Box 5003
Tonawanda, NY 14151-5003
http://sciencekit.com/  [order a free CD-ROM catalogue]
Phone: 1-800-828-7777
Fax: 1-800-828-3299
He passed around booklet #65305-05, Electric Motor, written by Dr Lawrence Lowery, University of California, Berkeley, which described several experiments. Then Earnest showed us the transparent box with a tunnel passing through it, containing iron filings suspended in mineral oil, for showing the magnetic field in 3 dimensions when a magnet is inserted into the tunnel. Then he showed us a simple kit for making an electric motor, which simplified the process of winding the wire and putting the loop into the armature.

Fred Schaal (Lane Tech HS, Math)

• He found rusty metal objects about 20 cm long on the streets.  It was conjectured that these objects were bristles from street sweeping vehicles.
• He reported that the glove data [handedness / chirality of lost gloves] were pretty even-handed so far.

Walter McDonald (CPS Substitute and Veterans Administration Diagnostic Radiation Technologist)
showed us this graph of the trigonometric functions [sine, cosine, tangent, cotangent, secant and cosecant] which was obtained from the Microsoft Encarta Encyclopedia.

Walter made the following points:

• The sine and cosine are periodic with period 360o or 2p radians.
• The tangent is periodic with periodic with period 180o or p radians.
• The tangent has an asymptote at ± p/2 radians.

Fred Farnell (Lane Tech HS, Physics)  What happens when waves meet?
put some paper cups on the floor and stretched a slinky™ across the floor, which was held at its ends by two assistants.  An assistant, by rapidly moving his end of the spring back and forth once {transverse to the direction of the stretched string), sends a transverse wave pulse toward the other end. When the other end was held fixed, the wave was seen to reverse its orientation and direction after reflection at the fixed end.  The cups were put parallel to the stretched slinky on both sides of it, and the goal was to set up waves that would knock down all of the cups.  This was seen to be difficult, if not impossible.  Then, the assistants set up waves coming in simultaneously from each end, so that we could see the slinky before, during, and after the overlapping intersection of the wave pulses in the middle.  Very interesting, Fred!

Bill Colson (Morgan Park HS, Math)
showed a workbook, Stretching Your Math Students' Achievement, Motivation, and Involvement: Grades 7 - 12 Resource Handbook by Irv Lubliner, recently obtained from the following source:

Bureau of Education and Research
915 118th Avenue, SE
PO Box 96088
Bellevue WA 98009
http://www.ber.org/
Tel: 1 - 800 - 735-3503
First he showed a clear exposition from that book of the rope trick (topological puzzle) that has been shown several times in SMILE: [ph101398.htm].

Then he showed us how to play a game called MAXIT, illustrating the point with a 4 ´ 4 square lattice. He put an ´ into one location, and had us to call out numbers between -10 and +10 for the other locations, with the result as shown:

 5 8 6 2 3 ´ -7 6 -1 -3 0 1 4 -9 9 9

People on one side of the room were the UP'S AND DOWN'S, whereas those on the other side were the RIGHT'S AND LEFT'S. After flipping a coin to see who moves first, the winner was allowed to move the ´ --- either up or down, or right or left, respectively, to another location. The number in that location is replaced by an ´, and they get the number of points corresponding to that number, and you cannot move into the location of an ´. The last team to be able to move ends the game---and the team with the largest point total wins.

The next game involved the creation of a magic square, such as the following one:

 14 20 3 12 7 32 38 21 30 25 19 25 8 17 17 11 17 0 9 4 17 23 6 15 16

The numbers in this table may look unrelated, but they have not been randomly chosen, because if you pick five numbers, each from a different row and a different column, and take the sum, you will get the total 79. For instance, the five numbers shown in bold give 14 + 25 + 25 + 0 + 15 = 79.  We have, in fact, generated 120 different combinations of numbers adding up to the total of 79Isn't that remarkable? Surprisingly, there is nothing unique about the number 79, and you can see how the table was made by adding another row and another column to it:

 ** 11 17 0 9 4 3 14 20 3 12 7 21 32 38 21 30 25 8 19 25 8 17 17 0 11 17 0 9 4 6 17 23 6 15 16

The inner numbers are generated by taking the sum of the corresponding numbers in the first row and first column. e.g. 21 + 17 = 38. The "magic number" 79---which is merely the sum of the ten numbers in the first row and the first column---can by changed by changing those 10 "seed numbers".

Don Kanner (Lane Tech HS, Physics)
announced that he was traveling to the Bay of Fundy this summer to witness the high tides for himself. There was a question of the meaning of the term tidal bore, a steep-fronted wave caused by the meeting of two tides or by the constriction of a spring tide as it passes up an estuary. The following image is from http://octopus.gma.org/undersea_landscapes/Bay_of_Fundy/

Here is a description from a Nova Scotia website, (http://fox.nstn.ca/~raftcamp/)
WHAT IS A TIDAL BORE?

"This is a natural phenomenon seen in very few parts of the world. The Bay of Fundy is particularly noted for its extremely high tides, the highest in the world, and for its tidal bores. In the funnel-shaped Bay of Fundy which is 48 miles wide at its mouth and narrows down along its entire length, the tide water enters the bay at its widest point. As it passes along toward the head of the bay it is, in effect, squeezed by the ever-narrowing sides and by the constant “shallowing” of the bottom....see a river change its direction of flow before your very eyes!  At the TIDAL BORE RAFTING PARK this advancing tide becomes a wave, varying from just a ripple up to 10 feet in height. This wave is referred to as a “TIDAL BORE”. Nowhere else in the world can a tidal bore of this magnitude be seen. This is where the bore rolls in, in its fullest ferocity, followed by 3 to 10 foot rapids. Thus, we have the phenomenon of a river changing its flow before your very eyes, created by the tidal wave, or bore, flowing in OVER the outgoing water. The size or height of the tidal bore varies according to the phases of the moon. Highest tidal bore occur around the full and new moons."

Don also mentioned that he had notified the publisher of a physics text of some errors in the drawings in the book.  Specifically, he mentioned a "strobe timed" drawing of a cannonball just shot from the cannon:

`____________            |   BOOM !!!   00 0  0   0    0     0      0       0        0 ____________|`

The diagram looks suspiciously like one for uniformly accelerated motion, which should not occur after the cannon ball leaves the cannon.

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):
h ¸ b = b ¸ (b + h)

or

h ´ (b + h) = b2

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

x2 = x + 1          or            x2 - x - 1 = 0

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

x = ( 1 + Ö5)/2

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

1 ¼ 2 ¼ 3 ¼ 5 ¼ 8 ¼ 13 ¼ 21 ¼34 ¼ 55 ¼ 89 ¼

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:

yn+1 = yn+ yn-1 .

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

The iteration formula

yn+1 = yn+ yn-1 .

is equivalent to

yn+1 / yn-1 = (yn+1 / yn) ( yn / yn-1 ) = (yn/yn-1) + 1

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

x2 = x + 1

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://www.cut-the-knot.org/do_you_know/fraction.shtml]:

A = a + 1 / (b + 1 / (c + 1 /( d + 1 / (e + ¼ ) ) ) )

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,

x = ( 1 + Ö5) / 2 = 1 + 1 / (1 + 1 / (1 + 1 /(1 + 1 / (1 +  ¼ )  )  )  ) =  (1; 1, 1, 1, 1, ¼ )

The golden mean is related to Penrose Tilings; see the website http://britton.disted.camosun.bc.ca/goldengeom/goldenpenrose.html.  By terminating this continued fraction after various steps we recover the ratios of Fibonacci numbers,

2 / 1 ¼ 3 / 2 ¼ 5 / 3 ¼ 8 / 5 ¼ 13 / 8 ¼21 / 13 ¼

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.

Notes taken by Porter Johnson.