High School Mathematics-Physics SMILE Meeting
08 March 2005
Notes Prepared by Porter Johnson
Information
• Earl Zwicker called attention to the article Quiet at the End on the American Physical Society webpage:  http://focus.aps.org/story/v15/st8. This article, which refers to a paper recently published in Physical Review Letters, contains the following statement:
"They say in space no one can hear you scream, but if you are falling into a black hole, you can't even hear yourself.  According to the so-called asymptotic silence hypothesis, spacetime becomes so contorted that signals can't travel any significant distance -- not even from your mouth to your ear. ..."

In addition, Earl mentioned the article Motoring Oil Drops, which also appeared on the American Physical Society webpage:   http://focus.aps.org/story/v15/st7.

• Earl Zwicker also announced the next meeting of Physics Northwest, at Cary-Grove High School on Thursday, 10 March 2005.  For details see their website:  https://www.edline.net/pages/Physics_Northwest.
• Finally, Earl showed off his new Motorola V-265 cell phone, with a 911/GPS locator and a voice recognition system. For details see https://www.motorola.com/moto_care/manuals/V_Series/V265_UG.pdfAren't these cell phones remarkable?

Fred Schaal [Lane Tech HS, mathematics]              RANDINT(1,14,4)+50
Fred
used the Pseudo-random Number Generator RANDINT, which is programmed into the TI-83 calculator. He generated a random, equally-distributed set of 200 integers from 1 through 14, and obtained the following number-of-occurrences of the generated numbers:

 Generated Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Number-of-occurrences 13 12 13 18 17 14 20 15 13 13 17 10 13 12
Does this appear to be a "random" set of numbers? The answer is "Yes", despite the fact that the number-of-occurrences ranges between 10 and 20. On statistical grounds, we would expect the average number of occurrences to be about 200/14 = 14.3, with a spread (standard deviation) of Ö14.3 = 3.8.  Thus, about 2/3 of the number-of-occurrences should lie between 11 and 17. That is consistent with the spread in the data. Curiously, only the number "6" occurs exactly 14 times.

Porter Johnson mentioned that "everybody knows" that it is unlikely for a randomly flipped coin to come up H (Heads) ten times in a row. However, not everybody realizes that the alternating sequence H T H T H T H T H T is equally unlikely. Furthermore, it is quite unlikely that in 1000 coin flips, Heads will occur exactly 500 times.

For a general discussion of Pseudorandom Number Generators see the Wikipedia webpage: http://en.wikipedia.org/wiki/Pseudorandom_number_generator.

Fred also brought in his metal candy box, for which we had taken exterior measurements last time mp022205.html to determine a volume of about 910 cm3.  We took a graduated cylinder filled with water, from which we were able to pour about 800 cm3 of water before the box became full. Our estimated volume was too large by over 10%. Why?

Fred also pointed out that the planet Mercury would be visible next to the New Moon just after sunset in the next few days. Thanks for the ideas, Fred!

Bill Colson [Morgan Park HS, mathematics]              Poetry
Bill
read us these two poems that were used in connection with a No Talent Show at a weekend school event:

Enjoyable, Bill!

Bill Shanks [retired, Joliet New Lenox environs]              Geometry
Bill
reminded us of the Pythagorean Theorem for a right triangle of sides (a, b, c = b + d), where c is the hypotenuse:

`                           /|               c = b + d  / |  b                         /  |                        /___|                          a `
c2 = a2 + b2
(b + d)2 = a2 + b2
2 b d + d2 = a2
2 b d = a2 - d2
b =  (a2 - d2 ) / (2d)
Bill pointed out, when a and b are chosen so that b is an integer, we obtain a right triangle with integer sides.  Bill found that Interesting Right Triangles were obtained for the cases d = 1, 2, 8, 9, 18, 25, 49, 50, ... .  These numbers are all of the form  d = 2p nq, where n is odd, and p is either zero or an odd number. For example, with  d = 1, we get b = (a2 -1) / 2, and a must be an odd number for b to be an integer.  We thus obtain these right triangles with integer sides:
 a b c=b+1 3 4 5 5 12 13 7 24 25 9 40 41 11 60 61 13 84 85 25 312 313 35 612 613 999 499000 499001
For d =2, we obtain  b = (a/2)2 -1, so that a must be an even number.  We thus obtain the table
 a b c=b+2 4 3 5 6 8 10 8 15 17 10 24 26 12 35 37 100 2499 2501 1000 249999 250001
For d = 8, we obtain b = (a/4)2 -4, so that a must be divisible by 4. We obtain these triangles:
 a b c=b+8 12 5 13 16 12 20 20 21 29 24 32 40 28 45 53 36 77 85 888 49280 48288
For d = 9, we obtain b = [ (a/3)2 - 9 ] / 2, so that a must be odd and divisible by 3. We obtain these triangles:
 a b c=b+9 15 8 17 21 20 41 27 36 45 33 56 65 39 80 89 111 680 6893 999 55440 55449
Bill finds this useful in making up exam problems, when he wants all the sides of the right triangle to be integers.
Puns du Jour: Word Origins
• Geometry (Earl Z): A little acorn grew eventually into a big oak. Then he looked out and remarked: "Gee, Ah'm a tree".
• Hypotenuse (Bill S): Poorly spelled sign on bathroom door, which should have read: "Hi; pot in use".
Fred Schaal mentioned Nearly Isosceles Integer Right Triangles, for which the the sides of the triangle are  (I, I+1, J), where I and J are integers.  Here are the first six examples of this infinite set:
 I I + 1 J 3 4 5 20 21 29 119 120 169 696 697 985 4059 4060 5741 23660 23661 33461

Note that the scale of subsequent triangles increases by a factor of about 6 each time.  For discussion see the solution to Problem #152 on this website:  http://www.dansmath.com/probofwk/probar16.html#anchor585356. For a general solution and a relation of these triangles to the Pell Equation, see the following PDF file:  http://hometown.aol.com/jpr2718/pell.pdf.

Thanks for the ideas, Bill!

Ann Brandon [Joliet West, physics]              Magnet Experiments
Ann
has a homebound student with MS this semester, so she developed a number of "take-home" experimental setups involving magnetism. Ann began by putting a bar magnet on the desk, and covered it with a large sheet of paper.  She place a small "toy" compass on the paper near the location of the magnet, and drew a short line on the paper to mark the direction of the compass needle (direction of the magnetic field).  Ann moved the compass in the direction of the field, marking the direction at each new location, and repeated the process several times.  She then connected the marks with a solid line, thus graphing a line of force of the magnetic field.  By starting at various positions, Ann traced out several lines of force for the magnetic field.  Next, Ann sprinkled iron filings out of a "salt shaker" and onto the paper.  The iron filings aligned along the lines of force that she had previously drawn. As the amount of fillings increased, the lines of force became three-dimensional.  Very interesting visual patterns!

Ann then placed two Ring Magnets on a wooden rod.  When the North (or South) poles of the magnets were adjacent to one another, the magnets repelled each another.  However, when the magnets had opposite poles adjacent, they attracted each other.  Ann then put several magnets on the wooden rod with the unlike poles adjacent, and held the rod vertically up. The magnets bobbed up and down slightly, coming to equilibrium positions.  Interestingly, the spacing between the magnets was greatest at the top, and least at the bottom. (Why?) Neato!

Ann pointed out that it is often possible to get End Rolls of Newsprint from local newspaper offices: e.g.

South Holland/Dolton Star - 6901 West 159th Street, Tinley Park, IL, 60477-1602
Phone: 708-802-8800 Fax: 708-802-8088
http://www.starnewspapers.com.
End rolls may or may not contain a lot of paper, depending upon what's left over after printing. The printers office will often give them gratis to school teachers, or else at minimal cost. Thanks, Ann!

Leticia Rodriguez [Peck Elementary School]              Ceragem Thermal Acupuncture Massager
Leticia
has been using a thermal massage bed, which is described on the Ceragem website:  http://www.ceragem.com/. The following is excerpted from that website:

What is Ceragem?
CERAGEM
is a thermal massager that helps soothe body aches and pains associated with daily stress, pressure, and bad posture. It combines the benefits of alternative medicine derived from traditional Eastern medicine with advanced technology to provide the most effective healing and relaxation. Ceragem is easy to use and highly effective, as proven by the positive feedbacks we received from our customers. Ceragem is available for free trial at our distribution centers.

The device consists of a bed with rollers for spinal alignment, with an IR light source to stimulate circulation.  Leticia asked that we each go to test this device (six times, without cost) to assess its effects on our health and happiness, at the following location:

5756 West Belmont Avenue, Chicago IL
(773) 205-1020

Notes prepared by Porter Johnson