High School Mathematics-Physics SMILE Meeting
12 October 2004
Notes Prepared by Porter Johnson

Bill Blunk [Joliet Central HS, recently retired]           Finding Metallic Stakes to Mark Property Lines 
Bill  was seeking to find the location of iron stakes driven into the ground to mark the edges of his property line.  Some of the white plastic caps on the stakes had disappeared, and the stakes themselves had presumably been pushed into the ground, and were no longer visible.  How could he find the (underground) stakes to establish the property lines?  A surveyor suggested using a metal detector, but Bill didn't have one.  [See the website How Metal Detectors Work:  http://home.howstuffworks.com/metal-detector1.htm.] Instead, he took out his trusty Boy Scout compass, and began to paw around in the ground in the probable location of one of the stakes.  Bill reasoned that the stakes would have been magnetized slightly as they were being driven into the ground.  He scanned over a 4 meter ´ 4 meter region. He found that the compass generally pointed in the direction of the earth's magnetic field --- 15 degrees East of North and at an inclination of about 60 degrees below the vertical, for this location in Montana.  However, in one small region the compass deviated from this direction, when the needle was free to rotate in the horizontal plane, as well as in a vertical plane.  He dug down at that location, and found the stake, about 10 cm below the surface.  In a similar fashion he found another boundary marker.  Using the survey map, he readily found the approximate location of the other markers. Two of them in relatively remote locations were above the ground, with the white plastic cap still present.  Good physics makes up for the lack of equipment, once again!  Very impressive, Bill.

Porter Johnson indicated that the Northwest Territory, consisting of the present states of Ohio, Indiana, Michigan, Illinois, and Wisconsin, was well-surveyed, and laid out into townships, sections, and plots, under the Articles of Confederation in the 1780's.  This tradition of careful layout of the land was generally followed for land settled in the West.  However, such a systematic layout has never been done in the Northeast, or in the South.  The street pattern of the city of Boston was laid out along old cattle trails -- but it has remained a mystery as to why the old trails were so crooked!  In the middle Atlantic states there is an elaborate network of boroughs completely surrounded by independent townships.  Throughout the Appalachian regions,  property boundaries were often set by the lay of the land, boundaries being determined by banks of streams and  rivers, crests of hills, big trees, and the like.  In that region the property lines are frequently in dispute to this day, even in populated areas.  The subject of geometry (earth measure) was invented by the Greeks and Egyptians living in the Nile Delta to re-establish property lines after the annual flooding of the river. 

Bill next took some strips of Mu-metal (permalloy), which is often used for magnetic shielding [http://www.fact-index.com/m/mu/mu_metal.html].  When held in a direction along an external magnetic field B, these strips become slightly magnetic, because of an induced internal field.  To test this assertion, he took two of these strips, and held them in a North-South direction at an inclination of about 45 degrees below the horizontal.  He showed that opposite ends of the strips would attract one another weakly, as an indication that the fields induced in the strips were in the same direction.  It was important to keep the strips in the proper orientation (parallel to the earth's field) while showing this attraction, since the effect quickly disappears when that orientation is changed.

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.amazon.com/Big-Book-Brain-Games-Mathematics/dp/0761134662] 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
The answer is 28, which is equal to the sum of its factors; 14, 7, 4, 2, 1:
14 + 7 + 4 + 2 + 1 = 28
In Euclid's Elements, Volume 9, Proposition 36: http://www.perseus.tufts.edu/hopper/text?doc=Euc.+9.Prop.36&fromdoc=Perseus%3Atext%3A1999.01.0086 the following assertion is made:

"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 2n-1 ´ (2n -1) is a perfect number.

PJ comment: Numbers of Euclidean form are very striking when expressed in base 2 --- the first "n" numerals are "1", and the next "n-1" are "0 . The property of perfection is more or less evident from this form :-- for example:
28 ® 11100 = 1110 + 1110 = 1110 + 111 + 111 = 1110 + 111 + 100 +10 + 1 ® 14 + 7 + 4 + 2 + 1
The following numbers, all of which are of this Euclidean form, are known to be perfect:
Perfect Number  n  2n-1  2n -1
6 2 2 3
28 3 4 7
496 5 16 31
8128 7 64 127
33,550,128  13  4096   8191 
The first four numbers were discovered by the ancient Greeks, and the fifth number was discovered in 1460. The cases n = 17, and n = 19 yield perfect numbers.  In 1782 the (already blind) mathematician Leonhard Euler [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html] found the eighth perfect number for n = 31,  which is 2,305,843,008,139,952,128 -- 19 digits long!  As of 2003, 40 perfect numbers were known -- all of Euclidean form; the largest corresponding to n = 20,996,011 and has 6,320,430 decimal digits. All even perfect numbers have been shown to be of Euclidean form, and there are no odd perfect numbers, up to 10300.

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:

N = 1 + 2 + ... + (p-1) + p = p (p + 1)/2.
Note that all Euclidean numbers are triangular numbers, with p = 2n -1. For details see the website Perfect Numbers from Mathworld:  http://mathworld.wolfram.com/PerfectNumber.html.

Fascinating stuff! Thanks, Walter.

Bud Schultz [West Aurora HS  Physics]           The Lightly Story 
Bud showed us a recording of the film The Lightly Story [http://www.landmarkmedia.com/videos_Detail.asp?videokey=11], which demonstrated transmission, reflection, and shadowing of electromagnetic waves.  The transmitter, with a frequency  in the GHz range, and antenna were of the type once used for radio communication with taxis.  The announcer (with a suspiciously Australian accent) showed that a receiver antenna had been connected in series with a small light bulb. When the length of the receiver antenna was appropriate (say, half a wavelength), the bulb would light when the transmitter was turned on and the receiver antenna was held near and parallel to the transmitter antenna.  By placing an identical antenna between it and the transmitter antenna, it was shown that the bulb did not light.  The intermediate antenna had produced a shadow, blocking the signal from the receiver antenna.  Furthermore, he showed that one antenna would reflect a signal to another antenna.  These phenomena of shadowing (lenses) and reflection (mirrors) are basic to all wave phenomena. But, where do we get such a transmitter?

Bud also showed us a rather intense Green Light Laser, which may be obtained for about $130 (including two batteries) at the Z-Bolt website at http://www.z-bolt.com/home.htm.  This laser has a rated intensity of  4.99 mW -- just under the 5.00 mW level at which registration with the state of Illinois is required. Very interesting, Bud!

Bill Shanks [Joliet Central HS,  happily retired]           Sale on Volt - Ohm Meters at Harbor Freight
Bill called attention to a sale on VOMs for about $2.00 each, at local franchise stores of Harbor Freight Tools [http://www.harborfreight.com/] in Arlington Heights IL, (940 W Dundee Rd);  phone: 847 - 392-1400 and West Aurora IL (904 N Lake St in Westgate Mall); phone: 630 - 966-9008.
Thanks for the heads-up, Bill!

Don Kanner [Lane Tech HS,  Physics]           In-Service Meeting for Mathematics and Science Teachers 
In a weak moment Don agreed  to direct an in-service meeting of teachers in the region at Lane Tech HS on (?Saturday? 29 January 2005 -- or "whenever").  He asked our group for help in assembling written materials for reproduction (before 05 Nov 2004!), as well as suggestions as to what Physics teachers would like their students to know.  We came up with the following suggestions, in short order:

Good luck on your courageous undertaking, Don! 

Notes prepared by Porter Johnson