High School SMILE Meeting
13 March 2001
Notes Prepared by Porter Johnson
Math/Physics

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

3 . 1 4 1 5 9 2 6 5 3 5 ...
 French Que j'aime a faire connaître ce nombre utile aux sages, immortel Archimède illustre inventeur quie de ton jugement peut priser la valeur. Pour moi ton problème est de pariels avantages ... Spanish Sol y Luna y Mundo proclaman al Eterno Autor del Cosmo English See, I have a rhyme assisting my feeble brain, its tasks oft-times resisting.
PJ Comment: Note that these messages constitute a code that loses meaning in translation. The French statement describes remembering the value of p, whereas the meaning of the Spanish phrase is the following:  Sun and Moon and World acclaim the eternal author of Cosmos.  The code is, of course, broken in virtually any translation. Many theologians and clerics argue that religious texts such as the Bible [http://www.biblediscoveries.com/129-News/Latest/42-the-original-bible-now-available.html], the Koran / Qur'an [http://www.usc.edu/dept/MSA/quran/], and the Baghavad-Gita [http://www.bhagavad-gita.org/]can be understood properly only in their original languages.

Bill also handed out copies of an article Test Yourself [SAT I questions from recent tests designed to measure verbal and math skills] from the 12 March 2001 issue of Time Magazine:[ http://www.time.com/time/education/article/0,8599,101063,00.html].

Fred Schaal (Lane Tech Park HS, Math)
considered the algebra problem of factoring the following sixth order polynomial z6 - a6. He first pointed out that, if you consider this as the difference of two cubes, and then consider z2 - a2 as the difference of two squares, you get the result

z6 - a6 = (z2)3 - (a2)3 = (z2 - a2)(z4 + a2z2 + a4) = (z - a)(z + a)(z4 + a2z2 + a4)

On the other hand, if you consider the  polynomial z6 - a6 as the difference of two squares, and then use the appropriate cube formula on each of the factors, you get

z6 - a6= (z3)2 - (a3)2 = (z3 - a3)(z3 + a3) = (z - a)(z2 + a z + a2)(z + a)(z2 - a z + a2)
Since both of the answers must be correct, it must be true that
(z2 + a z + a2)(z2 - a z + a2) = z4 + a2z2 + a4
Ann Brandon verified this assertion by straightforward and tedious algebra.  Note:  Since the LHS is symmetric under the transformation z ® - z, all odd powers of z must cancel on the RHS as well.  Porter Johnson commented on the related problem of finding all solutions of the sixth order polynomial equation z6 - a6 = 0.  The solutions z0, z1, z2, z3, z4, z5, consist of the factor a multiplied by each of the 6 sixth roots of unity; these complex numbers may be expressed in polar notation as
zk= a e2ikp/6 ; where k = 0. 1. 2. 3, 4, 5.
Here are the roots:
 Root Number Root Name Root Value 0 z0 a 1 z1 a/2 [1 + i Ö3 ] 2 z2 a/2 [1 - i Ö3 ] 3 z3 - a 4 z4 a/2[-1 - i Ö3 ] 5 z5 a/2 [-1 + i Ö3 ]
The factors zk form  the following regular hexagonal pattern in the complex plane
```                                   |                       z2 *        |       * z1                                   |                                   |                                   |                                   |                 z3                |                z0                 ---*--------------|---------------*----                                   |                                   |                                   |                                   |                                   |                                   |                       z4 *        |        * z5
```
The polynomial in question may thus be factored in the form
z6 - a6 = (z - z0)·(z - z1)·(z - z2)·(z - z3)·(z - z4)·(z - z5 )
The roots z0 and z3 are real, whereas (z1, z5) and (z2, z4) are complex conjugate pairs.  Thus the products (z - z1) · (z - z5) and (z - z2) · (z - z4) are real. In fact;
(z - z1)·(z - z5)= z2 - a z + a2 ;
(z - z2)·(z - z4) = z2 + a z + a2 .
You can also use these relations to get other identities easily:
(z - z1)·(z - z2)= z2  + a2 - a z ;
(z - z4)·(z - z5) = z2 + a2+ a z .
So that
(z - z1)·(z - z2)·(z - z4)·(z - z5 ) = z4 + 2 a2z2 + a4 - a2z2 =  z4 + a2z2 + a4

For the fascinating history of the Mathematician Evariste Galois, who studied roots of polynomial equations, see the website http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Galois.html.  Here is a comment, taken from that website, by one of his teachers:
"It is the passion for mathematics which dominates him, I think it would be best for him if his parents would allow him to study nothing but this, he is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments."

Marilynn Stone (Lane Tech HS, Physics)
showed a circuit board system containing a battery, 3 light bulbs, and 4 single switches, and one double switch [S5] attached to a plywood sheet [roughly 2 feet by 2 feet], as shown.

She then asked the students to accomplish these tasks with the battery:

 Connect B1 in series. Connect B2 and B3 in series. Connect B1, B2, and B3 in series. Connect B1 and B2 in parallel Connect B1, B2, and B3 in parallel. Connect B2 in Series. Connect B3 in series. Connect B1 and B3 in series. Connect B1 and B3 in parallel Connect B1 and B2 in series.
The last exercise may be rather difficult.

Don Kanner (Lane Tech HS, Physics)
began by taking a plastic straw, flattening one end to make a double reed, and blowing into it to make a sound.  He cut pieces off the other end  and blew to demonstrate that, as the straw becomes shorter, the pitch of the sound goes up.  He made a very interesting presentation at the Elementary SMILE Meeting on 06 March [Section B] , which is described in detail on the website em030601.htm.

Don then announced that he would give us a surprise test -- which turned out to be a hearing test. To that end, he put a chart on the board that looked like this one:

Average Sensitivity of Human Hearing at Various Frequencies
Source: http://www.bcm.tmc.edu/oto/studs/aud.html
He started to give us the hearing test, but we were saved because there was too much hum in his audio oscillator [perhaps because of a problem in the amplifier driven by  it]. The hearing test was deferred, and Don concluded by making the following general comments:
• The human ear is most sensitive in the region around 2 KHz.
• The threshold of hearing goes up by 20 dB [a factor of 100 in sound pressure] at around 100 Hz.  In other words, we are much less sensitive to low-frequency [base] notes than to high frequency [treble] musical notes.  Consequently, it is routine to amplify the low frequencies on stereo systems. One inevitably amplifies low frequency background in this process--- Dolby™ and certain other systems are designed to reduce the enhancement of low frequency noise, while preserving the signal.
• Many teachers cannot hear well at frequencies much beyond 8 KHz, whereas most students can detect frequencies up to 20 KHz.  Comment by Porter Johnson: "There is little difference in hearing thresholds between young male and female children. Between ages 10 and 20, males begin to show reduced high-frequency auditory sensitivity relative to females. Women continue to demonstrate better hearing than men into advanced age. These gender differences are probably due to greater exposure of males to noise rather than to their inherent susceptibility to its effects."  Source: [http://www.nidcd.nih.gov/health/hearing/noise.asp]
•  At about 20 KHz the "hearing threshold" and the "threshold of pain" are both about 120 dB---in effect we cannot hear higher frequency sounds.  Smaller animals can hear well at much higher frequencies [dogs and cats can hear well at about 40 KHz, as demonstrated by mechanical or electronic whistles], whereas elephants are very sensitive to low frequency rumble [10 - 20 Hz] not heard by smaller animals.

Porter Johnson commented that our region of maximum hearing sensitivity, 2 KHz, corresponds to a wavelength of about 10 cm---the distance between our ears. This rough correspondence also works pretty well for animals.  He also commented on transmission of sound in the oceans, which enables marine mammals to communicate over great distances.  "Acoustic thermometry, however, capitalizes on the presence of sound channels present in the deep sea capable of trapping and transmitting sound over very long distances. The channels are created by the variation of pressure and temperature with depth. Located at a depth of about 3,000 feet, these deep sea super-highways act almost like a lens in focusing the sound and guiding it over thousands of miles." Source: http://www.sio.ucsd.edu/scripps_news/pressreleases/ATOC98.html.

Porter Johnson (IIT BCPS Department) Handout:  Day and Night
used a globe and a small bed light and showed how the length of day, zenith angle of the local noon, and the direction of the rising and setting sun depends on the latitude, as well as the time of year.  The handout, which contains technical details for calculating these quantities [neglecting atmospheric refraction, the finite size of the solar disk, and the eccentricity of the earth's orbit around the sun] is located on his website at URL http://www.iit.edu/~johnsonp/daylight.html. He made the following qualitative observations:

• The axis of rotation of the earth is tilted at an angle of about 23o to the direction of its orbital angular momentum around the sun. This axis points roughly in the direction of the star Polaris in the Little Dipper / Ursa Minor constellation.
• At the Winter solstice [21-22 December] the earth's axis is tilted away from the sun.  The sun appears lower in the sky and  it is Winter in the Northern Hemisphere, whereas it is and Summer in the Southern Hemisphere.
• The earth rotates counterclockwise [as seen from the North Pole].  A period of about 23 hours and 56 minutes [sidereal day] is required for one rotation, [or about 366 1/4 rotations occur in one year].  The tracking motor on a telescope must be designed so that it will make a full rotation in a sidereal day, rather than in the more familiar solar day.
• At the poles there are 6 months of continuous darkness, followed by 6 months of continuous sunlight.  The transitions occur at the Vernal Equinox [21-22 March] and the Autumnal Equinox [21-22 September].
• At any point above the Arctic circle or below the Antarctic circle [latitude about 67o] there are 24 hour periods continuous sunlight [summer] and of continuous [darkness].  The sun lies very close to the horizon and its horizontal projection lies in every direction at some point of the day.
• Everywhere on the equator there are precisely 12 hours of daylight, and the sun rises exactly in the East and sets exactly in the West---at any time of the year.
• On the Vernal and the Autumnal Equinox there are precisely 12 hours of daylight and the sun rises precisely in the East and sets in the West, at all latitudes North and South.
• Between the Autumnal and Vernal Equinox [fall and winter] the sun rises South of East in the Northern Hemisphere, whereas for spring and summer it rises North of East in the Northern Hemisphere.  The times are reversed in the Southern Hemisphere.

Betty Roombos (Gordon Tech HS, Physics)
handed out copies of a worksheet activity entitled Exploring Life Science:  Measuring Liquid Volume with a Graduated Cylinder.

Arlyn VanEk (Illiana Christian HS, Physics)
illustrated the behavior of electric motors and generators, using the camera probe [http://www.allelectronics.com/] with the large-screen TV.  First, he reminded us that the directions of Current and Magnetic Field are related by the right-hand rule. Using your right-hand:

• Point your thumb in the direction of the conventional current, I [labeled by v in the diagram!]
• Curl your fingers into a half-circle around the wire. They point in the direction of the magnetic field, B.
[Source:  http://micro.magnet.fsu.edu/electromag/electricity/generators/]. He showed how to give a large-scale visualization of the convention by putting long paper cylinders on the fingers of his right hand.  He then used visuals to illustrate how a current-carrying loop of wire is made to rotate in a magnetic field, and noted that if the current does not vary with time [Pure DC], the wire will not continue to rotate, but will come to rest at an equilibrium position.  The changing current necessary to make an electric motor can be obtained with a split ring [commutator],  so that the current changes direction every half-rotation. The current will be of fixed magnitude, but its direction will change every half-cycle.

Next he showed how to make an electric generator to convert mechanical energy into electrical current.  The device is a rotating split-ring coil in an external magnetic field, so that the direction of flow of current reverses at every half-turn.  The output voltage of 3-5 volts was monitored on a standard digital oscilloscope, available in the Radio Shack Catalog [ http://www.radioshack.com/], such as this model:

\$1,199.99 Reg. Price; Brand: Instek
Cat #: 910-5360 Model: GOS-6103
As seen from the trace, the output voltage is about 5 Volts with full wave rectified AC structure; that is, I(t) = Io |sin w t|.  In addition, the oscilloscope showed spiked voltage pulses that correspond to the openings in the split ring commutator.  These pulses are similar to high frequency ignition interference that may be heard on AM channels on the car radio, which correspond to opening and closing of distributor points.

Ann Brandon (Joliet West HS, Physics)
passed out 4 sheets with various circuit problems designed to emphasize the basic approach in analysis of circuits involving resistors in series and in parallel. These sheets, which she prepared in the "draw" program in Microsoft™ Word /Office 95 or 97, had the following titles:

• Find the Total Resistance in Each Circuit
• Voltages in a Circuit
• Currents in a Circuit
• Currents, Voltages, and Resistances
Here is an image of one of the sample circuit problems from the third sheet:
The full set can be obtained from Ann Brandon: llbrandon@aol.com

Notes taken by Porter Johnson and Earl Zwicker.