High School SMILE Meeting
07 March 2006

Bud Schultz (Aurora Middle School)        Gonzo Gizmos
shared a book he bought at American Science and Surplus -- Gonzo Gizmos: Projects & Devices to Channel Your Inner Geek by Simon Field (http://www.kk.org/cooltools/archives/000667.php).  This book is full of great ideas, explanations, and projects -- it contains a lot of interesting activities involving electricity. Thanks for the Info, Bud!

Don Kanner (Lane Tech HS, physics)            The Sound of Physics
Don suggested modifying the simple discussion of open ended and closed ended organ pipes, as made at the last meeting by Larry Alofs. The equations l = 4L (for a pipe with one end closed)  and l = 2 L (for a pipe with both ends open) gives the fundamental frequency only for pipes under certain geometrical restrictions.  In fact, it is an oversimplified way of describing the vibrations. It is not just the length of the pipe that determines the pitch; we tested this for various pipes, across which we blew air to try to make standing waves. Another exercise involves a Florence flask and an Erlenmeyer flask of equal heights and volumes. They produce sounds of rather different pitch when air is blown across  them. The size of the neck and opening of the vessel is also important in determining what tone is made in this way.  For additional information see The Resonance of Common bottles and Jugs by Don Kannerhttp://www.iit.edu/~smart/kanndon/lessonb.htm. Hermann Helmholtz actually found out that the ideal shape for a resonating volume is a sphere.  For additional discussion see the comments at the 25 February 2003 HS Math-Physics SMILE meeting:  http://www.iit.edu/~smile/weekly/mp022503.html.

Sounds good!  Thanks, Don.

Fred Schaal (Lane Tech HS, math)               Parabolic Points
In an extension of his presentation at the last meeting, Fred used a similar procedure to trace out the points on a parabola using only his (chalk) compass, a meter stick and the blackboard. He chose a focal point  (focus) at random above a horizontal line (directrix).  He used the compass to draw a portion of a circular arc with an arbitrary radius, with the center at the focus . Two arcs are then made with the compass held at the same radius, with their centers on the line. A tangent to these two arcs intersects the first arc at two points, which lie on the parabola. The process is repeated using the same focal point but different radii, generating points to trace out a parabola.  For additional information see the interactive webpage The Parabola by Alex Bogomolnyhttp://www.cut-the-knot.org/ctk/Parabola.shtml,

Neat, Fred!

Earl Zwicker (IIT Physics)                       Mr Angry is on the left, and Mrs Calm is on the right
Earl had gotten an e-mail from Rudy Keil, including the remarkable image shown here. There are two images of a face, one with a calm look and one with an angry look. The two images seem to switch depending on whether they are viewed from close in (about 1 foot away) or far away (about 8 feet). It works completely!! But no one knew the reason for this! We will have to look for one!! One way to investigate it would be to try to find out if there is a consistent distance (for the members for the class) at which the transition occurs. We tried this. Fred tried it and the transition (where the images looked roughly the same) occurred at a distance of about 6 floor tile widths and the switch was complete at about 10 tiles. Don tried it and got 8 and 10 for the same figures. Walter got 8 and 10; Ed got 6 and 9. Fairly consistent results which did not seem to depend upon whether or not the observer was wearing glasses.  For additional discussion see the website http://cvcl.mit.edu/gallery.htm#hsflsf, from which the following has been excerpted:

"This impressive illusion created by Dr. Aude Oliva and Dr. Philippe G. Schyns, illustrates the ability of the visual system to separate information coming from different spatial frequency channels. In the right image, high Spatial Frequencies (HSF) represent a woman with a neutral facial expression, mixed with the low spatial frequency (LSF) information from the face of an angry man. On the left, the face of the angry man is represented in fine details whereas the underlying female face is made of blur only."
Thanks, Earl!

Porter Johnson (IIT, Physics)                  Sangaku-Followup
Porter continued the discussion of the “Circle Inscribing Sangaku”, which was introduced at the last class by Walter McDonald. This problem is discussed on the Mathworld Website on the web page http://mathworld.wolfram.com/CircleInscribing.html. However, that discussion is incomplete, in that it does not prove that the inscribed circle centered at O3 is tangent to the isosceles triangle ACB.

According to the statement of the problem, the  large circle of diameter 1 (unity)  is centered at point O, and a smaller circle of diameter r is centered at O2.  The smallest circle, which is of radius a and centered at O3, is tangent to the other two circles, and its center lies on the line O3A that is perpendicular to the major diameter XB.  The Mathworld website uses the fact that the right triangles OO3A and O2O3A have a common side, O3A, to determine the  length y of  that side O2A, as well as the radius a of the inscribing circle.  Their results are
(1 + r) a = r (1 - r)   ;

(1 + r)  y = r Ö[2 (1 - r)]    .

Let the symbol j represent the angle ACD.  Because the point C lies on the largest circle, its distance to the center O is 1/2. Furthermore, the right triangle ADC, has these side lengths:

[AD, DC, CA] = Ö(1 - r) / 2 ´ [ Ö(1 - r) , Ö(1 + r) , Ö 2 ]  .
Thus, we can show that
sin j = Ö[(1 - r) / 2]  .
Because the alternate interior angles O3AC and ACD are equal, we can compute the distance from the center O3 of the inscribed circle to the straight line AC:
y sin j = a
Consequently, the inscribed circle, with radius a and centered at O3, is tangent to the straight line AC. The result is thus established.

Porter then told us about Morley’s Theorem. Start with any triangle and trisect all three angles. Pairs of the trisecting lines from adjacent angles will intersect to make three points inside the original triangle. Connection of these three points will always produce an equilateral triangle!! Fred then illustrated this by laying out a carefully drawn figure on the board. For more details see the website  http://www.cut-the-knot.org/Curriculum/Geometry/Morley.shtml, which contains an adjustable triangle showing the result. See also http://www.jimloy.com/geometry/morley.htm, which contains the following comment:

"One of the interesting side results of some of the proofs is that the side of the equilateral triangle is equal to 8R sin(A/3) sin(B/3) sin(C/3), where A, B, and C are the angles of the larger triangle, and R is the radius of the circumcircle."
Fascinating, Porter.

Lee Slick (Morgan Park HS, retired)                        Cricket Temperature
described how to estimate the temperature (in degrees Fahrenheit) from the frequency of cricket chirps. (handout by Tom Skilling, Chicago Tribune, February 5, 2006http://wgntv.trb.com/news/weather/weblog/wgnweather/archives/ATW020506SUN.jpg ) Count the number of chirps  a cricket makes during  a 15 second interval, and add 39 to that number.  You get a remarkably accurate reading. This works because crickets are “cold blooded” (poikilothermic), so that their metabolism (and thus frequency of chirping) will increase as the temperature increases. For more details see the website Oecanthus:  Pulse Distribution and Temperature Effectshttp://facstaff.unca.edu/tforrest/ASA 98 Seattle/sld005.htm.
Keep on chirping!  Thanks, Lee!

Bill Colson (Morgan Park HS, math)               Assorted Literature
shared several items with us:

Thanks, Bill!

Our next SMILE meeting will be on Tuesday March 21, 2006. See you there!

Notes prepared by Ben Stark and Porter Johnson.