Mathematics-Physics High School SMILE Meeting
7 November 2000
Notes Prepared by Earl Zwicker

Fred Schaal (Lane Tech HS)
sketched out on the board a kind of geometry problem he will do next time using the TI 92 calculator. He drew two straight lines (representing mirrors), intersecting at some angle. Then he drew a triangle with vertex points A, B, C and sketched its reflection across the nearest of the mirrors. He pointed out that the order of the points, A, B, C was ccw (counter clockwise) for the triangle, but cw (clockwise) for the vertex points - A' ,  B' , C' - of its reflection. Sketching the reflection of the those points across the second mirror (straight line) located vertex points A", B", C" - which followed a ccw order. It could be interesting to see how Fred will do this with the TI 92. But this provoked discussion about reflections in mirrors, and brought out that it takes two mirrors to see your reflection as others see you!

Bill Colson (Morgan Park HS)
started off with a funny story about Sherlock Holmes and Dr Watson, who were out camping in the wilderness. The view of the night sky was magnificent, and Holmes wondered how one could explain it. Watson responded that the stars at one time might have been explained as holes in the celestial sphere. Or others might see the stars as outlining the shapes of various animals. More recently the big bang theory attempts to explain the origins of the stars. But Holmes appears impatient with Watson, who then asks what Holmes thinks. Holmes responds with, "I deduce that someone has stolen our tent!" Hah!

Bill passed out copies of a Chicago Tribune newspaper article (Nov 2, 2000) titled Making Science Accessible, Fun Defies the Laws of Physics. He also discussed the following books

It was interesting to see how our insight has changed.

Bill passed out several small world globes (about 1.3 in diameter) and then placed an inflatable globe about 16 inches in diameter on the table. It was transparent, but showed the various countries and the lines of longitude and latitude as well. He explained that when he introduces the topic of spherical geometry to his students, he likes to let them convince themselves that plane geometry is not always sufficient to deal with the problems one may face in real life.

Sailors, ships, planes which follow the curved surface of earth will have trouble if they assume they are traveling on a flat plane instead of a spherical earth. Euclid set forth postulates for the flat plane, which must differ from those for the geometry of a spherical surface. A point is the same in either geometry. But what about a straight line? What does that mean on the surface of a sphere? And what about parallel lines never meeting in a plane? Is there such a thing as parallel lines on a spherical surface? What are great circles? Bill wrapped a string around the globe to show a great circle. We saw that the equator is a great circle, but the smaller circles marking a constant latitude are not. He held up a map of the world which was a Mercator projection, showing earth on a flat surface. [See the website] The longitude and latitude lines look parallel on a that flat map, but not on the globe. The distortion in size of various continents could be seen when comparing with how they appeared on the globe.

Porter Johnson stretched a string on the Mercator map to show the distance between Frankfort, Germany and Fairbanks, Alaska. And then between Frankfort and Chicago - which appeared to be a much shorter distance. But when the same comparison with the string was done on the global map, the distances were very nearly the same!

Bill passed out a copy of an article by Cecil Adams in the Chicago Reader about the QIBLA, which is the Arabic name given to the direction toward Great Mosque in Mecca.  Muslims should pray while facing in that direction, the QIBLA, and it is often marked by an arrow in locations for prayers.  This need to determine the "prayer direction" catalyzed interest in astronomy, navigation, and geography, and put the Muslim world far ahead of western societies in these areas.  Most Muslim authorities agree that the QIBLA should lie along the "great circle" geodesic from one's location to Mecca.  In North America that direction is generally North of due East.  For more information see the websites, and

Good, phenomenological math, Bill!

Walter McDonald (CPS Sub and X-Ray Technician)
set up a transparent vertical sheet of plastic on the table, with a fluorescent lamp as illumination from behind, and he then displayed some large X-ray images of the spine, one after another, pointing out how he had enhanced some of them for detail using computer processing. According to Walter, the images are not produced by using a chemical liquid developer of a film emulsion, but rather through a process which reads the x-ray exposure on a plate and stores the result directly into computer memory. Interesting, Walter!

Ann Brandon (Joliet West HS)
held up three 6 inch wood blocks with cup hooks on end, and a small spring scale. She hung one of the blocks from the scale, which read a weight mg = 1.3 N. Then Ann used the scale to pull one of the blocks across the table horizontally at a steady speed. The pull on the scale read 0.2 N. When she stacked another block on top of the first, the pull went up to 0.4 N, and with three blocks, pull was 0.55 N. It was clear to see that the pull, which was due to sliding friction, went up nearly linearly with the combined weight of the blocks. But repeating the experiment did not produce the same results! The pull went from 0.4 N, to 0.9 N, to 1.5 N as the number of blocks stacked up went from 1 to 3.

Clearly, the friction was much higher the second time. Why? Ann showed us the reason. The second time, she had turned over the block in contact with the table, and it had a coating of silicone sealer on that surface; the first time its other (non-coated) surface was in contact, so friction was lower. She said that Roy Coleman - years ago - had showed that a silicone sealer coating drastically increases friction. Even though the block with the sealer had been used for many years, it still worked. She placed the non-coated block on the surface of a chair arm (desk), and tilted the arm up until the block just began to slip. Then she turned the block over with its coated side on the arm, and the chair arm had to be tilted at more than twice the angle before the block slipped, showing the high coefficient of friction. What a great way to get students' attention, and to learn something about friction! Thanks, Ann!

Notes taken by Earl Zwicker and Porter Johnson