Don Kanner [Lane Tech HS, physics]
Signs of the times + p a la mode
Don described the history of the multiplication symbol ´, as described in the references History of Mathematics and History of Mathematical Notation by Florian Cajori. The original representation of multiplication of one and two digit decimals was written as follows
3 3 7 7 8 | | X | | X | 3 3 7 5 6 ----- ... --------- ... ---------- 9 4 9 4 8 2 1 4 2 2 1 4 0 9 3 5 --------- --------- 1 3 6 9 4 3 6 8The notation was then simplified by leaving off the vertical lines:
7 8 X 5 6 ------ 48 42 40 35 ------ 4368The division problem 3/7 ¸ 2/5 was written in ancient times as follows:
3 2 3 ° 5 15 --- X --- = ----- = -- 7 5 7 ° 2 14Don then discussed various formulas involving the number p, which are all discussed in the ST Andrews (Scotland) website: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html, from which the following has been excerpted:
"The European Renaissance brought about in due course a whole new mathematical world. Among the first effects of this reawakening was the emergence of mathematical formulae for p. One of the earliest was that of Wallis (1616-1703)
2/p = (1°3°3°5°5°7° ...)/(2°2°4°4°6°6° ...)
and one of the best-known is
p/4 = 1 - 1/3 + 1/5 - 1/7 + ....
This formula is sometimes attributed to Leibniz (1646-1716) but it seems to have been first discovered by James Gregory (1638- 1675)."
Don also mentioned the formula used by Francois Vičta (1540-1603), for which the following information appears on the ST Andrews website:
"Vičte used Archimedes method with polygons of 6 ´ 216 = 393216 sides to obtain 3.1415926535 < p < 3.1415926537. He is also famed as the first to find an infinite series for p."
Don is engaged in an investigation to see which of the expressions converges most rapidly to p. Place your bets now!
Larry Alofs [Kenwood HS, physics]
Larry showed us this device, which is a substantially modified version of one presented by Tom Senior. He used a 2 meter PVC pipe of inside diameter about 38 mm (1.5 inches), with plastic caps for the ends. Near one end there was a T-connection to a vacuum pump. He inserted a ping-ping ball, tilted the pipe so that the ball went down to the end near the vacuum pump connection, and capped both ends. He found that the new, translucent caps on Pringles™ cans worked very well for capping. After he turned on the vacuum pump and let it run for a minute or so, the cap was visibly deformed. When he punctured the cap at the lower end, there was an explosive POW! --- the ping-pong missile shot out the other end, and SMASHed against the opposite wall. Very impressive display of launching power! The launch velocity of the ping-pong ball would surely be less than the velocity of sound, but it appeared to be quite fast, since we could not follow its trajectory, and the POW! and SMASH! seemed simultaneous. Larry then set up the apparatus with the bazooka aimed directly at a cardboard box, and launched it again. The ping-ping ball shot through both sides of the box, and smashed against the wall. Now, that's a really powerful serve!. Thanks for the powerful display of forces arising from air pressure, Larry!
Ann Brandon [Joliet West, physics]
Masses for the Masses
Ann showed us a set of brass slotted weights (including calibrated hook) that she had recently obtained (cheap!) for $9.00 from a Science Kit catalog, although the package referred to Edmond Scientific. Since the science supply houses don't actually manufacture very many of their products, the same items are often available from several of them.
Bill Blunk compared the mass of a 5 gram brass weight with that of a Nickel coin -- approximately 5 grams. He placed the brass weight at one end of the meter stick, placed a Nickel at the other end, and used a pencil as a fulcrum to balance the stick at the middle. The balance was good, and the stick remained balanced when the positions of the Nickel and brass weight were switched. Nice weights, Ann! Nice test, Bill!
Bill Blunk [Joliet area, retired]
Flying Rocks: Angels in the Centerfold?
Bill gave each of us a copy of a photograph taken in Glacier National Park (while he was hiking by a lake about 12 km off the road), which was published as a centerfold of the April 1 1983 issue of The Physics Teacher. In the image, rocks appear to float above a lake. (It was explained in the blurb, that the rocks contained Helium gas as a consequence of radioactive decay.) Unfortunately, the magazine printers were too alert, and the picture appeared right-side-up. (The rocks were in the lake!) Bill did not explain why the reflections of the rocks did not appear in the sky (HA!), but he did mention that hiking is a somewhat dangerous hobby. Four of Bill's personal friends have been mauled by grizzly bears! Thanks for the safe shot, Bill!
Charlotte Wood-Harrington [Brooks HS, physics]
Series and Parallel Resistances
Charlotte gave each of us three Christmas Tree bulbs with wire leads that she had cannibalized from old sets, as well as tape and a stick of chewing gum. We were permitted to chew the gum, with strict instructions to use its metallic wrapper to make electrical contact with the lights. We were able to make various series and parallel combinations of lights, which we tested with a 9 Volt battery. The lights could be made to burn dim or brightly, with various combinations.
As an extension of this lesson, Roy Coleman asked the following question:
Q: Why should batteries be purchased using a credit card?Enthusiastic and cheap! Thanks, Charlotte.
A: So that they get charged.
Roy Coleman [Morgan Park HS, physics]
Foot - Hand Rotation
Roy asked us to move our right foot in a clockwise circle. Then, simultaneously try to write a "6" in the air. Astonishingly, as we wrote the 6 in the air, the right foot reversed its clockwise motion to counterclockwise. How come? Thanks, Roy!
Monica Seelman [ST James Elementary School]
Fractals and Sierpinski's Pyramid
Monica passed out some bedtime reading material on Sierpinski's Pyramid [see http://en.wikipedia.org/wiki/Sierpinski_triangle and http://library.thinkquest.org/26242/full/fm/fm32.html]. Monica gave us each a one-page template that had a circle of radius about 20 cm with an inscribed equilateral triangle. We cut around the circle, and then folded the sheet along the triangle sides, and then folded it to make a regular tetrahedron. The tetrahedron was held together with tape for stability. Then, the various tetrahedrons were taped together to form the three-dimensional pyramid. We had 12 small pyramids inside our Stage 2 super-pyramid.
Now we know about Sierpinski's Pyramid. Thanks, Monica!
Notes prepared by Porter Johnson