High School Mathematics-Physics SMILE Meeting
23 April 2002
Notes Prepared by Porter Johnson
Announcements:
- The last SMILE meeting of the academic year will be Tuesday, 07 May. You may register for the Fall 2002 class at that time.
- Note that, beginning Fall 2002, free parking for SMILE participants will be available in the fraternity lot [33rd Street west of the El Tracks]. We will pass out an application form on the first day of class next Fall. You must sign the form and record your VIN [Vehicle Identification Number; it should appear on your automobile insurance card] and auto license plate number. In any event, the system will be fully operational by then!
- There will be a retirement party for Melanie R Wojtulewicz [CPS Manager of Science] on Friday 21 June [5-9 pm] at the Garfield Park Conservatory. For details contact Gloria Dobry, science facility at the Medill TPDC -- Room 315A; Mail Run 80. Melanie has been a dear, dear friend of many of us in the SMILE program, and we will miss her!
Monica Seelman (St James) -- Digital Numbers, Geometrical Shapes,
and Pouring
Water from a Coffee Pot
Monica handed out copies of an article: The Wonderful World of
Digital
Sums by M V Bonsangue, G E Gannon, and K L Watson, which appeared
in the
January 2000 issue of the periodical Teaching Children Mathematics,
and
discussed some interesting applications. First she made put
a
multiplication table on the board [PJ: here is a 9 ´ 9 version, like the ones on
spiral notebooks in schools a few decades ago, which some of us
remember very well.]
one-sies | two-sies | three-sies | four-sies | five-sies | six-sies | seven-sies | eight-sies | nine-sies |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 |
4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 |
5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 |
6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 |
7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 |
8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 |
9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 |
Monica patiently explained that a digital number is simply the sum of digits of a number, as illustrated here
Number | Digital Number |
6 ® | 6 |
13 ® | 1 + 3 = 4 |
27 ® | 2 + 7 = 9 |
38 ® | 3 + 8 = 11®1 + 1 = 2 |
In other words, a digital number is just the sum of the digits of a number, which becomes a number: 1, 2, ... , or 9 [more technically, the number modulo 9 + 1]. Here is our 9 ´ 9 multiplication table , expressed in terms of digital numbers
one-sies | two-sies | three-sies | four-sies | five-sies | six-sies | seven-sies | eight-sies | nine-sies |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
2 | 4 | 6 | 8 | 1 | 3 | 5 | 7 | 9 |
3 | 6 | 9 | 3 | 6 | 9 | 3 | 6 | 9 |
4 | 8 | 3 | 7 | 2 | 6 | 1 | 5 | 9 |
5 | 1 | 6 | 2 | 7 | 3 | 8 | 4 | 9 |
6 | 3 | 9 | 6 | 3 | 9 | 6 | 3 | 9 |
7 | 5 | 3 | 1 | 8 | 6 | 4 | 2 | 9 |
8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 9 |
9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
Monica then used these columns of digital numbers to specify the order of connecting the vertices of a regular nine-sided polygon (nonagon), which was made by drawing a circle and then dividing it into nine equal segments (arcs), as shown:
Monica next showed us how to connect the points in the order given by the columns in the table. We get the following types of figures in the nine cases:
Sequence Name |
Sequence |
Figure Name |
One-sies: | 1 ® 2 ® 3 ® 4 ® 5 ® 6 ® 7 ® 8 ® 9 | Regular Nonagon |
Two-sies | 2 ® 4 ® 6 ® 8 ® 1® 3 ® 5 ® 7 ® 9 | 2-star Nonagon |
Three-sies | 3 ® 6 ® 9 ® 3 ® 6 ® 9 ® 3 ® 6 ® 9 | Triangle |
Four-sies | 4 ® 8 ® 3 ® 7 ® 2 ® 6 ® 1 ® 5 ® 9 | 4-star Nonagon |
Five-sies | 5 ® 1 ® 6 ® 2 ® 7 ® 3 ® 8 ® 4 ® 9 | 4-star Nonagon |
Six-sies | 6 ® 3 ® 9 ® 6 ® 3 ® 9 ® 6 ® 3 ® 9 | Triangle |
Seven-sies | 7 ® 5 ® 3 ® 1 ® 8 ® 6 ® 4 ® 2 ® 9 | 2-star Nonagon |
Eight-sies | 8 ® 7 ® 6 ® 5 ® 4 ® 3 ® 2 ® 1 ® 9 | Regular Nonagon |
Nine-sies | 9 ® 9 ® 9 ® 9 ® 9 ® 9 ® 9 ® 9 ® 9 | POINT |
Comment by PJ: Note that the Nine-sies are all nines [one point in the figure], whereas the Three-sies and Six-sies involve the same 3 numbers, forming an equilateral triangle, but with the process of connection being done in opposite directions. Similarly, the One-sies and Eight-sies form the same regular nonagon, but involve tracing it in opposite directions. The Two-sies and Seven-sies trace the same 2-star nonagon, which hits alternate numbers and makes two revolutions before closing. Finally, the Four-sies and Five-sies produce the same 4-star nonagon, which closes on itself after making four revolutions.
Monica poured water from a coffee pot and raised the question as to why the coffee stream forms a twisting spiral when it comes off the lip of the coffee pot. We will investigate this more in a future meeting!
Fascinating stuff, Monica!
Fred Schaal (Lane Tech HS Mathematics) -- Top-ological Theory and
Planetary
Lineups
Fred dealt with "top"-ology at an extremely applied level,
showing a
molded plastic top that has a smooth curved bottom, with a projecting
shaft on
top. Holding the top by its stem, Fred set it spinning
about its axis of symmetry
with its bottom on the table. To the amazement of many of us, the top
turned
itself over, so that it was spinning on its stem! How come?
Fred also showed us that the top would
initially rotate "upside down" when set into motion in that
orientation, and would stay that way. This physics toy, which is
called a
"tippy top", has identical moments of inertia about directions
perpendicular to its symmetry axis. The top continues to rotate in the
same
sense when its flips, so that the angular momentum of the top does not
change
direction. This is different from the "rattleback",
which has three different moments of inertia, and for which the
direction of
rotation may change. For details concerning the tippy top, see http://www.youtube.com/watch?v=xu_Dp9IfgSU
and especially the American Physical Society page
http://www.aps.org/units/fed/newsletters/fall2001/kamishina.html,
which contains the following excerpt:
"Among a variety of tops, a tippy top is most popular. At a glance, a tippy top is hardly distinguishable from normal tops. A top usually rotates steadily around the rotational axis and the rotational axis rotates around the vertical axis as everyone knows. However a tippy top turns upside down while rotating [to see image click: http://www.aps.org/units/fed/newsletters/fall2001/images/k10b.jpg]. The big difference between them is that the usual top falls down when at rest while a tippy top doesn't. It is stable at rest. This means that the center of mass of a conventional top is situated higher and it is therefore unstable at rest, while on a tippy top the center of motion is at the lowest position at rest. Roughly speaking, rotational motion progressively lifts the center of mass of a tippy top, and finally turns it over. The mechanism by which the axis of rotation gradually moves up or down in addition to a precession, moving in a circular cone about the vertical axis, is in large part connected with the action of friction at the point of contact with the floor.""The quantitative explanation of this mechanism is too difficult for students to understand. The qualitative explanation is more suitable for children. To reproduce the motion of a tippy top, I showed a 2-dimensional tippy top consisting of a large ring and a small ring both made of metal wire The two rings are attached at a point with the small ring inside the large one on the same plane. The role of the smaller ring is to shift the center of mass of the system away from the center of the large ring. When you rotate the large ring around the vertical axis connecting two centers of both rings with the small ring at the bottom, the system acts like a tippy top. While when you do the same thing but with the small ring at the top it acts like a conventional top. The difference in behavior is the position of the center of mass of the system."
Fred also alerted us to the fact that the planets Jupiter, Saturn, Mars, Venus, and Mercury are aligned in the Western sky just after dusk. For the next two months, 8:30 pm is about the best viewing time For details see http://www.adlerplanetarium.org/index.shtml. If you miss this planetary display, you can catch it again for a repeat performance in about 40 years! Thanks, Fred!
Walter McDonald (Bowen HS and Chicago Veterans
Administration) -- Higher Dimensional Geometry
Walter described his efforts at tutoring students on
visualizations of
spaces of various dimensions. He presented the following table of
characteristic figures [called simplexes by mathematicians] in various
spatial
dimensions:
Number of Dimensions | Characteristic Figure |
0 | Point |
1 | Line |
2 | Triangle |
3 | Tetrahedron |
4 | Figure with Tetrahedron Faces |
Walter asked what is the difference in 4 dimensional (Euclidean) space and what physicists call "space-time"? Porter Johnson commented that the time interval Dt between two events and the spatial interval DL between the same two events can be regarded in terms of a unified "space-time", and that because of the constancy of the speed of light, v, it is required to define the invariant-interval-squared between two events as [DL]^{2} -[v Dt]^{2} . This space is called Minkowski space in Special Relativity, and which is a four dimensional (Euclidean) space, when expressed in terms of real space variables, with the time variable multiplied by i = Ö(-1). Incidentally, the development of non-Euclidean geometry in mathematics and its applications in physics are an outgrowth of the examination of whether Euclid's fifth postulate [parallel lines never meet] is a consequence of the other four. Non-Euclidean geometry is the mathematical framework for General Relativity, our (classical) theory of the Gravitational Field. For an interesting discussion of non-Euclidean geometry, see the St Andrews University web page: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.html.:
Very good, Walter!
Bill Blunk (Joliet Central HS Physics) -- May the Force be Wilber!
Bill set up a Wilberforce Pendulum [mentioned at the last
SMILE meeting], in which there are two degrees of freedom,
corresponding to
"up-down" motion of the mass suspended by a spring, as well as its
"torsional" motion. When he started the pendulum in an
"up-down motion", its motion gradually became torsional, and then went
on to switch slowly but steadily back and forth between "up-down" and
"torsional" motions. How come? Bill assured
us
that, since April Fool's Day has passed, this was not a trick, and
showed us
that there was nothing up his sleeve. We discussed the
matter at
length. A vibrating system with two degrees of freedom with
normal modes
lying at slightly different frequencies will execute periodic motion
only for
very specific initial initial conditions. Otherwise, one observes
"beats" between the two normal modes, in the same spirit as two tuning
forks of slightly different frequencies. There is coupling
between
"up-down" and "torsional" motions in this case, so that
neither of these motions corresponds to a normal mode of the system, as
one
might think. We decided to look for this coupling in the
"static" case in which the mass was not moving, making for slightly
different values of the suspended mass. To our surprise, the
equilibrium
position of the marker on the mass could be seen to rotate as the
suspended mass
was slightly changed.
You no longer have to beware the dark slide, Bill!
Leticia Rodriguez (Peck School) -- Tesselations; Mathematical
Applications; Scientific Method
Leticia passed around the following book, which contains various
tesselations
[which are regular periodic patterns, or periodic and quasi-periodic
"tilings"
of space]: Tesselations Teaching Masters; Dale Seymour
Publications, 1989; ISBN 0-88661-462-7. Leticia's primary
students color
these tesselations to make elaborate designs, and use them as a means
to learn
elementary ideas in mathematics [shapes, patterns, graphs, counting,
geometry,
etc] and elements of the scientific method [observing, estimating,
collecting
data, predicting, classifying, investigating, comparing, contrasting,
problem
solving, inferring, and drawing conclusions]. For further details
see her
website on the SMART home page: http://www.iit.edu/~smart/.
Porter Johnson mentioned that intricate, symmetric patterns are
employed
in many religions to convey a sense of spirituality in their
cathedrals,
chapels, churches, mosques, pagodas, shrines, and temples. One
beautiful example of these patterns is the
Baha'i Temple
in Wilmette Illinois; see the websites: http://members.core.com/~fphayes/bahai.htm
and http://www.sacred-destinations.com/usa/chicago-bahai-house-of-worship.htm.
Leticia also pointed out that teachers are entitled to a 15% discount on educational and school supplies for classroom use (with proper identification) from April 15 to May 31, 2002 at Amazing Savings stores, located in Morton Grove (Harlem & Dempster) , Wheeling (Elmhurst & Dundee), Chicago (McCormick & Lincoln), Broadview (17th and Cermak), and Bloomingdale (Springbrook Shopping Center). Thanks, Leticia!
Larry Alofs (Kenwood HS Physics) -- New "Physics Toy": Accurate
Digital Thermometer
Larry showed his new Infrared Thermometer providing a digital
readout accurate to
± 0.2 °C, which he
recently obtained at Radio Shack for
around $40. The device does require a non-standard 12 V battery, which
costs about $3.
The device is shown on the Radio Shack on-line catalog [http://www.radioshack.com/],
and for convenience navigating around their site, it is helpful to use
their
Catalog Number: #22-325. We used this device to measure the
following temperatures in our classroom:
Location | Temperature [°C] |
Room air | 23.2 |
Aim at Lights (fluorescent) |
22.6 |
Aim toward floor | 22.6 |
Aim toward Ceiling | 21.4 |
Coffee pot lid | 64.4 |
Aim at mouth | 32.6 |
Between cupped hands: before uncupping hands just after uncupping |
- 34.6 34.2 |
Blackboard: before rubbing after rubbing |
- 22.6 24.8 |
A beautiful gadget, Larry!.
Arlyn VanEk (Illiana Christian HS Physics) -- Standing Waves
Using Scroll Saw
Apparatus and a Marimba
Arlyn brought in a scroll saw [form of jigsaw with flat table]. He
tied one
end of a string to the top end of the blade. He played out 2-4
meters of string, pulled
the string taut, and turned on the saw. By varying the tension in
the
string, he could produce various standing waves. These transverse
waves,
with nodes at the ends, corresponded to N half-wavelengths,
with N-1
internal
nodes. That is, the length L of the string and the
wavelength
l are related by the relation L = N /2
l . We could see the fundamental mode N = 1,
as well as the first two harmonics,
N = 2; N = 3. The frequency, n, of
the waves is fixed by the
scroll
saw frequency; presumably, something like 60 Hz. The velocity,
v, of waves on the string is given
in terms of the tension T and the mass per unit length m as
v = Ö (T/m
); in turn the velocity
v is given by v = ln. Thus,
the tension required to excite the Nth mode is inversely
proportional to N^{2}:
T_{N} = 4mn^{2}
L^{2} / N^{2}. In order to increase the
mode number N, one must decrease the tension T.
Arlyn next described a Marimba [like a xylophone, except perhaps more so; see http://en.wikipedia.org/wiki/Marimba], which typically consists of wooden pieces with supporting members arranged to tune to a pentatonic scale. The avid Marimba player then plays the instrument by striking the pieces one at a time, as required by the melody. Arlyn illustrated the operation with two different [old pine construction] boards [2" ´ 8" ´ 4'] supported transversely by pieces of (slit) hard rubber hose. By adjusting the distance between supports; then striking them with a rubber mallet, Arlyn tuned the resonant frequencies of these boards. By sprinkling sand on top of one of the boards, he showed that nodes occur at the support locations, so that the distance between them was about a half wavelength. The distance between supports was 0.5 meters, the resonant frequency was be around 400 Hz. This simple "two note marimba" sounded quite nice, Arlyn! He went on to strike the end of a metal rod against the desk, holding it at various points to excite various normal modes. The "punch line" is that if you hold the rod at the location of a node of a low-lying resonance of the longitudinal vibrations of the bar, you will excite that mode when striking the bar.
Beautiful Physics, Arlyn [and especially enlightening for scroll saw operators and marimba players].
Notes taken by Porter Johnson