High School Mathematics-Physics SMILE Meeting
23 April 2002
Notes Prepared by Porter Johnson

Monica Seelman (St James) -- Digital Numbers, Geometrical Shapes, and Pouring Water from a Coffee Pot
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
2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 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
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 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 1 8 6 4 2 9
8 7 6 5 4 3 2 1 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


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
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
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!
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
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
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
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
before rubbing
after rubbing

A beautiful gadget, Larry!.

Arlyn VanEk (Illiana Christian HS Physics) -- Standing Waves Using Scroll Saw Apparatus and a Marimba
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 N2 TN = 4mn2  L2 / N2.  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