High School Mathematics Physics SMILE Meeting
19 November 2002
Notes Prepared by Porter Johnson
Betty Roombos [Gordon Tech HS, Mathematics]     The Ballistic Cart Put on Television 
presented a 21st century adaptation of the standard ballistics car demonstration, for which apparatus is available from, say, Sargent-Welch: http://www.sargentwelch.com/ [for a description of a ballistics car, see http://www.arborsci.com/Data_Sheets/P3-3527_DS.pdf]. The idea is that, when a ball is shot straight up from the cart while the cart is in uniform motion, it lands back in the cart, just where it came from.

As a modern variant of this standard demonstration, Betty used her digital camera, a Sony Mavica Model FD-88 with an 8X lens. It records 5, 10, or 15 second video images directly onto the 1.44 MB diskette that serves as the "film".  She recorded the video sequence before class, and then played it back on her MacIntosh computer, using the "freeze frame" option to show that the ball left the cart, went up, "stopped" in mid-air, and returned to the cart.  She used a grid on a transparency sheet to obtain quantitative information on the position of the cart at various time intervals.  To show us how simple this is, she played her recorded image back to us on our large TV Monitor, with impressive results. It was suggested that she could relate the "frame rep rate" to real time by recording the image of the second hand of a clock with her camera, either separately or as part of the apparatus.  You showed us how it really should be done,  Betty!

Fred Schaal [Lane Tech High School, Mathematics]      Coding / Decoding and Matrices
showed us how to code and decode a message, based upon matrix multiplication.  He illustrated the procedure for encoding and decoding, using the following highly significant message:

This simple message consists entirely of letters (to form words) and spaces (between words), without punctuation, capital letters, numbers, or other symbols.  Fred first encoded the message by identifying the space [_] and the 26 letters of the alphabet with the numbers 0 through 26, as follows:
_ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
The coded message can thus be converted into a list of numbers:
G O _ L A N E _ H O B B L E _ T H E _ M U S T A N G S _
 7 15 0 12 1 14 5 0  8 15 2 2 12 5 0 20  8 5 O 13 22 19 20 1 14  7 19 0
The next step is to parse the numbers in pairs, to obtain the following sequence of pairs:
| 7 15  |  0 12 |  1 14  | 5 0  |  8 15  |  2 2  | 12 5  | 0 20  |  8 5  | 0 13  | 22 19  | 20 1  | 14 7  | 19 0  |
Fred then asked us to invent a 2 dimensional matrix B of positive integers, with non-zero determinant. We came up with the following modest example:
    B =   | 6  3 |
| 5 2 |
Then we multiplied each of the parsed pairs in the message [treated as a rows] by this matrix, to obtain new pairs. For example:
    [ 7 15 ] | 6  3 |  =  [ 117 51]   AND  [ 0 12 ]  | 6  3 | = [ 60 24]
| 5 2 | | 5 2 |
We continue with to multiply to obtain the coded message [first two words shown]: 
Original Alphabet Message G O | _ L | A N | E _ First Row
Original Message as Numbers    7 15 | 0 12 | 1 14 | 5 0    Second Row
Coded Message 117 51 | 60 24 | 76 31 | 30 15 Third Row
How do we decipher the coded message contained in the third row? To do so, we first construct the inverse of the matrix B, which is
A =  | -2/3   1  | = B-1  
| 5/3 -2 |
Note that A B-1 = I, the 2-dimensional identity or unit matrix.  To decipher the message, multiply the matrix A by the each of coded column pairs. For example
    [ 117 51 ] | -2/3  1 |  =   [7 15]     AND  [ 60 24 ] |-2/3  1 | = [ 0 12]
| 5/3 -2 | | 5/3 -2 |
In summary, we recover the decoded numbers on the second row, and then may convert it back to the letters of the first row. Fred said that it would be unwise for potential spies to use anything below a 5-dimensional matrix for transmitting coded messages. Here, as in many contexts, bigger is better. This is a standard "matrix code" which is simple to decode if you know the matrix, but takes some time to crack for a large matrix. The matrix must frequently be changed, of course, to guard against cracking. Relatively primitive codes of this type were used for communications in World War II. Porter Johnson mentioned the book Between Silk and Cyanide: A Codemaker's War 1941-1945 by Leo Marks [Free Press 1999]  ISBN 0-6848-64223, which describes the experiences of a British cryptographer.  His codes for communicating with operatives behind Axis lines, based upon limericks and poems, were printed on parachute silk (You can guess what the cyanide pills were for!).  A great way to motivate students to learn matrix operations, Fred!

Professor Eduardo De Santiago [Civil and Architectural Engineering, IIT]    Bridge Design
Eduardo De Santiago
made his fourth annual presentation before SMILE and guest students and teachers on "How to be a structural engineer in one lesson"! He began by posing the following difficult question:

When and where will a given contest bridge fail?
He remarked that the answer to this question depends upon the details of the contest rules, craftsmanship in constructing the bridge, and other factors, although it seems that all good bridges up to now have been truss bridges. We will not repeat the discussion of why truss bridges are good, but refer to the relevant SMILE write-ups of 1999 [ph120799.htm], 2000 [mp112100.htm] and 2001 [mp112001.htm].  [See also the Bridge Building Contest Home Page: http://bridgecontest.phys.iit.edu/] Instead, we will simply list the relevant points that he made, in bullet form. A great deal of information is provided at the West Point Bridge Design Contest website, http://bridgecontest.usma.edu/index.htm.  In particular, you can design your bridge, and test it to find how and when it will fail. Also, you can download the following packet from that website:
Designing and Building File-Folder Bridges:  A Problem-Based Introduction to Engineering by Stephen J Rossler
This book provides students with an opportunity to learn how engineers use math, science, and technology to design real structures. It is intended primarily for high school students, but those in lower grades should be able to complete all but Learning Activity #3, which requires the application of geometry, algebra, and some basic trigonometry. A windows-based software package is also available at that website; see  http://bridgecontest.usma.edu/download.htm.

Ann Brandon, Larry Alofs and Bill Blunk were unable to do their presentations due to lack of time, but will be scheduled for next time.  See you then!

Notes taken by Porter Johnson