Undergraduate Projects

Many thanks to everyone who sent me problems (unfortunately I didn't keep a record of all names so I hope cmath-l members will forward this to members of their departments).

I now have about 60 projects (Latex file attached). I would appreciate it very much if people could take a look through - make corrections, improve what is there, add some references, and if you get a new idea add that too.

This will then be distributed to schools during the year and hopefully next year we will have quite a few math projects at science fairs across the country - and here's the next request.

Offer to be a judge at your local science fair - we will need mathematicians available to evaluate the projects.

Again, thanks a lot!!! Kathy

Note: In order to run this LaTeX file, use your browser to see the source file and copy the portion starting with the word \documenstyle and continue to the words \end{document}. Save it as text. If you do not have 12pt,fullpage.sty then just erase the words 12pt,fullpage leaving \documentstyle[]{article} and it will run in LaTex, at least it did for me. If you have trouble, e-mail to me. Afton

\documentstyle[12pt,fullpage]{article} \begin{document} \begin{center} POSSIBLE SCIENCE FAIR MATHEMATICS PROJECTS \end{center} What follows is a selection of ideas for science projects. In most cases only a very brief outline is presented (sometimes with a reference) in order to leave students plenty of scope for what they do. It is not expected that the problem stated will necessarily be the project. These are ideas intended to get people thinking (they are in no particular order). \vspace{.5cm} \begin{enumerate} \item At certain times charities call households offering to pick-up used items for sale in their stores. They often do a particular geographical area at a time. Their problem, once they know where the pick-ups are, is to decide on the most efficient routes to make the collection. Find out how they do this and investigate improving their procedure. A similar question can be asked about snow plows clearing city streets, or garbage collection. {\em References}: Euclidean tours, chinese postman problem - information can be found in most books on graph theory but one of particular interest is ``Introduction to Graph Theory'' by G. Chartrand. \item How should one to locate ambulance stations, so as to best serve the needs of the community? The reference given above may help. \item An International Food Group consists of twenty couples who meet four times a year for a meal. On each occasion, four couples meet at each of five houses. The members of the group get along very well together; nonetheless, there is always a bit of discontent during the year when some couples meet more than once! Is it possible to plan four evenings such that no two couples meet more than once? There are many problems like this. They are called combinatorial designs. Investigate others. \item How does the NBA work out the basketball schedule? How would you do such a schedule bearing in mind distances between locations of games, home team advantage etc.? Could you devise a good schedule for one of your local competitions? \item How do major hospitals schedule the use of operating theatres? Are they doing it the best way possible so that the maximum number of operations are done each day? \item Investigate ``big'' numbers. What is a big number? The following examples might guide your investigation. A bank is robbed of 1 million loonies. How long it would take to move that many? How much it would weigh? How much space would it take up? How big a swimming pool do you need to contain all the blood in the world? Is $10^100$ very big? What is the biggest number anyone has ever written down (check the Guiness book of world records over the last few years)? How did this number come about? \item Build a phsical model based on dissections to prove the Pythagorean Theorem. Build an exhibit on the Pythagorean theorem but with "The semicircle on the hypotenuse ..." \item What is the fewest number of colours needed to colour any map if the rule is that no two countries with a common border can have the same colour. Who discovered this? Why is the proof interesting? What if Mars is also divided into areas so that these areas are owned by different countries on earth. They too are coloured by the same rule but the areas there must be coloured by the colour of the country they belong to. How many colours are now needed? {\em Reference:} Joan Hutchinson, ... \item Study the golden mean, its appearance in art, architecture, biology, and geometry, and it's connection with continued fractions, fibonacci numbers. What else can you find out? What is the Golden Mean? \item Study the regular solids (platonic and Archimidean), their properties, geometry, and occurance in nature (e.g. virus shapes, fullerene molecules, crystals). Build models. \item Study the cycloid curve: its tautochrone and brachistochrone properties and its history. Build models. \item Infinity comes in different ``sizes''. What does this mean? How can it be explained? {\em References:} Refer to either of the Dover paperbacks, ``Theory of Sets'' by Kamke, and ``The Continuum and other types of Serial Order'' by Huntington, or any book on Set Theory. \item Investigate visual representations of different finite numbers. For example, if p is a prime with 100 digits, then if 1 and p are on the same line segment, with p say 6 inches to the right of 1, then $\sqrt{p}$ is about $10^{-50}$ inches to the right of 1, less than one atom away. (And it's by inspecting the lattice points in the $\sqrt{p} \times \sqrt{p}$ array that one proves that p is the sum of two squares!) Investigate further. \item Discover all 17 ``different'' kinds of wallpaper. (Think about how patterns on wallpaper repeat.) How is this related to the work of Escher? Discover the history of this problem. {\em references:} G.C. Shephard, ``Additive Frieze patterns and multiplication tables'', Math. Gaz. 60(1976) p179-184; H.S.M. Coxeter, ``Frieze Patterns'', Acta Arithmetica, XVIII(1971) p297-310; and J.H. Conway and H.S.M. Coxeter, ``Triangulated Ploygons and frieze patterns'', Math. Gaz. 57(1973) p87-94 (questions), 175-183 (answers). \item Study games and winning strategies - maybe explore a game where the winning strategy is not known. Analyze subtraction games (nim-like games in which the two players alternately take a number of beans from a heap, the numbers being restricted to a given subtraction set). {\em References:} E.R. Berlekamp, J.H. Conway, R.K. Guy, ``Winning Ways'', Academic Press, London (this book contains hundreds of othr games for which the complete analysis is unknown eg. Toads and Frogs) ; R. Guy (editor), ``Combinatorial Games'', Proceeedings of Symposia in Applied Math, AMS publication (pay special attention to the last section where lots of questions are asked). \item Most computers these days can handle sound one way or another. They store the sound as a sequence of numbers. Lots of numbers. 40,000 per second, say. What happens when you play around with those numbers? eg. Add 10 to each number. Multiply each number by 10. Divide by 10. Take absolute values. Take one sound, and add it to another sound (i.e. add up corresponding pairs of numbers in the sequences). Multiply them. Divide them. Take one sound, and add it to shifted copies of itself. Shuffle the numbers in the sequence. Turn them around backwards. Throw out every third number. Take the sine of the numbers. Square them. For each mathematical operation, you can play the resulting sound on the computers speakers, and hear what change has occurred. A little bit of programming, and you can get some very bizarre effects. Then try to make sense of this from some sort of theory of signal processing. \item Investigate self-avoiding random walks and where they naturally occur. {\em Reference}: : G. Slade, ``Random walks'', American Scientist, March-April, 1996. \item Investigate the space shuttle's failed attempt to put a tethered satellite into orbit. \item Draw, and list any interesting properties of, various curves: evolutes, involutes, roulettes, pedal curves, conchoids, cissoids, strophoids, caustics, spirals, ovals, ... {\em Reference}: Cundy and Rollett, ``Mathematical Models'', Oxford (which has lots of other ideas, too); E.H. Lockwood, ``A Book of Curves'', Cambridge; and there's also a book by Yates, ``Curve Tracing''(?). \item Make a family of polyhedra, e.g., the Archimedean solids, or Deltahedra (whose faces are all equilateral triangles), or equilateral zonohedra, or, for the very ambitious, the 59 Isocahedra. {\em Reference:} See any Coxeter revision of Rouse Ball's ``Mathematical Recreations and Essays'' (which is full of many ideas). There's also Coxeter, DuVal, Flather and Petrie, ``The 59 Icosahedra'', U of Toronto Press; Magnus J Wenninger, ``Polyhedron Models'', Cambridge, 1971; and Doris Schattschneider and Wallace Walker, ``M.C. Escher Kaleidocycles'', Pomegranate Art Books, 1987. \item Find as many triangles as you can with integer sides and a simple linear relation between the angles. What about the special case when the triangle is right-angled? \item Find out all you can about the Fibonacci Numbers, 0, 1, 1, 2, 3, 5, 8, ... \item Find out all you can about the Catalan Numbers, 1, 1, 2, 5, 14, 42, ... \item What is Morley's triangle? Draw a picture of the 18 Morley triangles associated with a given triangle ABC. Find the 18 more for each of the triangles BHC, CHA, AHB, where H is the orthocentre of ABC. Discover the relation with the 9-point circle and deltoid (envelope of the Simson or Wallace line). \item What is a hexaflexagon? Make as may different ones as you can. What is going on? {\em Reference:} Martin Gardner, ``Hexaflexagons and other Mathematical Diversions'', Univ. of Chicago Press, 1988. \item Investigate trianglar numbers. If that's not enough, do squares, pentagonal numbers, hexagonal numbers, etc Venture into the third and even the fourth dimension. {\em Reference:} Conway and Guy, ``The Book of Numbers'', Springer, Copernicus Series, 1996, Chapter 2. \item Ten frogs sit on a log - 5 green frogs on one side and 5 brown frogs on the other with an empty seat separating them. They decide to switch places. The only moves permitted are to jump over one frog of a different colour into an empty space or to jump into an adjacent space. What is the minimum number of moves? What if there were 100 frogs on each side? Coming up with the answers reveals interesting patterns depending on whether you focus on colour of frog, type of move, or empty space. Proving it works is interesting also - it can lead to recursion, there is also a simple proof that is not immediately obvious when you start. Look for and explore other questions like this - one of the most famous is the Tower of Hanoi. \item Investigate the creation of secret codes (ciphers). Find out where they are used (today!) and how they are used. Look at their history. Build your own using prime numbers. {\em Reference:} M. Fellows and N. Koblitz. ``Kid krypto.'' Proc. CRYPTO '92, Springer-Verlag, Lecture Notes in Computer Science vol. 740 (1993), 371--389. \item There is a well-known device for illustrating the binomial distribution. Marbles are dropped through the top and encounter a number of pins before dropping into cells where they are distributed according to the binomial distribution. By changing the position of the pins one should be able to get other kinds of distributions (bimodal, skewed, approximately rectangular, etc.) Explore. \item Build rigid and nonrigid geometric structures. Explore them. Where are rigid structures used? Find unusual applications. This could include an illustration of the fact that the midpoints of the sides of a quadrilateral form a parallelogram (even when the quadrilateral is not planar). Are there similar things in three dimensions? \item Build a true scale model of the solar system - but be careful because it cannot be contained within the confines of an exhibit. Illustrate how you would locate it in your town. Maybe even do so!! \item Build models to illustrate asymptotic results such as Stirling's formula or the prime number theorem. \item What is/are Napier's bones and what can you do with it/them? \item Covering a chessboard with dominoes so that no two dominoes overlap and no square on the chessboard is uncovered. Consider (a) a full chessboard, (b) a chessboard with one square removed (impossible - why?), (c) a chessboard with two adjacent corners removed, (d) one with two opposite corners removed (possible or impossible?), (e) A chessboard with any two squares removed. What about using shapes other than dominoes (eg 3 $1 \times 1$ squares joined together)? What about chessboards of different dimensions? {\em Reference:} ``Polyominoes'' by Solomon W. Golumb, pub. Charles Scribner's Sons \item Build models showing that parallelograms with the same base and height have the same areas (is there a 3-dimensional analogue?). This can lead to a purely visual proof of the Pythagorean theorem. The formula for the area of a circle can also be presented in this way. {\em R eference:} H. R. Jacobs,`` Mathematics a Human Endeavor'', 3rd ed, p 38) \item Use Monte Carlo methods to find areas (rather than using random numbers, throw a bunch of small objects onto the required area and count the numbers of objects inside the area as a fraction of the total in the rectangular frame) or to estimate pi. \item Find pictures which show that $1 + 2 + \cdots + n = \frac{n(n+1)}{2}$; that $1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$; and that $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$. How many other ways can you find to prove these identities? Is any one of them ``best''? \item What is fractal dimension? Investigate it by exaomining examples showing what happens when you double the scale to (a) lines (b) areas (c) solids (d) the Koch curve. \item Knots. What happens when you put a knot in a strip of paper and flatten it carefully? When is what appears to be a knot really a knot? Look at methods for drawing knots. \item Is there an algorithm for getting out of 2-dimensional mazes? What about 3-dimensional? Look at the history of mazes (some are extraordinary). How would you go about finding someone who is lost in a maze (2 or 3 dimensional) and wandering randomly? How many people would you need to find them? \item Investigate the history of pi and the many ways in which it can be approximated. Calculate new digits of Pi - see: http://www.cecm.sfu.ca/people/Peter\_Borwein/ to discover what this means. \item What is game theory all about and where is it applied? \item Construct a Kaleidoscope. Investigate its history and the mathematics of symmetry. \item Consider tiling the plane using shapes of the same size. What's possible and what isn't. In particular it can be shown that any 4-sided shape can tile the plane. What about 5 sides? Look for books and articles by Grunbaum and Shepherd, and check the Martin Gardber books. \item Explore Penrose tiles and discover why they are of interest. \item Investigate the Steiner problem - one application of which is concerned with the location of telphone exchanges to minimize costs. \item Look for new strategies for solving the travelling salesman problem. \item Explore egyptian fractions. \item How do computer bar codes (the ones you see on everything you buy) work? This is an example of coding theory at work. Find others. Investigate coding theory - there are many books with titles like ``an introduction to coding theory'' (this is not about secret codes). {\em references}: Joe Gallian, `` How computers can read and correct ID numbers'', Math Horizons, Winter, 1993, p14-15; Joe Gallian, ``Assigning Driver's License Numbers'', Mathematics Magazine, 64 (1991), 13-22; and Joe Gallian, ``Math on Money'', Math Horizons, November, 1995, p10-11. \item The Art Gallery problem: What is the least number of guards required to watch over all paintings in an art gallery? The guards are positioned at specific locations and collectively must have a direct line of sight to every point on the walls. {\em reference}: Alan Tucker, ``The Art Gallery Problem'', Math Horizons, Spring, 1994, p24-26 \item The Parabolic Reflector Microphone is used at sporting events when you want to be able to hear one person in a noisy area. Investigate this; explaining the mathematics behind what is happening. \item There is a traditional Chinese way of illustrating the Pythagorean theorem using paper. Investigate and make models. \item Use PID (proportional-integral-differential) controllers and oscilloscopes to demonstrate the integration and differentiation of different functions. \item Try the "Monty Hall" effect. Behind one of three doors there is a prize. You pick door \#1, he shows you that the prize wasn't behind door \#2 and then gives you the choice of switching to door \#3 or staying with \#1, what should you do? Why should you switch? Make an exhibit and run trials to ``show'' this is so. Find the mathematical reason for the switch. \item Look at the ways different bases are used in our culture and how they have been used in other cultures. Collect examples: time, date etc. Demonstrate how to add using the Mayan base 20, maybe compare to trying to add with Roman numerals (is it even possible?) \item Explore the history and use of the Abacus. \item Investia]gate card tricks. Some of the best in the world were designed by the mathematician/statistician Persi Diaconis. {\em Reference:} Don Albers, `` Professor of (Magic) Mathematics'', Math Horizons, February 1995, p11-15 \item Explore magic tricks based in Mathematics (again see the article about Persi Diaconis). \item Investigate compass and straight-edge constructions - showing what's possible and discussing what's not. For example, given a line segment of length one can you use them straight edge and compass to ``construct'' all the radicals? \item Chaos and the double pendulum. \item There are several books that have a variety of articles that can be used to generate projects: John Mason with Leone Burton and Kaye Stacey, ``Thinking Mathematically'', revised edition, Addison-Wesley , 1985 Cliff Sloyer, ``Fantastiks of Mathematics: Applications of Secondary Mathematics'', Janson Publications, Inc., Providence, R.I., 1986. ISBN 0- 939765-00-4. Paul Hoffman, ``Archimedes Revenge'', Ballantine Books T.F Banchoff (in Steen, L.A. , ed.) (1990). ``On the shoulders of giants: new approaches to numeracy'', National Academy Press, Washington, D.C Nancy Casey and Mike Fellows (1993). `` This is mega-mathematics: stories and activities for mathematical thinking, problem-solving and communication'', The Los Alamos National Laboratory, Los Alamos, New Mexico Arthur L Loeb, "Concepts and Images, Visual Mathematics" Paulos, J.A. (1991). ``Beyond numeracy: ruminations of a numbers man'', Alfred A. Knopf, New York. And then of course there are all the Martin Gardner books. \end{enumerate} \end{document}

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