Mathematics Department Extra Credit Writing Projects for Winter 2007
Below are some ideas for writing projects for which your math instructor might give you extra credit.
We continue to publish the best projects in The Right Angle, monthly newsletter of the Mathematics Department. You should feel free to contact the editor, Randy Schwartz, who can offer advice and sources of information such as those mentioned below (office LA-563, telephone/voicemail 462-4400 ext. 5290, e-mail email@example.com).
A number of the topics would be suitable for all students, even those in lower-level courses (including Math 45 and 53):
- Women and Mathematics
- Climate change and global warming
- gelosia (lattice) multiplication
- double-entry bookkeeping
- Euler’s polyhedral formula.
In writing your paper, you shouldn’t be satisfied with a general description of the topic. You should actually dig into it, explain what is involved, and preferably get into some actual mathematics using examples that help clarify it. Otherwise, what you and your readers learn will be superficial.
Also, remember that writing a research paper, even a brief one, doesn’t mean splicing together phrases and sentences that you find in other people’s books, articles, and websites. Of course, you’ll need to gather information and to list the sources where you found it, but you’ll also need to:
- analyze the information (break the facts and ideas down into their parts in order to see how they relate to one another)
- synthesize the information (put facts and ideas together in a systematic way in order to get a comprehensive understanding of the topic)
- organize and write the paper (figure out the best way, using your own words and understanding, to unfold what you’ve learned so that the readers get a clear understanding, too).
The Writing Fellows website at http://www.schoolcraft.edu/fellows has useful guidance in how to go about doing this.
WOMEN AND MATHEMATICS
In conjunction with Women’s History Month, which is celebrated every March, we invite you to investigate the life and contributions of an important woman mathematician. Some examples include:
- Gabrielle Emilie du Châtelet (1706-1749) was famous for her work on the dynamics of fire, and for her translation of Newton’s work in physics from English into French. There have been a number of biographies, including two recent ones: David Bodanis, Passionate Minds: The Great Love Affair of the Enlightenment, Featuring the Scientist Emilie Du Châtelet, the Poet Voltaire, Sword Fights, Book Burnings, Assorted Kings, Seditious Verse, and the Birth of the Modern World (New York: Crown Publishers, 2006) [Bradner Library PQ 2103 .D7 B63 2006]; and Judith P. Zinsser, La Dame d’Esprit: A Biography of the Marquise Du Châtelet (New York: Viking, 2006).
- Mayme I. Logsdon (1881-1967) was a specialist in algebraic geometry who taught at the University of Chicago between 1921 and 1946. A starting point for reading about her is the chapter by Judy Green and Jeanne LaDuke, “Contributors to American Mathematics”, in Gabriele Kass-Simon, Patricia Farnes and Deborah Nash, eds., Women of Science: Righting the Record (Bloomington : Indiana University Press, 1990).
- Anneli Cahn Lax (1922-1999) was a professor at New York University and distinguished editor at the Mathematical Association of America. Start with the remembrance written by Mark Saul in Notices of the American Mathematical Society 47:7 (August 2000), pp. 766-769, available at http://www.ams.org/notices/200007/mem-lax.pdf
- Ingrid Daubechies (born approx. 1954) is a Belgian-born Princeton University mathematician who has been a pioneer in the new field of wavelet analysis. Wavelets are being applied wherever masses of data must be compressed (such as in high-definition television), or where visual or audio patterns must be recognized (such as in fingerprint identification). The best place to start is an annotated interview of Daubechies published by Deana Haunsperger and Stephen Kennedy, “Coal Miner’s Daughter”, in Math Horizons (April 2000, pp. 5-9, 28-30).
Your report should answer these questions:
- What were the mathematical contributions made by this woman?
- What motivated her to pursue mathematical work?
- How did she surmount all of the obstacles that traditionally barred women from pursuing advanced mathematical study?
MATHEMATICS AND THE ENVIRONMENT
Earth Day on April 22 reminds the world of its responsibility to protect the Earth, and to conserve and wisely manage its natural resources. We invite you to investigate one of the following ways in which mathematics is used to study these environmental resources.
Climate change and global warming
How have earth climates changed over geologic time? Are greenhouse gas emissions and other human industrial activities responsible for an overall warming of the earth’s atmosphere in modern times? Answering this controversial question is difficult, in part because climate varies naturally over time, even without human interference, and in part because the underlying mechanisms of climate change are very complex. Mathematical modeling is an important part of answering the challenge of global warming. University of Michigan professor of geological sciences Dr. Henry N. Pollack is a nationally known expert in this field. He and his colleagues have gathered geothermal temperatures at sites around the world to reconstruct the history of Earth’s surface temperatures and to help predict future trends. For information, see:
- Henry N. Pollack and Shaopeng Huang, “Underground Temperatures Reveal Changing Climate”, Geotimes 43:8 (August 1998).
- Univ. of Michigan press release, “New Evidence of Global Warming”, San Diego Earth Times Jan. 1998, available at http://www.sdearthtimes.com/et0198/et0198s2.html.
- Henry N. Pollack and David S. Chapman, “Underground Records of Changing Climate”, Scientific American June 1993, pp. 16-22.
The importance of biodiversity
Why does life in a given area become precarious when the number and variety of species fall below a certain point? Mathematics helps us answer this question. Here are some resources to investigate:
Lord Robert May, formerly of Oxford University, studies communities of interacting plants and animals, especially their food chains, then applies mathematical concepts to understand how species diversity affects the stability and complexity of these ecosystems. Articles about his work are available at:
- Dr. G. David Tilman at the Univ. of Minnesota uses mathematics extensively in analyzing the results of his field experiments in biodiversity. The New York Times has carried many articles about his work, such as William K. Stevens, “Ecologist Measures Nature’s Mosaic, One Plot at a Time”, New York Times 6 October 1998 (Science section).
Models of environmental pollution
Environmental analysts use a variety of mathematical methods to assess the dangers of, and losses caused by, air, water, and land pollution. These methods include cost-benefit analysis, the environmental Kuznets curve, and quantitative risk assessment (QRA).
- An article that summarizes the use of cost-benefit curves and the environmental Kuznets curve to model the relation between societal wealth and pollution, and the appropriate level of sensitivity to environmental threats, is S. W. Pacala et al., “False Alarm over Environmental False Alarms”, Science 301 (Aug. 29, 2003), pp. 1187-1188.
- A textbook that explains algebraic and computer techniques in investigating the threat of groundwater contamination, air pollution, and hazardous material emergencies in Charles Hadlock, Mathematical Modeling in the Environment (Washington, D.C.: Mathematical Association of America, 1998).
- The EPA’s use of quantitative risk assessment to gauge the risk of cancer from pollutants emitted in the air by the coke ovens of Ostrava (Czech Republic) is discussed in Jeff Wheelwright, “The Air of Ostrava”, Discover 17:5 (May 1996), pp. 56-67.
Sustainable harvests with linear algebra
Models based on matrix algebra are used to predict the growth rate of a population and the long-term trends for its age distribution. This can allow forest and livestock rangers, and the managers of fish, game and wildlife preserves, to design sustainable harvesting policies. A discussion is provided in Howard Anton and Chris Rorres, Elementary Linear Algebra: Applications Version, 9th edition (New York: John Wiley & Sons, Inc. 2005), pp. 648-657, 743-761. [This is the textbook used for Math 230 at Schoolcraft. Similar discussions are provided in other Linear Algebra textbooks]
Balancing predator and prey with differential equations
In 1925-26, population ecologists Alfred J. Lottka and Vito Volterra independently formulated an important model to describe the population fluctuations between a predator species and a prey species, like foxes and rabbits, or bass and sunfish, or ladybugs and aphids. The Lottka-Volterra model is a system of two nonlinear differential equations, and it predicts that in many cases, the populations of predator and prey will mutually stabilize by cycling between two extremes. See if you can address these questions in your paper:
- What practical problems prompted Lottka and Volterra to formulate their model?
- Explain the assumptions that were made and the variables and equations that were used to define the model.
- Describe the different mathematical solutions of the equations. What ecological behavior is predicted by these solutions?
- See if you can implement this model numerically or graphically with a graphing calculator or a computer.
- What are some of the limitations of this model? How has it been extended to apply to other situations?
Among the many good discussions of this model are:
- John Vandermeer, Elementary Mathematical Ecology (New York: Wiley & Sons, 1981), pp. 158-233.
- C. Henry Edwards, Jr., and David E. Penney, Differential Equations and Boundary Value Problems: Computing and Modeling, Third edition (Upper Saddle River, NJ: Prentice-Hall, 2004), pp. 393-406. [This is the textbook used for Math 252 at Schoolcraft. Similar discussions are provided in other Differential Equations textbooks]
MATHEMATICS IN EUROPEAN CULTURE
In conjunction with the Focus Europe project this year at Schoolcraft College, we invite you to write about one of the following topics that highlight ties between mathematics and other facets of European culture, and help explain how modern Europe became pre-eminent in this field.
The Mathematics of Trade and Commerce in Medieval Italy
Capitalism arose in Italy before any other place in the world. As trade and commerce began to flourish there, how did the quantitative problems faced by merchants and bankers push mathematicians like Leonardo Fibonacci to adopt new techniques?
- Joseph and Frances Gies, Leonard of Pisa and the New Mathematics of the Middle Ages (New York: Crowell, 1969).
- Frank Swetz, Capitalism and Arithmetic: The New Math of the 15th Century, Including the Full Text of the Treviso Arithmetic of 1478, Translated by David Eugene Smith (Chicago: Open Court, 1987)
- Laura Ackerman Smoller, website on gelosia (lattice) multiplication, http://www.ualr.edu/lasmoller/medievalmult.html
- Anthony Phillips, “The Romance of Double-Entry Bookkeeping”, http://www.ams.org/featurecolumn/archive/book1.html
Mathematics in Church Architecture
How did the builders and designers of medieval and Renaissance churches use practical geometry, involving ratios and circular arcs, in their work?
- Hugh McCague, “A Mathematical Look at a Medieval Cathedral”, Math Horizons April 2003, pp. 11-15, 21.
- Benno Artmann, “The Cloisters of Hauterine”, The Mathematical Intelligencer 13 (Spring 1991), pp. 44-49.
Geometric Perspective in Renaissance Art
Going back to ancient Greek and Roman times, European artists and designers have used mathematics in a variety ways. What methods were used by Renaissance artists when they invented concepts of visual perspective?
- Erwin Panofsky, “Dürer as a Mathematician” in James R. Newman, ed., The World of Mathematics (Redmond, WA: Tempus, 1988) [Bradner Library QA 3 .W67 1988], Vol. 1, pp. 591-612.
- Anthony Phillips, “Alberti’s Perspective Construction”, http://www.ams.org/featurecolumn/archive/alberti1.html
- Judith Veronica Field, The Invention of Infinity: Mathematics and Art in the Renaissance (New York: Oxford University Press, 1997)
- Judith Veronica Field, Piero Della Francesca: A Mathematician’s Art (New Haven: Yale University Press, 2005); review of this book by Anthony Phillips, http://www.ams.org/notices/200001/rev-phillips.pdf
- Judith Veronica Field, “Piero Della Francesca and Painting as a Science”, http://halshs.archives-ouvertes.fr/view_by_stamp.php?label=REHSEIS&langue=fr&action_todo=view&id=halshs-00004274&version=3 (see 05 Field.tif.pdf near bottom of page)
- Morris Kline, “Projective Geometry”, Scientific American 192:1 (January 1955), pp. 80-85; widely reprinted
Mathematics in European Warfare
The use of mathematics and science to devise the most powerful gunnery, the most accurate rangefinding, and the strongest fortifications, utterly transformed the battlefields of early modern Europe. How did geometry, trigonometry, algebra, and calculus play a role in this new military technology?
- Oxford University’s Museum of the History of Science, “The Geometry of War, 1500-1750”, http://www.mhs.ox.ac.uk/geometry/content.htm.
- C. W. Groetsch, “Tartaglia’s Inverse Problem in a Resistive Medium”, American Mathematical Monthly 103:6 (Aug-Sep 1996), pp. 546-551.
Mathematics in Revolutionary France
The French Revolution that began in 1789 was inspired in part by the scientific revolution, and had a profound impact on mathematicians and scientists. Why did such mathematicians as Joseph Fourier, Gaspard Monge, and Alexandre Vandermonde become strong allies of the Revolution, some even accompanying Napoleon on his conquest of Egypt?
- Carl Boyer, “Mathematicians of the French Revolution” in Frank Swetz, From Five Fingers to Infinity: A Journey Through the History of Mathematics (Chicago: Open Court, 1994), pp. 560-573
- Ken Alder, The Measure of All Things: The Seven-Year Odyssey and Hidden Error that Transformed the World (New York: Free Press, 2002) [Bradner Library QB 291 .A43 2002]
The Marvelous Bernoullis of Switzerland
The Bernoullis were an amazing Swiss family that included at least eight prominent mathematicians (one Daniel, two Jacobs, two Nicolauses, and three Johanns) during the 17th and 18th Centuries. What were their various contributions to math and science, and how did this aptitude get nurtured across three generations?
- Brief biographies of all eight Bernoullis can be found in the MacTutor History of Mathematics Archive, http://www-history.mcs.st-andrews.ac.uk
- James R. Newman, “Commentary on the Bernoullis” in his The World of Mathematics (Redmond, WA: Tempus, 1988) [Bradner Library QA 3 .W67 1988], Vol. 2, pp. 759-761, followed by Daniel Bernoulli’s “Kinetic Theory of Gases”.
- Howard Eves, “The Bernoulli Family” in Frank Swetz, From Five Fingers to Infinity: A Journey Through the History of Mathematics (Chicago: Open Court, 1994), pp. 523-526, followed by William Dunham’s essay “The Bernoullis and the Harmonic Series”
Two Rival Programs for Calculus
The key concepts of modern calculus were developed more or less simultaneously by Isaac Newton in England and Gottfried Leibniz in Germany. Soon, there was a dispute over “who got there first” and, more importantly, the best way to think about and record the findings in this new branch of mathematics. In England, mathematicians kept favoring Newton’s “fluxional calculus”, even though methods developed in Germany and France had some clear advantages. As a result, British mathematics became isolated and was held back for many decades. How did this happen? And what were the problems that led Newton and Leibniz to invent calculus in the first place?
- C. H. Edwards, “The Calculus According to Newton and Leibniz”, in Douglas M. Campbell and John C. Higgins, Mathematics: People, Problems, Results (Belmont, CA: Wadsworth International, 1984), Vol. 2, pp. 104-111
- Dorothy V. Schrader, “The Newton-Leibniz Controversy Concerning the Discovery of the Calculus”, in Frank Swetz, From Five Fingers to Infinity: A Journey Through the History of Mathematics (Chicago : Open Court, 1994 ), pp. 508-522.
The Principle of Maximum Goodness
An interesting dialog between mathematics and philosophy was sparked by a series of natural observations in early modern Europe. In one set of observations, Pierre de Fermat and other mathematicians found that the laws of reflection and refraction of light can be deduced from the basic principle that a ray of light follows the path that minimizes the amount of time required for its travel (the “principle of least time”). Later, Pierre de Maupertuis showed that Newton’s laws of motion can be deduced from the basic principle that a process follows the path that minimizes the amount of energy required for its completion (“the principle of least action”). Maupertuis used this as a launchpad to formulate a more sweeping principle of “optimism” or maximum goodness: God created the best of all possible worlds, and the mathematics of optimization is the key to understanding not only nature but society, history, and morality. Voltaire and others argued strongly against thisidea. What were the terms of this debate, and what were the mathematical underpinnings?
- Ivar Ekeland, The Best of All Possible Worlds: Mathematics and Destiny (Chicago: University of Chicago Press, 2006)
- Kevin Brown, “Stationary Paths” in Reflections on Relativity, http://www.mathpages.com/rr/s3-04/3-04.htm
EULER: THE MASTER OF US ALL
April 15, 2007 will mark the 300th anniversary of the birth of Swiss mathematician Leonhard Euler, one of the greatest of all times. The title of a biography by William Dunham fits well: Euler: The Master of Us All (Washington, D.C.: Mathematical Association of America, 1999). Write a paper about Euler’s life, and describe one or two of his greatest accomplishments, such as the following (sample resources are given, too).
- Euler’s Polyhedral Formula, V – E + F = 2. See Courant and Robbins, “Euler’s Formula for Polyhedra” in James R. Newman, ed., The World of Mathematics (Redmond, WA: Tempus, 1988) [Bradner Library QA 3 .W67 1988], Vol. 1, pp. 573-577; also Joseph Malkevitch, “Euler’s Polyhedral Formula”, Part 1 at http://www.ams.org/featurecolumn/archive/eulers-formula.html, Part 2 at http://www.ams.org/featurecolumn/archive/eulers-formulaII.html.
- Euler’s formula for the volume of a tetrahedron. See Heinrich Dörrie, 100 Great Problems of Elementary Mathematics (New York: Dover, 1965), pp. 285-289.
- Euler’s solution of the Seven Bridges of Königsberg problem. See Gerald L. Alexanderson, “Euler and Königsberg’s Bridges: A Historical View”, Bulletin of the American Mathematical Society 43:4 (Oct. 2006), pp. 567–573, also available at http://www.ams.org/bull/2006-43-04/S0273-0979-06-01130-X/S0273-0979-06-01130-X.pdf. The original work by Euler can be found in James R. Newman, ed., The World of Mathematics (Redmond, WA: Tempus, 1988) [Bradner Library QA 3 .W67 1988], Vol. 1, pp. 561-571.
- Euler’s proof that every perfect number is of form 2n-1(2n – 1)
- Euler’s formula exi = cos(x) + isin(x), and the resulting Euler Identity, eπi + 1 = 0
- Euler’s solution of a variety of differential equations by exact or approximate methods
- Euler’s invention of the gamma function. See Philip J. Davis, “Leonhard Euler’s Integral: A Historical Profile of the Gamma Function”, American Mathematical Monthly 66:10 (Oct. 1959), pp. 849-869.
FOURIER’S GREAT DISCOVERY
The year 2007 marks the 200th anniversary of the publication of Joseph Fourier’s great treatise on heat, in which he first described the technique that would come to be known as Fourier analysis. Write a paper on Fourier’s life and work. A good starting point is the online biography by O’Connor and Robertson (see below), which also lists many references. You should also try to focus on answering the following questions:
- What was the impact on Fourier’s life and work due to the French Revolution and Napoleon’s actions in France and in Egypt?
- What is Fourier analysis? How was this technique useful to Fourier in analyzing heat flow? What other applications to mathematics, engineering and physics has it seen in the past two centuries?
Some helpful resources are:
- Bochner, S., “Fourier Series Came First”, American Mathematical Monthly 86:3 (Mar. 1979), pp. 197-199.
- O’Connor, John J., and Edmund F. Robertson, “Jean Baptiste Joseph Fourier”, St. Andrews Univ., available at http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fourier.html
- Shoucair, F. S., “Joseph Fourier’s Analytical Theory of Heat: A Legacy to Science and Engineering”, IEEE Transactions on Education 32 (1989), pp. 359-366.
CAUCHY: A PRODIGIOUS LIFE IN MATHEMATICS
May 23, 2007 will mark the 150th anniversary of the death of Augustin Louis Cauchy, one of the most important and prolific mathematicians of all time. Born in Paris during the first year of the French Revolution, Cauchy would go on to write an astounding total of 789 mathematics papers— his collected works filled 27 volumes! He contributed to virtually every branch of mathematics and to many related fields in physics. Write a paper on Cauchy’s life and work. A good starting point is the online biography by O’Connor and Robertson (see below), which also lists many references. You should also try to focus on answering one or more of the following questions:
- What effect did social and political events have on Cauchy’s life and work?
- What is calculus, and how did its character change as a result of Cauchy’s work?
- Summarize Cauchy’s work in the theory of matrices and their eigenvalues.
Some helpful resources are:
- Grabiner, Judith V., “Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus”, American Mathematical Monthly 90:3 (Mar. 1983), pp. 185-194.
- Hawkins, Thomas “Cauchy and the Spectral Theory of Matrices”, Historia Mathematica 2:1 (Feb. 1975), pp. 1-29.
- Nahin, Paul J., Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills (Princeton: Princeton University Press, 2006)
- O’Connor, John J., and Edmund F. Robertson, “Augustin Louis Cauchy”, http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cauchy.html
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