Math in Everyday Lives

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    Consider this physical experiment: Start

    with a circle of wire that has been twisted,bent, and stretched into some new shape.If we dip it i nto soapy water and pull it outagain, there will be a soap film stretchingacross it. Physical tensions make the filmwant to have the least possible area while

    still spanning the wire f rame.Here is the mathematical Plateau

    Problem: You are given a bent circularcurve in three-dimensional space, like thewire. There are many different possibletwo-dimensional surfaces touching the

    entire given curve, like attached sheets ofplastic wrap. Show that there must be onethat has the smallest total area.

    Solving the Plateau Problem requiresproving that there exists an optimal least areasurface for every set of parameters, i.e. the

    given curve. It is similar to showing the exis-tence of a solution to the soup can problem.The trouble is that instead of the solutionbeing two numbers (the radius and height ofthe best soup can), the solution comes froman infinite dimensional space of surfaces.

    Various precise versions of this questionhave been answered, starting in the 1930swith the awarding of the first ever FieldsMedal, and it has inspired vast generaliza-tions and beautiful mathematics. Currentlythere are hundreds of mathematicians around

    the world (including undergraduates) workingon problems in minimal surfaces (soap films)and bubble clusters, with many questionsboth big and small still unresolved.

    To learn more about soap films, visitmy Web site at http://www.smcm.edu/users/

    ammeadows/create/.

    Paper Cranes and Satellites

    by Susan Goldstine, Assistant Professor of Mathematics

    Page 10 River Gazette

    Now boarding rows 25 to 30

    boarding rows 20 and up. Wethis monotonous call as we patto board a plane. When your rwins this airplane boarding loshuffle down the gangway onlyget to the plane, wait while you

    sengers stow their luggage, andmore for those in aisle seats to

    allow the window and middle-gers to slide i nto their crampedquarters. The whole process seforevercouldnt it be faster?

    Actually, it could!Using a mathematical mod

    a computer simulation, we comvariety of boarding strategies. Bpassengers in a random order wboarding group and running a

    of simulations (3,000 for each sdetermined the mean, best-, anboarding times. The results wegiven our experiences with theefficiency-conscious airline ind

    The fastest way to get pas

    a plane is to board them from in, starting with the window pthen the unlucky middle-seat finally the aisle passengers. Susave about five minutes on avepared to the back-to-front stra

    How can it be that mathemat-ics, being after all a product ofhuman thought independentof experience, is so admira-bly adapted to the objects ofreality? asked Albert Einstein.Its a remarkable question thatcaptures the dual wonders ofthat discipline we call mathe-matics: its eternal mystery andits supreme practicality.

    On these pages, fac-ulty from St. MarysDepartment of

    Mathematics share stories thatillustrate some of the ways in

    which complex mathematicalideas have a decidedly realimpact on our everyday lives.Ever wondered if there was amore efficient way to board anairplane? What about thoseroadmaps that, once unfurled,seem to defy our best attemptsto refold themisnt there abetter way? Mathematics, itseems, holds the answers.

    Best of all, you dont haveto be a math whiz to appreci-ate how mathematics trans-forms our worldin both big

    ways and smallor to marvelat the purity and beauty ofideas that are miracles in andof themselves.

    Origami is the traditional Japanese artof paperfolding. One of its most com-

    mon manifestations is the paper crane, along-necked bird with tapered wi ngs thatcan be folded by a novice with a littleorigami training. Masters of the art formcan take a single square sheet of paperand fold it without cuts into an endless

    variety of forms: gardens of flowers, swarmsof insects, animals, seashells, and almostanything else real or imagined.

    Within the past few decades, math-ematicians have developed a rigorousstudy of the mathematical properties of

    paperfolding and its applications to suchdiverse fields as art, algebra, and industry.One area of interest is the investigation ofrigid origami, which poses the question:when can a creased sheet of paper be foldedalong its creases while the paper itself stays

    perfectly flat? Put another way, could agiven origami model be folded from piecesof stiff metal with hinges between them?

    How does mathematics help us

    understand the behavior of a physical

    phenomenon like a tsunami? Mathema-

    ticians often will talk about an equation

    describing the phenomenon. This lan-

    guage means that the equation provides

    a relationship between variables such as

    wave speed or wavelength and physical

    parameters such as gravity and ocean

    depth. Such a relationship can be used

    to approximate the future behavior of

    the wave itself.

    For example, the usual approxima-

    tion for wave speed in the open ocean

    is a formula; namely, speed is propor-

    tional to the square root of the force

    of gravity times the ocean depth. Such

    quantifiable relationships allow scien-

    tists to generate warnings of approach-

    ing tsunamis, with predictions of both

    arrival times and wave size, so that

    coastal communities may be able to call

    for evacuations.

    The description process is compli-

    cated by the fact that the shape of the

    wave can change in response to changes

    in the physical environment. A tsunami

    in the open ocean is a barely percep-

    tible change in sea surface elevation,

    over a range of miles. As the tsunami

    approaches the shore, however, the

    increasingly perceptible presence of thesea floor begins to affect the shape of the

    wave, leading to the development of the

    wall of water that most people associ-

    ate with the word tsunami.

    Hence, the equations used to

    describe tsunamis are applicable in dif-

    ferent situations: one kind of equation

    applies for tsunamis out in the open

    ocean, where the depth is very large, but

    a second set of equations applies to near-

    shore situations, where the shape of the

    wave is affected by the drag across the

    sloping, shallower sea floor. One current

    applied mathematics research problem

    is to develop a good description of the

    transition between the open ocean and

    the near-shore zone.

    Math in Our Everyday Lives

    A key innovation in this area is theMiura map fold, discovered by Japanese

    astrophysicist Koryo Miura. A cunning varia-tion on the traditional accordion-pleat mapfold, the Miura fold allows a large map to becollapsed into a very compact package andreopened instantly without twisting the paperas it opens. In fact, the Tokyo subway sells

    maps that are folded with the Miura method.However, the fold was originally developedfor operation on a grander scale. Miuradesigned his map fold so that satellites couldcarry their massive solar panels into outerspace in a packet small enough to fit inside a

    rocket capsule and deploy the panels in orbit.

    Source:Thomas Hull, Project Origami, A KPeters, Ltd., 2006

    Tsunami Predictionby Katherine Socha, Assistant Professor of Mathematics

    Soup Cans and Soap Filmby Alex Meadows,

    Assistant Professor of Mathematics

    Here is a typical problem

    given to first-year calcu-lus students: A soup company wants topackage its soup efficiently using cylindricalcans. Each can must be 20 cubic inches involume. The metal for the top and bot-tom discs of the can costs fifteen cents per

    square inch, while the metal for the sidecosts twelve cents per square inch. What

    shape is the cheapest can?This is called an optimization problem

    Such quantifiable relation-ships allow scientists to

    generate warnings of ap-proaching tsunamis, withpredictions of both arrivaltimes and wave size, sothat coastal communities

    may be able to call for

    evacuations.

    Aby David Ku

    Simon Read

    because the solution is somehow the best.

    Sometimes optimization problems do nothave solutions. For example, if we elimi-nate the above requirement that the canhold 20 cubic inches of soup, then there isno cheapest can because smaller cans arecheaper and there is no smallest can.

    One of the great mathematical ques-tions that existed at the turn of the twenti-

    eth century was a question about geometricoptimization, called the Plateau Problem.

    Within the past few decades, mathematicians have developed arigorous study of the mathematical properties of paperfolding andits applications to such diverse fields as art, algebra, and industry.

    Brainteas

    PhotobyBarbaraWoodel

    PhotobySusanG

    oldstine

    The Miura map fold (see below), discoveredby astrophysicist Koryo Miura, enables abulky sheet of paper to be neatly collapsed andeasily opened and refolded.

    The fastest way to gefrom the outside in, sthe unlucky middle-s