Equations can help computers solve geometric problems

first_imgShareCONTACT: B.J. AlmondPHONE:(713) 348-6770EMAIL: [email protected] CANHELP COMPUTERS SOLVE GEOMETRIC PROBLEMSMathematician at Rice Universityis optimistic his research will have practical applicationsA Rice Universitymathematician is on the hunt for equations that represent countless geometricproblems. When these equations areidentified, computers can be used to find solutions to complicated problems thatmight be encountered in engineering and other industries.“To understand thegeometry of something, you need ways to analyze it systematically and learn moreabout the properties it might have,” said Brendan Hassett, the Edgar OdellLovett Assistant Professor of Mathematics at Rice in Houston. “Trying tovisualize the abstract properties of something can be a struggle, but if you canrepresent those properties with equations, you can use a computer to perform ananalysis.”Hassett specializes intranslating geometric problems into algebraic equations. Mathematicians usecoordinates to study lines, circles and other geometric objects, just as sailorsuse longitude and latitude to map a coastline. The relations among thecoordinates of the points on a line or circle are expressed concisely asequations.The geometric problemsthat Hassett is most interested in involve the fourth dimension, in whichobjects are identified by four coordinates, or numbers. That increases thecomplexity of the equations as well as the number of possiblesolutions.“For a computer to solvean equation, it needs clear, step-by-step instructions on how to calculate thedefinitive answer,” Hassett said. “So it’s important to come up with a systemfor identifying all the possibilities and expressing each with aformula.”The complexity of suchproblems, which can entail thousands of equations, means that their solutionsmight not find immediate uses in industry. But the vast quantity of informationcoursing through today’s digital world means that intricate mathematics arisesmore and more in everyday life.Hassett cites the compact disc as an example,where sophisticated algebraic formulas are used to safeguard the integrity ofthe music.“If you scratch a musicCD, the disc will keep playing,” he said. “Some redundancy has been built intothe disc. The extra information allows the CD player to reconstruct the musicwhen some of the information is lost. “Mathematically, youwant to repeat information so it can be recovered in more than one way,” Hassettsaid.The geometricinspiration for the algorithms used to store information on CDs is more than 120years old. But that’s par for the course in mathematics, he said.“When you work inmathematics, you need to be confident that your findings will continue in thefuture,” said Hassett, whose research is supported by the National ScienceFoundation. “The long-term development of science, the things you study out ofpure curiosity, will lead to very useful applications, even if you can’t predict120 years from now how your work will benefit society. The historical trend isthat mathematics tends to be useful, even in areas where it wasn’texpected.” AddThis Rice University is consistently ranked one of America’sbest teaching and research universities. It is distinguished by its: size-2,700undergraduates and 1,700 graduate students; selectivity-10 applicants for eachplace in the freshman class; resources-an undergraduate student-to-faculty ratioof 5-to-1, and the fifth largest endowment per student among Americanuniversities; residential college system, which builds communities that are bothclose-knit and diverse; and collaborative culture, which crosses disciplines,integrates teaching and research, and intermingles undergraduate and graduatework. Rice’s wooded campus is located in the nation’s fourth largest city and onAmerica’s South Coast. last_img read more

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