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Unformatted text preview: Supersolids: Solids Having Finite Volume and Infinite Surfaces William P. Love Mathematics Teacher, January 1989, Volume 82, Number 1, pp. 60–65. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). Founded in 1920 as a not-for-profit professional and educational association, NCTM has opened doors to vast sources of publications, products, and services to help teachers do a better job in the classroom. For more information on membership in the NCTM, call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright January 1991 by the National Council of Teachers of Mathematics. All rights reserved. A curious and interesting class of geometric solids exists that I have named supersolids. A supersolid is a bounded solid that has finite volume and infinite surface area. A bounded solid is one that may be contained inside a sphere having finite radius. Supersolids are interesting because they defy our intuitive sense of reality. How can an object have a finite volume and have an infinite surface area? Such paradoxical concepts fascinate students. If it were physically possible to construct a supersolid, one could create a planet using a finite amount of material and have unlimited surface area. This planet might solve the population problem, since everyone could have an infinite amount of land. The study of supersolids is useful for the first-year calculus teacher. These solids are definitely nonroutine and are rarely seen in any calculus textbook. Because of their curious properties, students are highly motivated to investigate these solids to show that their volume is finite and their area is infinite. The more advanced and creative students may be encouraged to discover new and unusual supersolids of their own. The traditional calculus curriculum introduces the concept of integration before the concept of infinite series. Students use integration techniques to find the areas and volumes of solids of revolution and other solids. Students are frequently confused when introduced to infinite series because they see no concrete need or application for them. Supersolids furnish an ideal introduction to this topic because students can visualize a situation requiring infinite series. Supersolids are therefore useful for combining two major concepts in calculus: integration and infinite series. Five examples of supersolids are presented here: • The supercone • The supercube • The superpyramid I • The superpyramid II • The supersine solid The first four are solids generated using elementary geometry and infinite series. Only a few basic properties of infinite series are required, and the concepts are easily understood by most students. The last example is a solid of revolution, which requires a number of integration principles as well as infinite series. number of integration principles as well as infinite series....
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