School of Science Indiana University-Purdue University Indianapolis

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School of ScienceIndiana University-Purdue University Indianapolis

Carl C. Cowen, Dean IUPUI School of Science President, Mathematical Association of America

NCTM Annual Meeting, St. LouisApril 27, 2006

The St. Louis Arch and Other Mathematically

Interesting Shapes

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• Introduction• The Catenary: physical description• Making a model of the St. Louis Arch• The Catenary: mathematical description• Comparison between catenaries, parabolas, and hyperbolas• Ruled Surfaces• The Challenges facing us in Math• Sources

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• The Catenary: physical description

A catenary is the curve formed bya flexible cable of uniformly distributed weight when hung from supports at two points.

Galileo thought this curve would be aparabola, but Leibniz, Huygens, and theBernoulli brothers showed otherwise. Jefferson is given credit for the English word ‘catenary’

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Catenary through (-1,1); (0,0); (1,1)

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•v

•p

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• The St. Louis Gateway Arch is an inverted catenary, 630 feet wide at the base and 630 feet tall: its equation is

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Catenary and parabola

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Catenary, parabola, and hyperbola

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• Ruled surfaces

A surface is called a ‘ruled surface’ if through each point of the surface, thereis a line lying entirely in the surface.

Clearly a plane is a ruled surface.

But also, so is a cylinder!

And more!!

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Hyperboloid of one sheet

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St Louis Science Center: Hyperboloid, one sheet

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park, Ann Arbor: part of Hyperboloid, one sheet

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• Not educating enough people in science, technology, engineering and math

• Need to diversify our math and science Need to diversify our math and science workforceworkforce

• Middle school and high school are the critical Middle school and high school are the critical timestimes

• Spread message of opportunity to parents and Spread message of opportunity to parents and counselorscounselors www.science.iupui.edu/ccowen/Careers.html

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• Middle school and high school are the Middle school and high school are the critical timescritical times


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• Spread message of opportunity to parents Spread message of opportunity to parents and counselorsand counselors www.science.iupui.edu/ccowen/Careers.html

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• Explosion in biological research and progress

• The mathematical sciences will be a partThe mathematical sciences will be a part

• Opportunity: Opportunity: few mathematical scientists are few mathematical scientists are biologically educated biologically educated few biological scientists are few biological scientists are mathematically educated mathematically educated

Dr. Rita Colwell: “We're not near the fulfillment of biotechnology's promise. We're just on the cusp of it…”

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Report Bio2010: “How biologists design, perform, and analyze experiments is changing swiftly. Biological concepts and models are becoming more quantitative…”

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NSF/NIH: “Emerging areas transcend traditional academic boundaries and require interdisciplinary approaches that integrate biology, mathematics, and computer science.”

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Main PointsMain Points• • Sophisticated math is everywhere around usSophisticated math is everywhere around us

• • New mathematics is being created every day New mathematics is being created every day and it is being used to improve our livesand it is being used to improve our lives

• • We must use our influence to help counselors, We must use our influence to help counselors, parents, and students understand the amazing parents, and students understand the amazing variety of careers that can be built on an variety of careers that can be built on an education in math and scienceeducation in math and science

• • We must dramatically increase participation We must dramatically increase participation of women and other underrepresented groups of women and other underrepresented groups in college and post-graduate math and sciencein college and post-graduate math and science

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SourcesSources• • Mathematical Association of America Mathematical Association of America

• • American Mathematical SocietyAmerican Mathematical Society

• • Association for Women in MathematicsAssociation for Women in Mathematics

• • National Association of MathematiciansNational Association of Mathematicians

• • SACNAS – Society for the Advancement of SACNAS – Society for the Advancement of Chicanos and Native Americans in ScienceChicanos and Native Americans in Science

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On-line Magazine that uses history to teach mathematics

Convergence

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Mathematical Moments: explanations of math used in the modern world

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MAA Online - Columns:

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www.science.iupui.edu/ccowenwww.maa.org convergence.mathdl.org/ www.maa.org/news/columns.htmlwww.ams.org www.ams.org/ams/mathmoments.htmlwww.awm-math.org/www.math.buffalo.edu/mad/NAM/www.sacnas.orgen.wikipedia.org

URL’s for the SourcesURL’s for the Sources

http://www.mwaa.com/_/Gallery%20Image/_/dca_washington.jpg























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Model of the St Louis Arch

3 feet bead/ball chainFast curing epoxy (comes in a pair of tubes)plastic cup to mix epoxy in spoon to mix epoxy withstand to hold “arch”thread to tie chain to stand

tie string to ends of chain to tie chain to standmix epoxy thoroughly, add chain to coat chain

with glue, trying to keep string out of gluehang chain on stand with string, let cureturn upside down


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Cylinder/Hyperboloid ruled surface device

2 disks of cardboard, about 6 inches diameter1 foot long dowel 3/8” diameter or a long TinkerToy rod35 feet of yarn or string, cut into pieces about 1 foot long

punch holes at center of cardboard disks to accommodate the dowel/rod snuglyslit disks about 3/8” in, along radius at 32 equally spaced points on rimsinsert rod into center holes and position disks near ends of rod insert lengths of yarn into slits so that each end of each yarn piece is held in one slit on top disk, one on bottompulling disks apart to slightly stretch yarn, with yarn pieces perpendicular to ends, get a cylinderbut by twisting end disks relative to each other so yarn pieces are not perpendicular, create a hyperboloid of one sheet.


School of Science Indiana University-Purdue University Indianapolis

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