1 School of Science Indiana University-Purdue University Indianapolis Carl C. Cowen, Dean IUPUI School of Science President, Mathematical Association of America NCTM Annual Meeting, St. Louis April 27, 2006 The St. Louis Arch and Other Mathematically Interesting Shapes

2 School of Science Indiana University-Purdue University Indianapolis 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

3 School of Science Indiana University-Purdue University Indianapolis The Catenary: physical description A catenary is the curve formed by a flexible cable of uniformly distributed weight when hung from supports at two points. Galileo thought this curve would be a parabola, but Leibniz, Huygens, and the Bernoulli brothers showed otherwise. Jefferson is given credit for the English word ‘catenary’

4 School of Science Indiana University-Purdue University Indianapolis Catenary through (-1,1); (0,0); (1,1)

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6 School of Science Indiana University-Purdue University Indianapolis Catenary through (-1,1); (0,0); (1,1) v p

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9 School of Science Indiana University-Purdue University Indianapolis The St. Louis Gateway Arch is an inverted catenary, 630 feet wide at the base and 630 feet tall: its equation is

10 School of Science Indiana University-Purdue University Indianapolis Catenary through (-1,1); (0,0); (1,1)

11 School of Science Indiana University-Purdue University Indianapolis Catenary and parabola

12 School of Science Indiana University-Purdue University Indianapolis Catenary, parabola, and hyperbola

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19 School of Science Indiana University-Purdue University Indianapolis Ruled surfaces A surface is called a ‘ruled surface’ if through each point of the surface, there is a line lying entirely in the surface. Clearly a plane is a ruled surface. But also, so is a cylinder! And more!!

20 School of Science Indiana University-Purdue University Indianapolis Hyperboloid of one sheet

21 School of Science Indiana University-Purdue University Indianapolis St Louis Science Center: Hyperboloid, one sheet

22 School of Science Indiana University-Purdue University Indianapolis park, Ann Arbor: part of Hyperboloid, one sheet

23 School of Science Indiana University-Purdue University Indianapolis Not educating enough people in science, technology, engineering and math Need to diversify our math and science workforceNeed to diversify our math and science workforce Middle school and high school are the critical timesMiddle school and high school are the critical times Spread message of opportunity to parents and counselorsSpread message of opportunity to parents and counselors

24 School of Science Indiana University-Purdue University Indianapolis Not educating enough people in science, technology, engineering and math Need to diversify our math and science workforceNeed to diversify our math and science workforce Middle school and high school are the critical timesMiddle school and high school are the critical times Spread message of opportunity to parents and counselorsSpread message of opportunity to parents and counselors

25 School of Science Indiana University-Purdue University Indianapolis Not educating enough people in science, technology, engineering and math Need to diversify our math and science workforceNeed to diversify our math and science workforce Middle school and high school are the critical timesMiddle school and high school are the critical times Spread message of opportunity to parents and counselorsSpread message of opportunity to parents and counselors

26 School of Science Indiana University-Purdue University Indianapolis Not educating enough people in science, technology, engineering and math Need to diversify our math and science workforceNeed to diversify our math and science workforce Middle school and high school are the critical timesMiddle school and high school are the critical times Spread message of opportunity to parents and counselorsSpread message of opportunity to parents and counselors

27 School of Science Indiana University-Purdue University Indianapolis Explosion in biological research and progress The mathematical sciences will be a partThe mathematical sciences will be a part Opportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educatedOpportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educated Dr. Rita Colwell: “We're not near the fulfillment of biotechnology's promise. We're just on the cusp of it…”

28 School of Science Indiana University-Purdue University Indianapolis Explosion in biological research and progress The mathematical sciences will be a partThe mathematical sciences will be a part Opportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educatedOpportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educated Report Bio2010: “How biologists design, perform, and analyze experiments is changing swiftly. Biological concepts and models are becoming more quantitative…”

29 School of Science Indiana University-Purdue University Indianapolis Explosion in biological research and progress The mathematical sciences will be a partThe mathematical sciences will be a part Opportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educatedOpportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educated NSF/NIH: “Emerging areas transcend traditional academic boundaries and require interdisciplinary approaches that integrate biology, mathematics, and computer science.”

30 School of Science Indiana University-Purdue University Indianapolis Main Points Sophisticated math is everywhere around us Sophisticated math is everywhere around us New mathematics is being created every day and it is being used to improve our lives New mathematics is being created every day and it is being used to improve our lives We must use our influence to help counselors, parents, and students understand the amazing variety of careers that can be built on an education in math and science We must use our influence to help counselors, parents, and students understand the amazing variety of careers that can be built on an education in math and science We must dramatically increase participation of women and other underrepresented groups in college and post-graduate math and science We must dramatically increase participation of women and other underrepresented groups in college and post-graduate math and science

31 School of Science Indiana University-Purdue University Indianapolis Sources Mathematical Association of America Mathematical Association of America American Mathematical Society American Mathematical Society Association for Women in Mathematics Association for Women in Mathematics National Association of Mathematicians National Association of Mathematicians SACNAS – Society for the Advancement of Chicanos and Native Americans in Science SACNAS – Society for the Advancement of Chicanos and Native Americans in Science

32 School of Science Indiana University-Purdue University Indianapolis On-line Magazine that uses history to teach mathematics Convergence

33 School of Science Indiana University-Purdue University Indianapolis Mathematical Moments: explanations of math used in the modern world

34 School of Science Indiana University-Purdue University Indianapolis MAA Online - Columns:

35 School of Science Indiana University-Purdue University Indianapolis convergence.mathdl.org/ en.wikipedia.org URL’s for the Sources

36 School of Science Indiana University-Purdue University Indianapolis Model of the St Louis Arch 3 feet bead/ball chain Fast curing epoxy (comes in a pair of tubes) plastic cup to mix epoxy in spoon to mix epoxy with stand to hold “arch” thread to tie chain to stand tie string to ends of chain to tie chain to stand mix epoxy thoroughly, add chain to coat chain with glue, trying to keep string out of glue hang chain on stand with string, let cure turn upside down

37 School of Science Indiana University-Purdue University Indianapolis Cylinder/Hyperboloid ruled surface device 2 disks of cardboard, about 6 inches diameter 1 foot long dowel 3/8” diameter or a long TinkerToy rod 35 feet of yarn or string, cut into pieces about 1 foot long punch holes at center of cardboard disks to accommodate the dowel/rod snugly slit disks about 3/8” in, along radius at 32 equally spaced points on rims insert 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 bottom pulling disks apart to slightly stretch yarn, with yarn pieces perpendicular to ends, get a cylinder but by twisting end disks relative to each other so yarn pieces are not perpendicular, create a hyperboloid of one sheet.