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School of Science Indiana University-Purdue University Indianapolis

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

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School of Science Indiana University-Purdue University Indianapolis

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

  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 2

  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 aparabola, but Leibniz, Huygens, and the Bernoulli brothers showed otherwise. Jefferson is given credit for the English word ‘catenary’ 3

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

  5. School of Science Indiana University-Purdue University Indianapolis 5

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

  7. School of Science Indiana University-Purdue University Indianapolis 7

  8. School of Science Indiana University-Purdue University Indianapolis 8

  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 9

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

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

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

  13. School of Science Indiana University-Purdue University Indianapolis 13

  14. School of Science Indiana University-Purdue University Indianapolis 14

  15. School of Science Indiana University-Purdue University Indianapolis 15

  16. School of Science Indiana University-Purdue University Indianapolis 16

  17. School of Science Indiana University-Purdue University Indianapolis 17

  18. School of Science Indiana University-Purdue University Indianapolis 18

  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, thereis a line lying entirely in the surface. Clearly a plane is a ruled surface. But also, so is a cylinder! And more!! 19

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

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

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

  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 workforce • Middle school and high school are the critical times • Spread message of opportunity to parents and counselorswww.science.iupui.edu/ccowen/Careers.html 23

  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 workforce • Middle school and high school are the critical times • Spread message of opportunity to parents and counselorswww.science.iupui.edu/ccowen/Careers.html 24

  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 workforce • Middle school and high school are the critical times • Spread message of opportunity to parents and counselorswww.science.iupui.edu/ccowen/Careers.html 25

  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 workforce • Middle school and high school are the critical times • Spread message of opportunity to parents and counselorswww.science.iupui.edu/ccowen/Careers.html 26

  27. School of Science Indiana University-Purdue University Indianapolis • Explosion in biological research and progress • The mathematical sciences will be a part • Opportunity: 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…” 27

  28. School of Science Indiana University-Purdue University Indianapolis • Explosion in biological research and progress • The mathematical sciences will be a part • Opportunity: 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…” 28

  29. School of Science Indiana University-Purdue University Indianapolis • Explosion in biological research and progress • The mathematical sciences will be a part • Opportunity: 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.” 29

  30. School of Science Indiana University-Purdue University Indianapolis Main Points • Sophisticated math is everywhere around us • 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 dramatically increase participation of women and other underrepresented groups in college and post-graduate math and science 30

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

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

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

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

  35. School of Science Indiana University-Purdue University Indianapolis URL’s for the Sources 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 35

  36. School of Science Indiana University-Purdue University Indianapolis Model of the St Louis Arch3 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 standtie 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 36

  37. School of Science Indiana University-Purdue University Indianapolis Cylinder/Hyperboloid ruled surface device2 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. 37

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