School of Science Indiana University-Purdue University Indianapolis

1 / 37

# School of Science Indiana University-Purdue University Indianapolis - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## School of Science Indiana University-Purdue University Indianapolis

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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

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

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

School of Science

Indiana University-Purdue University Indianapolis

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

4

School of Science

Indiana University-Purdue University Indianapolis

5

School of Science

Indiana University-Purdue University Indianapolis

p

v

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

6

School of Science

Indiana University-Purdue University Indianapolis

7

School of Science

Indiana University-Purdue University Indianapolis

8

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

School of Science

Indiana University-Purdue University Indianapolis

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

10

School of Science

Indiana University-Purdue University Indianapolis

Catenary and parabola

11

School of Science

Indiana University-Purdue University Indianapolis

Catenary, parabola, and hyperbola

12

School of Science

Indiana University-Purdue University Indianapolis

13

School of Science

Indiana University-Purdue University Indianapolis

14

School of Science

Indiana University-Purdue University Indianapolis

15

School of Science

Indiana University-Purdue University Indianapolis

16

School of Science

Indiana University-Purdue University Indianapolis

17

School of Science

Indiana University-Purdue University Indianapolis

18

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

School of Science

Indiana University-Purdue University Indianapolis

Hyperboloid of one sheet

20

School of Science

Indiana University-Purdue University Indianapolis

St Louis Science Center: Hyperboloid, one sheet

21

School of Science

Indiana University-Purdue University Indianapolis

park, Ann Arbor: part of Hyperboloid, one sheet

22

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

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

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

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

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

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

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

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

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

School of Science

Indiana University-Purdue University Indianapolis

Convergence

On-line Magazine that uses history to teach mathematics

32

School of Science

Indiana University-Purdue University Indianapolis

Mathematical Moments: explanations of math used in the modern world

33

School of Science

Indiana University-Purdue University Indianapolis

MAA Online - Columns:

34

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

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

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