Connections Between Mathematics and Biology

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

• p

• v

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

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School of Science Indiana University-Purdue
University Indianapolis
Catenary through (-1,1) (0,0) (1,1)
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School of Science Indiana University-Purdue
University Indianapolis
Catenary and parabola
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School of Science Indiana University-Purdue
University Indianapolis
Catenary, parabola, and hyperbola
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School of Science Indiana University-Purdue
University Indianapolis
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School of Science Indiana University-Purdue
University Indianapolis
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School of Science Indiana University-Purdue
University Indianapolis
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School of Science Indiana University-Purdue
University Indianapolis
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School of Science Indiana University-Purdue
University Indianapolis
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School of Science Indiana University-Purdue
University Indianapolis
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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!!
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School of Science Indiana University-Purdue
University Indianapolis
Hyperboloid of one sheet
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School of Science Indiana University-Purdue
University Indianapolis
St Louis Science Center Hyperboloid, one sheet
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School of Science Indiana University-Purdue
University Indianapolis
park, Ann Arbor part of Hyperboloid, one sheet
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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
  counselors www.science.iupui.edu/ccowen/Careers.h
  tml

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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
  counselors www.science.iupui.edu/ccowen/Careers.h
  tml

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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
  counselors www.science.iupui.edu/ccowen/Careers.h
  tml

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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
  counselors www.science.iupui.edu/ccowen/Careers.h
  tml

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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
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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
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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.
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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
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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
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School of Science Indiana University-Purdue
University Indianapolis
Convergence
On-line Magazine that uses history to teach
mathematics
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School of Science Indiana University-Purdue
University Indianapolis
Mathematical Moments explanations of math
used in the modern world
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School of Science Indiana University-Purdue
University Indianapolis
MAA Online - Columns
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School of Science Indiana University-Purdue
University Indianapolis
URLs for the Sources
www.science.iupui.edu/ccowenwww.maa.org
convergence.mathdl.org/ www.maa.org/news/colu
mns.htmlwww.ams.org www.ams.org/ams/mathmomen
ts.htmlwww.awm-math.org/www.math.buffalo.edu/mad
/NAM/www.sacnas.orgen.wikipedia.org
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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 archthread 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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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.
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