Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
 
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Medium level:

Ellipsoidal Paths

Figure 1
Figure 1

Given an ellipse, and a smaller ellipse strictly inside it, start at a point on the outer ellipse, and in a counterclockwise fashion (say), follow a line tangent to the inner ellipse until you hit the outer ellipse again. Repeat. Figure 1 shows an example.

Now it is quite possible that this path will never hit the same points on the outer ellipse twice. But if it does "close up" in a certain number of steps, then something amazing is true: all such paths, starting at any point on the outer ellipse, close up in the same number of steps!

This fact is known as Poncelet's Theorem.

Presentation Suggestions:
Intuition may be gained by presenting special cases, such as where the ellipses are concentric circles.

The Math Behind the Fact:
This process that produces this path may be thought of as a dynamical system on the outer ellipse, and is related to the study of circle maps and rotation numbers in dynamical systems. You can learn more about Poncelet's theorem in any classical text on algebraic geometry.

How to Cite this Page:
Su, Francis E., et al. "Ellipsoidal Paths." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    algebraic geometry
Subjects:    calculus, analysis
Level:    Medium
Fun Fact suggested by:   Jorge Aarao, Johannes Huisman
Suggestions? Use this form.
3.95
 
current
rating
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!