patrick honner

CDs and Cavalieri’s Principle

I finally got around to shedding my library of CD cases (I know, I’m quite behind), but it got me thinking about Cavalieri’s Principle.

Cavalieri’s Principle essentially states that if two prisms have the property that all corresponding cross sections have the same area, then those prisms have the same volume.

For instance, here we have two stacks of CD cases. Every cross-section of each “prism” here is a single CD case. Since corresponding cross-sections always have equal area, Cavalieri’s Principle tells us that these prisms have equal volume, even though one of the stacks is oblique.

That the stacks have equal volume is made clearer with a simple transformation of the stack on the right.

Here’s another demonstration of Cavalieri’s Principle using the CDs themselves.

Challenge NYT Sports

Math Quiz: NYT Learning Network

Through Math for America, I am part of an on-going collaboration with the New York Times Learning Network. My latest contribution, a Test Yourself quiz-question, can be found here:

http://learning.blogs.nytimes.com/2011/04/4/test-yourself-math-april-4-2011/

This question is based on the financial troubles facing the New York Mets, and their contributions to baseball’s revenue sharing pool.

By patrick honner, 14 years8 years ago

Challenge Geometry Teaching

Paper Pyramids

Another installment from my Fun With Folding series: paper pyramids!

First, start with a triangular cut-out. Construct this triangle’s medial triangle by connecting the midpoints of each side. If you don’t have a ruler handy, just fold corner to corner, and crease in the middle to find the midpoint of each side!

Now, fold up the sides and tape them together!

The best part about this activity is that it doesn’t always work! Finding out which triangles this will work for, and which it won’t, leads to lots of good mathematical questions to explore!

Appreciation Geometry Photography

Math Photo: Light Parallelogram

Playing with some shadows, and non-shadows, falling on my carpet, I caught this:

$Math Photo -- Light Parallelogram$

Challenge Geometry

Great Graph Game

Wow, is this fun: a graph theory-based game called Planarity.

http://www.planarity.net/

Given a set of vertices and edges, your job is to untangle the graph so that no two edges intersect. In essence, you are proving that the given graph is indeed a planar graph.

If it takes you a long time to succeed, just tell your friends you got caught up thinking about questions like “How can I prove it’s always possible to untangle this graph?” and “I wonder how many more edges I could add while keeping this puzzle solvable?” That’s what I tell people.