Circle of Differences

Joshua Zucker took the lead in developing this circle session on Circles of Differences for the 2011 Circle on the Road workshop.

Abstract:
Place n numbers at the vertices of a regular polygon. At the
midpoint of each side, put the absolute value of the difference
between the neighboring numbers, creating a new regular polygon.
Repeat. What values of nn are interesting? What starting sets of
numbers last the longest before becoming boring? What does it mean
for a set of numbers to become boring?

Audience: Strong Middle School students, High School students and Teachers.

Apprentices: Ina Loobeek, Vinton Geistfeld, Gloria Brown Brooks

Notes:
What values of nn are interesting? In most ways, 4 (a square) shows all the interesting behavior without getting too complicated. Experiment a bit with other values of nn if time permits.

What does it mean for a set of numbers to become boring? Students should be able to come up with a definition on their own. The definition might have to change with a triangle instead of a square, though!

What starting sets of numbers last the longest before becoming boring? This is a very difficult question. It should lead to ideas of symmetry (in the obvious ways, like flipping over your square, as well as non-obvious transformations like multiplying your numbers by a constant or adding a constant to all of them). It should also lead to good general strategies like working backwards, though here that strategy has limits. Perhaps it will even lead to the idea that the operation of taking absolute differences is itself a transformation, and then hunting for fixed points of that transformation.

Audience: Strong Middle School students, High School students and Teachers.

Apprentices: Ina Loobeek, Vinton Geistfeld, Gloria Brown Brooks

A hand out for this session is attached.

Attachment
Difference Engine student handout.pdf