Chaos To Order

The animation at this website (<— link) demonstrates the following mathematical fact about polygons:

If you connect the midpoints of the sides of a polygon to form a new polygon and continue the process indefinitely, you will eventually end up with an ellipse (oval).

Starting figure:                                                Eventually becomes:

Screen Shot 2015-09-10 at 7.32.14 PM  Screen Shot 2015-09-10 at 7.32.24 PM

Huh.  Cool stuff!

BADGING:  Go to the link above and try running the animation 5 different times with a 10-sided polygon, and once with a 100-sided polygon.  Write a paragraph comparing/contrasting what happened in the different instances.  Then, print off this shape (an irregular heptagon):

drawingveryirregularheptagon2

Complete the next three iterations of the activity by hand, estimating the midpoint of each segment (trace your new figures on a new sheet of paper each time — use a window or other light source to help you see the midpoints and connect them into a new polygon).

Write a second paragraph hypothesizing why you think this occurs.  Why does an orderly ellipse form out of a seemingly chaotic mess no matter what that chaotic mess looks like to begin with?

You should turn in your two written paragraphs and your final (third) iteration of the above heptagon.

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