Pentagrams and quartic polynomials

I’m still enjoying my new-found freedom that comes with the end of the semester. I’ve gotten some research done and I’ve been able to catch up on some reading.

One article that I found particularly interesting was “Quartic Polynomials and the Golden Ration,” by Harland Totland, from the June 2009 issue of Mathematics Magazine.

This is how the article begins. Draw a pentagram in the usual way. Find the unique quartic polynomial that has minimum values at the two lowest points of the pentagram and a relative maximum at the lowest crossing point (as shown below).

Picture 1

It turns out that the graph of this polynomial also goes through two of the other vertices of the pentagram. Moreover, the two remaining points of intersection of the curve and the pentagram lie directly below two crossing points of the pentagram (although it may look like it, these are not the points of inflection). Beautiful!

As a side note, I made the figure above with GeoGebra. I finally had a chance to sit down and play with GeoGebra. Wow! What a wonderful piece of software. Easy to use, powerful, well thought-out, and free! What took me so long to give this a try? I will definitely be using it in my classes.


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