Exploration: Heart to Bell

The goal of this exploration is to use graphing tools, such as the TI-8x, Graphinc Calculator 3.5, Maple, or other equation driven graphic programs to explore the pattern as n is varied over some suitable range -- in this case - 5 to 5

The polar equation

produces the shape of a heart when n = - 5.

and the shape of a bell when n = 5.

What patterns can be generated in varying n from -5 to 5? Try using some graphing tool to examine such equations.

The picture below was generated with Algebra Xpresser. The multiple graph of 11 different equations, corresponding for n = -5 to n = 5, was published by Michael W. Chamberlain, U. S. Naval Academy, Annapolis, MD, in the January 1994, Vol 25, No. 1, issue of The College Mathematics Journal. An active Algebra Xpresser file would show the graphs in 11 different colors and would also show them being produced. In 2008, Algebra Xpresser is no longer available. Use Graphing Calculator 3.5 to produce this set of graphs.

Animation

A further perspective on the transformation from a heart shape to a bell shape can be obtained in an animation over the variable n using Graphing Calculator 3.5. Enter the equations

Then n for animation to range from -5 to 5.

Can you find other patterns? What if the variation is from -10 to 10?

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