A Mathematical Valentine

Cardioid

The cardioid is a mathematical curve formed by the envelope of circles with centers on a given circle, and passing through a fixed point on the given circle. The word cardioid comes from the Greek word "kardia", for heart, as in "cardiac surgery, cardiology and cardio-pulmonary, medical words relating to the heart . Here is what a cardioid looks like:

The heart shape has become a symbol for Valentine's Day, when many of us give cards, flowers and balloons to our "sweethearts"!

Would you like to make a mathematical valentine for your sweetheart?

If so, follow the instructions below, to construct a mathematical curve, the Cardioid, using The Geometer's Sketchpad software.

1) Construct a circle with center A and passing through a point B. Then hide point B (click on point B, then choose Hide Point in the Display menu).

2) Draw a free point on the circle and label it P:

3) Divide the circle into some number of equal arcs. In this example, there are 18. (Select point A and Mark it Center in the Transform menu. Then select P and Rotate it 20 degrees in the Transform menu. Repeat this process)

4) Select Q, then P, and use the Construct menu to construct a circle with center Q and passing through P.

5) Using the Construct Menu, construct a circle with center R and passing through P.

6) Continue this process until you've completed the cardioid. It will look better if you select the original circle and hide it (use the Display menu)

Now you can create a Valentine's Card for your sweetheart! Type in a nice message, and color the cardioid in GSP, or copy and paste it into a painting program and color it there . . . or print it out on paper and color it with pens or pencils. For extra credit in Connecting Geometry, send your GSP file of the Cardioid to your TA.

This valentine's card is dedicated to my loving husband.

Happy Valentine's Day!

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