Teaching About Hearts

This topic is a great one to introduce around valentines day since there are multiple equations one can use to generate different hearts.  I realize hearts is not the proper mathematical term but it is one that students can relate to and once the topic is introduced, we can sneak in the proper term of cardioid. 

A cardioid is defined as a fixed point on the circumference of one circle with radius r that is rolling around the circumference of a second circle with the same radius and does not slip thus forming something similar to the cross section of an apple. 

For some math classes we teach, we don't need to get that into the explanations but I found a site from Denmark that explores the heart curve in such a way, we can introduce students to the topic in a way they understand. The site defines a heart curve as a closed curve which is the shape of a heart such as the hearts found on a deck of cards. Furthermore, many of the directions they give, combine geometric shapes to produce the final shape.

In the first part of the site, they show how to easily draw three different types of hearts including one that is closer to the human heart and is a good way to integrate art into the math classroom.  The easiest is using a square that stands on a point with a half circle whose diameter is the same length as the side of the square on two sides.  Then they give five different instructions for drawn hearts including two that use restricted sin waves, reflection, rotation, and semicircles. These use solid geometric vocabulary. 

The next section is where they talk about calculated heart curves where they list how to use certain formulas and restricted domains to create a dozen or so hearts and at least one three dimensional heart. They show the equation and the heart the equation produces so students can use a program such as demos to play around with the equations and experiment.  Furthermore, there is information on the Mandelbrot set and cardioid or weaving things to produce heart shaped final products. Finally, it offers some real life places one can find hearts and how they were created such as in window grills in Venice, or the heart shaped rosettes in Venice.  

If on the other hand, you want to provide the equations so students can graph them to see what happens, head for Wolfram because they have a lovely page with seven different equations with drawings of the final product and more detail on the actual equations.  It is more of a reference page but it provides teachers with what they need to create a lesson or two on heart shapes.

The New York Times ran a nice article on the Perfect Valentine back in 2019 and in the article, they have a link to a site where students can change the sliders to change the heart or they can change parts of the equation to see what happens when you change squares to cubes or changes the constants to see how it all changes the beginning heart. The best thing about the widget is that the creation is three dimensional.  

Valentines and Cardioids are related and such a great place to throw in some art work, serious mathematics and fun.  Let me know what you think, I'd love to hear.  Have a great day.