C’mon, Have a (Magic) Heart
The Zen of Magic Squares, Circles and Stars by Clifford Pickover is chock full of magic arrangements. On page 55, Pickover discusses Dürer’s method for creating a 4 × 4 magic square:
- Starting with the upper left corner and proceeding horizontally to the right, number the squares of a 4 × 4 grid with the consecutive integers 1‑16.
- Starting with the lower right corner and proceeding horizontally to the left, number the squares of a different 4 × 4 grid with the consecutive integers 1‑16.
- From the first grid, keep the integers that occur on the main diagonals. From the second grid, keep the integers that do not occur on the main diagonals.
A visual representation of the process might help to clarify:
The result is a 4 × 4 magic square. In fact, it is a slightly modified version of the magic square that appears in Albrecht Dürer’s Melencolia I. (Note that if the other 8 numbers from each grid were combined in a similar fashion, they would form a magic square, too.)
Serendipitously, my sons and I recently completed an art project that can be combined with Dürer’s method to form a “magic square heart.” The project my sons completed is as follows:
- Draw a square with a semicircle on top. Repeat to create two of these figures, preferably on paper of two diffferent colors, and cut them out.
- Cut from the bottom of each figure to the diameter of the semicircle, to divide the squares into equal‑width strips.
- Finally, “weave” the strips to form a checkerboard pattern.
This idea can be combined with Dürer’s method to create a magic square heart. But instead of dividing the squares into equal‑width strips, divide them into three strips whose widths are in the ratio 1:2:1. Then, draw the outlines for 16 squares, and number the squares as described in Dürer’s method above. The two pieces will look like this:
Then, weave the three strips into a pseudo‑checkerboard pattern. When woven together, the result will be the following magic square heart:
To complete this project with students, you can use the template below.
- Magic Heart Template:
That said, it’s my belief that students will have maximum mathematical fun if they are allowed to create the heart from scratch. It’s an exercise in geometric construction to draw a square with a semicircle on top; weaving the strips into the appropriate configuration can lead to a discussion of geoemtric symmetry; investigating the patterns formed by the numbers can lead to a discussion of numerical symmetry; and, investigating the square to find that the rows, columns, and diagonals have a constant sum may inspire young minds in the same way that it inspired Albrecht Dürer.