Archive for October 18, 2011
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.