Geomagic Squares

The magic square is an ancient and well-known mathematical object.  In the figure at the right, the sum of every row, column, and diagonal is the same, namely, 30.  This is a basic magic square.

Magic squares have been around for thousands of years, and there many variations have been explored.  Which makes the invention of a new kind of magic square all the more amazing:  the geomagic square.

http://www.newscientist.com/gallery/magical-mathematics/2

These geomagic squares turn the algebraic magic square into something geometrical:  instead of numbers summing up to the magic constant in every direction, polygonal tiles can be put together to form the same shape in every direction!  Here’s an example.

With some flipping and rotating, every sequence of tiles in this “square” can be arranged to make the same figure, namely, a 4 x 4 square with one small square missing. The example at the right shows the middle row being assembled to form the “magic constant”:  the white square in the middle is the missing square.

The natural questions:  how do you construct geomagic squares?  Can you make a geomagic square for any given “magic constant”?  For a given “magic constant”, how many geomagic squares can you create?  What others can you think of?