Geomagic Squares

 

We've all heard of magic squares but not as many have heard of geomagic squares. One person, Lee Swallows, took the idea of magic squares further by investigating geometric magic squares, aka geomagic squares for short. A geomagic square is defined as whose cells contain spacial elements of certain dimension.  Magic squares are considered to be a special case of geomagic squares where all elements are one dimensional.

Think of it this way.  The numbers in the magic square represent the segment lengths in a straight line that add up to a total distance. In addition, a geomatic square might be made up of 2 dimensional areas that when put together create the same area with the same shape such as squares.

The same applies to if the square is made up of individual volumes that when combined create the same 3 dimensional shape with the same total volume. 

In regard to two dimensional shapes, they might be geometric shapes both regular and irregular or sections of a circle that when combined in any direction will form the same circle.  The circle ones will be based on the standard magic square.  Think of it this way.  360 /15 = 24 degrees.  The 15 is from the total that each row, column, or diagonal adds up to.  Thus if you look at the magic square, the first row first cell 

would have part of a circle that is 8 x 24 or 192 degrees, the second cell would show one segment or one that was 24 degrees while the last would be 6 x 24 or 144 degrees.  If you add them all up, you get 192 + 24 + 144 which adds up to 360 degrees,  If you repeat this for for the second and third rows, you get a full complement of segments that do add up to full circles.  

There is a general formula that can be used to describe magic squares is :

Now we can use it with numbers such as C=5, A = 3 and B = 1 you end up with the magic square as referenced above but what if you assigned A, B, and C to shapes.  Think of A = a small rectangle.  The C represents a square with sides equal to the length of A and B is a semicircle of diameter equal to the length of A, you can use those shapes to form a visual representation of magic squares. The idea is that you may have to rotate the shapes to get them to match up to form the shape.  

Now there are so many applications of this basic idea that I don't have the space to go into it, so you might want to read this article by Lee Swallows.  I was impressed with the different ways one could represent the basic magic square referenced above.  Read the article then check out the gallery at  this site by Lee Swallows to see what all one can do with the geomagic squares.