Geometric magic squares use shapes to take numerical magic squares into higher dimensions. In 2001, Lee Sallows discovered that the numerical magic squares known of since antiquity are actually just one-dimensional versions of something much greater.

In a geometric magic square, or geomagic square, the target is not a number, but a shape. In the square above, the 3×3 grid is made up of 9 different shapes composed of equilateral triangles. Each of the three rows, each of the three columns, and the two major diagonals are made up of three shapes that can be combined to form a diamond. The shapes may be flipped or rotated. Below is a geomagic square with a square as the target shape.

The geomagic square below has a triangle made of hexagons as its target. Since there are 15 hexagons making up the triangle, this is the geometric equivalent of the famous Lo Shu numerical magic square, in which the numbers 1 through 9 occupy the 9 grid boxes, and the magic sum is 15.

The examples above are all geometric equivalents of existing numerical magic squares, but the example below demonstrates that there are geometric magic squares with no numerical equivalent.

We are not limited to squares. Below is a geometric magic pentagram using ten shapes: one at each point of the star and each vertex of the inner pentagon. A hexagon is the target shape.

And here we have a magic hexagram with a diamond target.

How to Construct Geometric Magic Squares

To make your own geomagic squares, you will probably want to start with colored pencils and graph paper, including isometric graph paper and hexagonal graph paper. On a computer, you’ll want a vector graphics program, such as Inkscape.

One method begins by picking a target shape and figuring out how it can be broken up into parts. The first three geomagic squares above were all constructed this way. The diamond is made up of 18 equilateral triangles, the square is composed of 36 squares, and the triangle is made up of 15 hexagons. When you know how many units your target shape contains, you can construct a numerical magic square as a starting point. One way of doing this is to use a formula devised by Édouard Lucas:

This formula will produce a numerical magic square as long as 0 < a < b < c − a and b ≠ 2a. The magic sum is 3c. For example, the Lo Shu square:

Every magic square can be flipped or rotated and it is still considered to be the same square.

It is easy to see that the geomagic square above with a triangle made of hexagons as its target shape is merely the Lo Shu square in geometric form. The first two geomagic squares above also have numerical equivalents. However, the fourth example, with a six-sided star or hexagram as its target shape, has no equivalent numerical magic square. Instead, it was constructed using Lee Sallows’s “substrate” method. To begin, a substrate is produced from a trivialized numerical magic square, for instance by having a or b equal to zero. Here is the substrate used for the hexagram geomagic square:

As you can see, this is not a true magic square, because numbers and shapes are repeated. The magic sum of this trivial numerical magic square is 6, but only the numbers 1, 2, and 3 are used. The corresponding trivial geometric magic square has a target shape of a hexagram, but only the diamond, chevron and boat are used. In order to “detrivialize” such a square and transform it into a true geomagic square, one must add geometric variables, such as the plug-shaped keys and keyholes used above.

For more information about geomagic squares, Lee Sallows’s website and book are highly recommended.

These designs are available as prints or on other products at RedBubble.