Deconstructing Magic Squares

Today I'm going to talk about deconstructing magic squares into smaller shapes. I think this information is helpful for both designing and tatting magic squares. Later on, I'll make a second post to show how I designed the onion ring magic square.

Muskaan also has some posts about magic squares, which you can read by clicking here, and here. If you are interested in the origin of the magic square, scroll down to the bottom her second post.

There are two ways that I like to visualize magic squares. The first involves looking at the pattern as a group of four smaller squares, connected in the middle.

Small Squares

This type of visualization is helpful for designing magic squares.

Let's use my recent onion ring square as an example. Here is one square by itself:

And here are four squares connected together:

For a magic square, the trick lies in redesigning the center, where all four squares meet. A magic square will have one continuous path that connects all four squares together:

Using a simple diagram, the path to tat a small magic square looks like this:

I find that it is best to begin tatting at the corner of the square. It's easier to finish the tatting on the outer edge, and this starting position also allows the square to be built up to any size.

Here are a few more examples of magic squares broken down into four smaller squares. I have boxed one small square in blue for clarity. Notice how the smaller squares connect in one continuous round in the center of each magic square:

So, what happens if you take four magic squares and connect them together, using the same method pictured above? You get an even larger magic square!

This large magic square can be visually broken down into 16 small squares (boxed in pink) or into 4 magic squares (boxed in green). All squares flow together in one continuous, and somewhat confusing round.

Triangles

This type of visualization is helpful for tatting magic squares.

As magic squares grow, the path to tat them becomes more and more complicated. For this reason, I find that it is extremely helpful to visualize magic squares in a second way: as a series of triangles.

If you begin tatting in the spot designated "A" on my diagrams, you will find that the pattern is built up in triangular sections.

I'll go through this step by step, using my onion ring magic square as an example. The same basic stitch count is used throughout. (Please note: in the following example, "clockwise" and "counter clockwise" refer to the direction of the tatting in the photos. In practice, because tatting is worked from the front and back side, actual directions may vary).

The first section of the pattern looks like this:

From here, I have a choice to make. I can turn counter clockwise to complete the square or I can turn clockwise to build a larger triangle.

A counter clockwise turn uses an onion ring to corner, and results in a completed small square:

On the other hand, if I had chosen to turn clockwise to build a larger triangle, I would need to tat an inward-outward facing ring combination to corner. Here is the resulting larger triangle:

After creating the larger triangle, I am faced with the same decision again. This time, tatting in a clockwise direction will finish the square:

While tatting in a counter clockwise direction will build a larger triangle:

Note that each clockwise turn uses inward-outward facing rings to corner, and each counter clockwise turn uses an onion ring to corner. This rule is consistent throughout the pattern.

Moving on from the expanded triangle, I can turn counter clockwise to form a square:

or I can turn clockwise to build a larger triangle:

I can keep building this way indefinitely, creating larger triangles until I feel like turning to make a square. For this particular pattern, I stopped at the image below, which involved a clockwise turn to complete the square:

Lines of Symmetry

When expanding magic squares, it can be tricky to keep your place in the pattern. Something that I've found to be helpful is to use lines of symmetry as a guide.

Let's look at some of the lines of symmetry in the large magic square pictured below:

Some of the lines deal with the overall square, whereas others are for smaller sections. There are more lines of symmetry than what I have drawn. Depending on where you are in the pattern, the most prominent lines will change.

This is easiest to visualize if we use the triangle expansions that I talked about earlier. Let's start with the smallest triangle and expand it into a larger triangle. I can use this edge as a guide:

First I have to tat the corner, and then I can tat a mirror image of my previous work. The result is a larger triangle:

To expand this into an even larger triangle, I can use the new edge as a guide:

I make an onion ring corner, and then tat the mirror image of my previous tatting to form a larger triangle:

If I want to turn this into a square, I can use the other edge as a guide:

Again, I need to tat a mirror image of my previous work. The result is a square:

Using this technique, you can memorize the basic stitch count to tat triangles and squares without referring to the diagrams. It takes some practice, but I've found that this method works much better than trying to keep my place in a diagram.

That's all for today's post. It contains a lot of information, hopefully not too confusing. If you have any questions or find that something isn't clear, don't hesitate to ask in the comments below! For my next post I will talk in depth about how I designed the magic square pictured above.