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Thursday, June 29, 2017

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.

17 comments:

mika said...

Niby wszystko wygląda tak prosto a efekt jest magiczny.

God's Kid said...

That is all very interesting!!! :)

Lace-lovin' Librarian ~ Diane said...

Robin, your explanation is fascinating! I can't imagine how much time it takes for you to figure out these techniques and then write about them. Thank you for sharing all you've learned. I am mesmerized!

Jane McLellan said...

Ok, I think I see, thank you. I tatted a magic square a while ago, http://janemactats.blogspot.co.za/2015/01/magic-square.html but I'm not convinced I ever got my head around how it worked.

GraceT said...

Oh, fascinating! I still haven't tried the original plain magic square - must put both it and your magic square on my list to tat. Thanks for sharing the pattern!

muskaan said...

What a great post and you have systematically laid it all out for us with superb visuals & explanation !
I prefer tatting the chains in reverse stitch in such serpentine patterns - it keeps the tatting facing front always, so that one can easily cross-check with the pattern/diagram. RW makes it very confusing for me.
Wondering whether my quatrefoil square can follow the same pathway - on the face of it it seems to follow the guidelines you've set up. But it is only rings, hence each triangle will not be 'symmetrical' when working.
After reading your post I am even more excited to give it a go !

Robin Perfetti said...

If you connect four of your quatrefoils into a larger square, and can find a path to connect them in one round, it can be made into a magic square. You can also piece the four squares together digitally, (using Inkscape or other programs) which would save some tatting time.

Hopefully I will have a post next week showing how I designed this magic square, which should also give you some ideas of how to proceed.

muskaan said...

I had identified one path earlier - square by square & was going to go with it (it is a simple connection using SR). But after reading this post I was curious to try the triangle route. Found it & it works :-)
Will share in a few days - my first sample has a few mistakes. I am eager to see how the negative spaces work out....

Michelle said...

Ow, I I see a not the eye of a designer, especially one whose work is so wonderful, is really exciting! I'm fascinated by the mathematical side of this too. Unfortunately I was turned off to math by a teacher who told me that I'd never need it(as a girl) and never really joined it again. You, however, inspire me to at least think about it. Gorgeous work as always!

Robin Perfetti said...

Thank you Michelle :) Luckily, the math isn't too tricky and is not a requirement for designing patterns. But I'm glad to see that it has renewed your interest!

S Snow said...

This is fascinating. I want to drop everything else (in life) and tat your patterns. I've just finished my first magic square. Now to try the onion ring. Thank you thank you.

Sarah said...

This is fascinating and clear - thank you.

Shirla said...

This is fascinating. You have taken tatting to a whole new level. Thank you.

Jenn said...

Thanks for sharing the process with us :). It's been great catching up on your blog posts and seeing all the new designs! Must dig out my shuttles!

anita said...

You carved the magic square into small pieces. You directed our eyes to certain areas. Thank you! You've given me understanding!

siamkitty said...

Tablecloth!

Robin Perfetti said...

Definitely, if you have the patience to tackle such a large project!