Magic squares

Have you ever wondered why a particular square medallion is called a 'magic square', a name which imparts certain traits which other such medallions don't have? This not because it:
* contains a particular motif, e.g. a heart
* will tessellate
* has sections which may be coloured to resemble intarsia work.

The design of a magic square is such that it:
* can be made in one pass CTM while
* allowing the original design to be reproduced multiple times without having to end off and begin again;
* follows a certain triangular formula (along the diagonal of the square).

There is one such design by Robin Perfetti in the Craftree patterns which may be called a 'magic square'.

The original medallion may be reproduced CTM to give by dividing the design along the diagonal.

In 2017 in 'Tatting by the Bay', Robin gave instructions for enlarging this CTM to create a lacy fabric. This involves the triangular formula mentioned above.
https://drive.google.com/file/d/0B8R5ZNar0YDdeHV5WVA4cndyUWc/view

Here's the formula for working along the diagonal:

Another important determinant is the specific pattern of motif involved.
Depending on the motif, there are multiple pathways one can follow, not merely diagonally.
Triangular formula is Not the only magic pathway.
In my Quatrefoil triangles, squares, pentagons, and hexagons, I have been able to trace simple circular and spiral type pathways or even side to side. Only split ring is required to move to another repeat or level.

But if you have to use a split ring, is it still a magic square? I thought that a magic square can be opened out just by turning the direction of tatting.

in reply to GraceT's post:

I prefer to call it a Magic Pathway.
Reasons :
1 & 2 similar to what Judith has listed , but

1. different pathways are possible besides the diagonal one
2. continuous tessellation not merely of a square motif, but other shapes as well

The term Magic Square is limiting, considering the range that is possible.

in reply to muskaan's post:

If I understand you correctly, the Magic Pathway would result in the tiling of the original square without altering it, and also without having to tie off and restart. But I understand the Magic Square to be something different. The original square is not reproduced exactly, but it's opened out and its elements are repeated to make up a larger square.

I'm not sure who first started using the term "magic square," although the Anne Orr pattern has been made into infinitely expanding squares for quite some time. Georgia Seitz referenced a tablecloth made from the Anne Orr pattern, all in one pass, and I believe the tablecloth was made in the 1980s. A snapshot of the Anne Orr pattern can be seen in one of Jon Yusoff's blog posts. In the second blog post you can see Jon's expansion of the square:

https://tatsaway.blogspot.in/2008/02/old-motif-revisited.html
https://tatsaway.blogspot.in/2008/05/update-on-anne-orr-square.html

When I designed my own magic squares, I followed the same triangular pathway that tatters have been using for the Anne Orr pattern. It is my understanding that the term "magic square" references this particular pathway. However, since I cannot trace the origin of the term, I'll never know what the first person who called it a magic square intended the name to mean.

@muskaan, I tend to think of them as "infinite pathways," referring to tatting that can be expanded indefinitely, in one pass. So far, I've only been able to find an infinite pathway for squares and rectangles. I would love to find infinite pathways for other shapes, but so far they all eventually reach a dead end.

That's why it's called a 'magic square'. Tatters can double and treble the surface area of the fabric, but it will still remain a square.

Sure, you can determine pathways in other designs by adding split rings and bridging chains, but they are just that - pathways. There's no magic.

Got it.
The 'magic' is in figuring out if there is or isn't; and if there is then how - love puzzling it out kind of like a maze.

in reply to Bay's post:

I do need to put those pathways up, don't I! But you will get an idea from the Fortuna Square pathway I am sharing in the other thread in response to Judy's question. The 2nd pathway will give you the 'spiral' sequence that works in the Quatrefoil shapes, besides the zigzag one.

One of my grandma's heirloom hankies had a corner tatted like post #2 by @Judith Connors . I spent time trying to figure out how it was done before I ever read a line of tatting instruction... so that one is "in my bones" as it were.

Yes, @Judy. That CTM design has been around for nearly a century. With decorative picots it's quite attractive, even using the triangular half-pattern. Which just goes to remind us that tatters of 'l-o-n-g ago' were interested in working with continuous thread. It's not a recent phenomenon.

@JudithConnors has shared an article on Magic Squares published in the Queensland Tatters' Talking Tatting.

For tatters who are mathematically inclined, here's Archimedes' geometric series which correlates with the magic squares: 1, 4, 16, 64, 256,..... A Google search will provide further information.

Usually the enlarged patterns stop at 16 (4x4), though it's possible to work beyond that if a continuous pathway is devised.

in reply to JudithConnors's post:

This would be very helpful for keeping track of a Magic Square while it's in progress.
Basingstoke

True, @GraceT. The pink units actually provide a series comparison: 1 unit (medallion), 4 units (continuously linked medallions), 16, 64, etc.

Robin's 1- and 4-unit squares, in post #1 above, are easy enough for tatters to create. For ease of working, they can be viewed as two halves (along the diagonals) always starting at the top L-hand corner, clockwise, for continuity of thread.

When you approach the 16-unit square (4x4), you need the formula given in post #2 for one half; rotate the work, and repeat for the other half to complete the square. Although, for simplicity, this matches the Anne Orr magic square, it can be applied to all 4x4 patterns, such as Robin's.

I have yet to see a tatter attempt the 64-unit (8x8) square, but it would be possible. Any takers?

in reply to JudithConnors's post:

and @GraceT

I agree with Grace. Thanks, Judith!
Rosencrantz

I haven't tried the Magic Square yet. It's on my list to do, but there are many doilies higher up on the list! A 64-unit square would certainly be beyond me as a first attempt. But, wasn't there a wedding shawl entirely tatted from the the Magic Square (original pattern)?

This would have involved hundreds of units. Possibly it involved the up-and-down pathway: even a number of smaller squares joined together? Or, maybe 'magic square' was vaguely used where a continuous thread was involved?

As you can see, magic squares form a very small subset, that produces an overall lacy fabric which appears tessellated on completion. @GraceT, most tatters start with the 4-unit square, so Robin Perfetti's onion-ring square is a nice challenge for you.

@JudithConnors the wedding shawl was made as a very large triangle (omitting the final turn that would have turned it into a square). The photos on Georgia’s site seem to be missing, but there is a result on Pinterest. Not sure if the link will work here. If not, it can be found on a Google image search for tatting wedding shawl, but you will have to scroll through the results.

https://www.pinterest.com/pin/134474738849004805/

Thanks, @Bay. This would certainly have produced a large magic square, a reasonably heavy one.

Thanks, @Bay. I knew I had seen it somewhere! So, I searched my hard drive, and found that I had downloaded a web archive of that page on Georgia Seitz's webpage, with all the pictures! It is a large wedding shawl indeed - there is a picture of it where the long side of the triangle spans a three-seater sofa.

I am not sure what to do now, though. It may not be kosher to share that page with you all here. I guess I should write to Georgia Seitz about the broken picture links instead...

in reply to GraceT's post:

Okay, I've just done that...

Here's another design from which a larger magic square could be created: https://www.craftree.com/patterns/9921?page=1#242473