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Old 12-29-2006, 01:16 PM
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Celtic Knots 101 - Working in a Circle

Ok! Lookit! I'm actually ahead of schedule! Ready?

Let’s start with working running borders in a circle. I tend to think of it as bending them. And while it appears really complicated, I find it easier than working in a straight line in a lot of cases. Reason #1, no corners. Reason #2, it’s easy to sneak in an extra repeat or curve if you need it. And last but not least, reason #3: It creates a big visual impact with very little effort.

The books on knotwork spend a fair amount of time & effort discussing how the ancient scribes did it, using minimal tools and using lots of geometrical terms that send this math-challenged girl into a tizzy trying to make heads or tails of what they’re talking about. They also spend a lot of time having you draw in “guides” that aren’t needed in the long run. Now, while I fully support the idea of understanding *how* it was done in the past, I am also very very glad I’m not living then (I like electricity, indoor plumbing, and having a protractor!). That being said, there are a few geometrical terms/definitions you’ll need to keep in mind:

--Circles have 360 degrees.
--The diameter of a circle is the measurement from edge to edge.
--The radius of a circle is the measurement from the center point to the outside edge (it’s *half* of the diameter)

There are couple of things to remember when working with curves: #1: the arcs of the knots on the outside edge are going to be longer than those on the inside, so make sure you aren’t working so tight & small that the inside edge will become impossible. #2: Any breaks in the pattern will be very obvious, so take that into consideration when drafting your initial composition.

In the interest of not re-inventing the wheel, I’ll start with a simple running border you should feel comfortable with already: The braid. Here it is in a straight line:



We won’t worry about trying to make it all one band at this stage. And to make it easier, I’ll show you how to plot the circle to resemble the graph paper. We’ll need to draw 4 concentric circles, 1/4” apart. In order to stay small enough you won’t be overwhelmed & big enough you’ll still be able to see what you’re doing, we’ll make the outside circle 4” in diameter. Here’s the step by step:

I’ve divided my paper in half, & picked a spot to be the center of the circle. Measure 2” along the line (this is the radius) from the midpoint and make a tic mark. Now, make 3 more marks 1/4” apart moving back toward the center. Using your compass, set the point in the midpoint, and the pencil end at the outside mark. Draw your circle. Now, keeping the point at the midpoint, set the pencil end to the next mark, etc. Once you’ve drawn in these 4 circles, the largest will be 4” in diameter, then 3 1/2, 3, and lastly 2 1/2 .



Next, take your protractor & line the 0 mark up with your center line and the center on your midpoint. Now make little tic marks every 10 degrees. At this size, 10 degrees is roughly the same size as the 1/4” grid of standard graph paper. I usually only make tic marks around 1 half of the paper & use the midpoint to line up the ruler. This way, if my tic marks are a little off, it’ll be consistently off and still pass through the midpoint (important when we get to the spirals). Use the ruler to draw in guidelines on both sides of the circle. I hope its looking a little familiar at this stage.



Now, just like you did on graph paper for the running borders demo, start drawing in the “bones” of the braid.



Draw in your curves & start the weave pattern.



Erase the “bones” and clean-up.



Now, remember the rule about braids? It won’t be one band of interlace when they connect if the number of curves on the outside edge can be evenly divided by 3. Since we used 20 degree segments per curve on the outside (tic mark every 10, and having the curves use 2 spaces like we did on the graph paper), our 360 circle has given us 18 curves. Which is evenly divided by 3. So, to make this into 1 band, if we make tic marks at 9 degrees, instead of 10, we would have 20 curves along the outside. Which *can’t* be divided by 3.



With me so far? Ready to move on?
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Last edited by TessDB : 12-29-2006 at 01:36 PM.
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Old 12-29-2006, 01:24 PM
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Re: Celtic Knots - Working in a Circle

It’s Challenge Time!

When we worked with the triangles & pretzels, I showed how to do knots without that center band. I did that for a few reasons. Primarily, as you start to design carpet pages and work with the knotwork in irregular spaces, grids and what-not are going to be harder and harder to come by. The other reason is that I find it really fussy to *always* work it out on graph paper and then try to adapt it to an actual piece. So here’s the assignment: Can you do a simple twist around a circle without using a center line? And I’d like you to start getting used to using as few guide marks as possible (less erasing). I’ll show you how to set it up. But lets practice a bit first:

On your sketch paper draw 2 fairly long parallel lines, 1/2” apart. Make tic marks (top & bottom) every 1/2”. Give yourself as much room as you need to start feeling comfortable. Now, draw in your top & bottom curves, centered between the tic marks and letting the right-hand side cross over into the space for the next arc (remember the discussions on which direction the knots will “pull”?). These will be the outer edge of your bands. Here’s how mine looks:



Next, draw in the inner edge of your bands on the top. Extend your lines a bit more. Notice how the inner edge is aligned. It should naturally become the outer edge of the lower curves & create the “over” portion of the twist. You may need to correct some angles & extend your lines. Don’t worry if it looks a little messy. That’s what erasers are for.



Now complete the lower curves. Try to keep the width of your bands consistent. You may need to correct the angles and refine a bit. That’s fine. Notice mine is *really* sloppy looking at this stage.



Clean it up, and there you go. Mine still looks pretty rough. This is about 3 times the size I usually do a twist in… hmmm… maybe the teacher needs more practice too!



Now for the fun part. Let’s bend it around a circle.

Just like you did for the braid, draw a center line, pick a midpoint, measure up 2” and draw a circle 4” in diameter. Now make a tic mark 1/2” from that circle, moving back toward the midpoint. Draw in another circle using the tic mark & the same midpoint. This one should be 3” in diameter.



Now with the protractor, mark around at 20 degree intervals. Using your ruler, make tic marks to guide you.



So now that you’ve had practice doing a twist in a straight line, apply the same steps to going around the circle. I’ve re-discovered that at this scale, I’m much neater if I draw in both edges of the bands at the same time. Do whatever works best for you, just make sure your weave pattern is constant.



So those of you who aren’t math challenged have probably realized that this is two bands of interlace (twist = 2 curves for a full repeat; 360 degrees in 20 degree segments = 18 segments. An even number of curves keeps this two bands). The question is: is that what we want? Let’s pretend the answer is no. There’s a bunch of ways to address/fix it:

#1 Figure it out before hand. We need an odd number of curves, but want to keep them relatively the same size as we have here. If we make it 19 segments, instead of the 18 we’ve done, we’d make a tic mark at *about* every 19 degrees on the protractor (360 / 19 = 18.94736etc) . Or we could make it 17 curves. That would be a tic mark *about* every 21 degrees (360 / 17 = 21.17647etc). Or we could sacrifice size and make it 15 curves, with tic marks every 24 degrees (360 / 15 = 24). Oy. Makes my head hurt…

#2 Break the pattern & join the bands in some way (only once!) like these:





#3 Rob Peter to pay Paul, so to speak. Pick an area & rework the curves, sneaking an extra one in (my preferred method if I didn’t think of it before I started the whole thing & I don’t want to break the pattern). This is essentially the same as fix #1, except you don’t make yourself crazy counting little lines on a protractor!



Still with me? Next, some rules for working with braids & plaits.
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Old 12-29-2006, 01:31 PM
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Re: Celtic Knots - Working in a Circle

MATH ALERT!
These are the rules for braids & plaits when working in a circle:

Twists – 2 bands
Any number of curves divisible by 2 = 2 bands (even numbers)
Any number of curves not divisible by 2 = 1 band (odd numbers)

Braids – 3 bands
Any number of curves divisible by 3 = 3 bands
Any number of curves not divisible by 3 = 1

4 band plait
any number of curves divisible by 4 = 4 bands (8, 12, 20, etc.)
any number not divisible by 4 but divisible by 2 = 2 (14, 18, 22, etc)
any number divisible by 4 plus *or* minus 1 = 1 (seems like the long way to say that, but it’ll make sense in a sec.) So 15 curves = 1 band. 17 curves = 1 band.

5 band plait
any number divisible by 5 = 5 bands
any number not divisible by 5 = 1

6 band plait
any number of curves divisible by 6 = 6
any number not divisible by 6 but divisible by 2 = 2
any number not divisible by 6 but divisible by 3 = 3
any # divisible by 6 + *or* - 1 = 1 So 17 or 19 curves = 1 band

See a pattern emerging? These rules hold true when working a border around a frame, too. Which is why symmetry is *not* your friend if you’re trying to make it all one band! When you’re working in a circle, it’s easy to sneak an extra curve in or remove one. The human eye doesn’t detect it very well. In a straight line, it’ll jump out.

Anybody care to guess what the rules for 8’s and 9’s are? And anybody notice the, um, easiness of the prime numbers (1,2,3,5,7,11,13,17, etc.)? And now I could get *really* esoteric and dense on you and launch into a discussion about sacred geometry… But I won’t. Although it’s tempting…

Questions? Lets see how you’re doing!

Next part will be knotwork around the circle (brush up on your triangles & pretzels!). I hope I’ll have it up in the next few days.

Tess
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Old 01-05-2007, 09:50 AM
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Re: Celtic Knots - Working in a Circle

Hopefully, y’all aren’t completely tangled in bending a running border around a circle!

To start with, we’ll use a pretzel knot & a triangle knot. And we’ll keep it simple. If you're aiming to make a border or a ring of a circle all one line, an easy short-cut is to create a repeat that uses only one band to begin with. Like a pretzel knot. Or opening a triangle knot (only once!) in some way. To show what I mean, here’s the pretzel knot we’ll be using. All I’ve done is replace the bottom curve of the pretzel with a twist.



And here’s the slightly modified triangle knot. In this case, I broke one of the curves & turned them into straight bands.



Now, the nice thing about this is that no matter how many of either of those you connect up, it’ll still be all one band as long as you’re using those open ends. If I go too fast or skip over something you’d like in more detail, definitely let me know!

Ready?

For this one, we’ll be doing a 9-pointed star ringed by a 4band plait. So just like for the previous section, we’ll start drawing in some concentric circles. Draw in your center line, pick your midpoint, & draw in a circle that’s 5” in diameter, then 4”, and finally 2”. With your protractor, make tic marks every 20 degrees. Going around the circle, draw in a solid line from your first tic mark to the midpoint. At the next tic mark, just make little marks across the lines for your circles to the midpoint. Once you’ve gone around, erase the midline that’s breaking the pattern. You should have something that looks like this:



Now from the tic mark on the outside edge of your 2nd ring, draw a straight line to the next solid line on the inner edge. From the same tic mark, draw in a straight line to the other solid line. After you’ve gone around the circle, you should have this:



So in your center ring, draw in the pretzel knot using the solid lines & your tic marks to help you get the spacing right. Pretty neat, eh?



Moving to the middle ring, flip the pretzel knot so the twist portion is pointing outward. Now, for this one, I won’t be joining the rings of knots. *But* if you want to, in order to keep your weave consistent, this 2nd ring will need to run the opposite direction as the center ring (it’s because these knots are only one band). So the center in my example is rotating clockwise. The 2nd ring rotates counter-clockwise.



Next comes the triangle knot. Work two of them in the spaces between the points like this:


After you’ve gone around the circle, you should have something that looks like this:



And last step, put a 4-band plait in that outside ring. For this example I’m not all that concerned about trying for all one line. So ‘cause its easy I’m doing 2 curves on the outside per section. And out of habit, I’ve got them pulling clockwise. If I wanted to connect the middle ring to the outside, the plait would need to pull counterclockwise. Use the tic marks to help you get the spacing right. I’ve done it here as a progression.



And here it is completed:



Thanks to a little photoshopping, here’s what it would look like if we had the middle ring rotating clockwise, too.



The difference between the two is very subtle, but it is there in how the viewer’s eye moves around the circle. This is something to keep in mind when you’re composing a larger piece. Sometimes, you need to run things in an opposite direction. It helps to hold the piece still for the viewer.

Ready to move on?
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Old 01-05-2007, 09:55 AM
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Re: Celtic Knots - Working in a Circle

For this next one, lets use the same knots, just in a slightly different layout and order.

Once again, we’ll do 3 concentric circles. First one is 5” in diameter, then 3”, and lastly 2”. And in the interest of not completely redoing the layout, tic marks at 20 degrees. So just like before, divide the smallest ring into 9 pie pieces. Leave the middle ring open (the plait will go here). In the outer ring, draw in the points of the star, so the fat end of the triangles line up with the ones on the very inside circle. Here’s what mine looks like:



Just like the previous example, work the pretzel knot in the middle.



Next, work the 4 band plait. I’ve done 2 curves per section, allowing it to pull clockwise.



This time, I’ll put the triangle knot in the “points”, and work 2 of the pretzel knots in the spaces between.



And here it is completed.



If you look at them side by side, even though they use the exact same elements used in a slightly different order, they look *very* different. So hopefully by now you’re getting some ideas of different ways you can take some really basic elements and make them look *hugely* complicated.





Ready to take in up a notch? I should have the next part posted later today. And please! If I’m going too fast or you need me to explain something, feel free to post or pm me.

Tess
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Old 01-05-2007, 10:28 AM
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Re: Celtic Knots - Working in a Circle

Tess, This is awesome! Looks like a challenge. I'd better find a protractor and get hopping (or knotting as the case may be). Great start for the new year!! - Chris
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Old 01-05-2007, 03:17 PM
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Re: Celtic Knots 101 - Working in a Circle

Great to have you along, Chris!

Buckle your seat belt, 'cause this next one's a doozy!

Tess
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Old 01-05-2007, 03:30 PM
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Re: Celtic Knots 101 - Working in a Circle

Ok, on this one, we’re going to shoot the moon. By the time we’re done, it *will* be all one band. We’ll be using 4 main elements. A braid in that outside ring, then 3 different knots. Two of the knots are variations of the same triangle knot. Here are the knots:



Knot A should be fairly familiar from the pretzels & triangles thread. Now, notice that B and C are essentially the same, except instead of a point at the bottom, I’ve substituted a curve. That’ll make sense in a sec. If you want to see a step-by-step of how to draw these, let me know.

So lets start drawing in our circles. Just like before, draw your midline & pick your midpoint. We’ll be drawing 3 concentric circles again (did I mention I have a thing about threes?). Outside circle will be 6” in diameter, then 5”, and lastly 3”. With your protractor, make tic marks every 10 degrees. We’ll start by dividing our circles into 12 pie pieces. So starting at your midline, skip the next 2 tic marks & draw a line at the 3rd mark (30 degrees) that passes through the midpoint to the other side. Make tic marks for the other degree points across your circle lines (10 & 20 degrees). Here’s what mine looks like:



In that middle ring, start at your midline. Draw a line from the inner edge of the ring to the next tic mark on the outer edge of the ring. Repeat on the other side of your midline. Now do the same around the circle at *all* of the solid lines. You’ll end up with something that looks like a bunch of chicken feet like this:



So starting with your center ring, draw in knot B. As usual, I’ve done mine pulling clockwise.



In the middle ring, work knot C with the curve snugged up to the outer edge of the ring. Make sure it’s pulling the same direction as the center ring. Once again, mine is going clockwise.



Now work knot A in the triangle spaces you have left. Unlike the previous examples, we don’t need to do two of them. If the line that divides that triangle in half is confusing you, go ahead & erase it. I like to leave it ‘cause it helps me get the points right. But do what works for you. In case you haven’t guessed it yet, this is our repeat unit.



Now for the braid in the outer ring. This is the tricky part. Getting it to line up correctly with the middle ring is crucial. So instead of starting to draw your curves for the bands on the outer edge of the ring (like usual), start on the inner edge. We need a curve to run next to that middle curve of knot A and the curve of knot C. So start with those (ignore the erased bits… got a little ahead of myself).



Now fill in the other curves you need on that inner edge like this:



Now do the outer edge. Remember that for a braid the curves don’t mirror each other. So for this I’ve got 4 curves on the outside edge of the ring like this:



Ok, because I’m neater if I do the whole thing at once, I’ll go ahead and fill in the braid the rest of the way around. If you want to go section by section, that’s just fine. Just make sure all of your “overs” are pulling the same way. Looking at the outside edge of the braid, mine are going *counter-clockwise*. This is of *huge* importance a bit later when we start joining the separate knots together.



So now that I’ve got the braid finished, I’ll repeat knots A, B, & C around the circle like this:



Here they are, all filled in:



Whew. Next, let’s connect them.
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Old 01-05-2007, 03:39 PM
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Re: Celtic Knots 101 - Working in a Circle

Alright, let’s figure out how to join the knots!

Before we break & join anything, let’s figure out what’s going on… some of you may have already realized that the braid on the outside ring is three strands (360 / 12 sections = 48; 48 / 4 (# of curves on our outside edge per segment) = 12. Which is evenly divisible by 3.). Remember the possible ways to “fix” the problem? In this case, because of where we need those inside curves to be, we can’t sneak in an extra curve & I *really* don’t want to change the layout. So we’ll need to break & re-join the braid in some way. Start thinking about it. We’ll come back to it.

Now the plan is to join knots B (from the center ring) to knots C in the middle ring. And then join our new BC knot to the braid. And because we don’t want it to feel left out, we’ll also join knot A to the braid. Once we’ve done that, we’ll be left with 3 bands. Here it is in color so you can see a little better what’s going to happen.



The trick is going to be connecting the red strand to the yellow strand. And then connecting the blue to the red. And then the yellow one to the blue… hmmm…

Ok, first things first. Let’s join up the knots in the inner two rings. Think of the tic marks where we drew the solid lines creating the “pie” pieces as numbers on a clock… This’ll make the next part easier. Because I like how it looks with all of those points coming together at the “12 o’clock” line, we’ll leave those & join them on the other edge (circled in red). And to maintain balance, we’ll mirror that when we join the next pairs (circled in blue). And let’s not forget connecting to the outer braid (circled in yellow)



So take the points & change them into curves on knots B & C. And replace the curves on the outer edge & the braid with an “X”. Here is a detail view of how to do it:



And here’s the full view:



This is still 3 bands. So… Here’s how to break & rejoin the braid into one band. Down at the 6 o’clock position, change the two curves on the outer edge into spade corners. Do the same thing at 2 o’clock & 10 o’clock.



See what happened? At 10 o’clock, the blue & red bands are connected, allowing the yellow to pass by. At 2 o’clock, the red & the yellow are joined & the blue slips through. 6 o’clock is where the magic happens… This is where the blue & the yellow get doubled back on themselves & get connected.



And now we have one band! There are other ways to do it. But I’ll let y’all figure them out on your own.!



Any questions? Let’s see how you’re doing!

I’m off to start getting the demos ready for the spirals…
Tess
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Old 01-05-2007, 08:20 PM
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Re: Celtic Knots - Working in a Circle

Ok, I have *no* idea what happened, but it's been brought to my attention that these two layout images I posted are *not* correct for the simpler pretzel/triangle circle demo. They are correct for the monster-make-it-all-one-line demo. I'll try to post the correct ones later tonight. So so sorry if I've confused anyone!

Quote:
Originally Posted by TessDB






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Old 01-05-2007, 10:50 PM
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Re: Celtic Knots 101 - Working in a Circle

Ok, these are the correct layouts for that first pretzel/triangle circle:





It was also pointed out to me (thank Hubby who was reading over my shoulder) that this:

Quote:
(360 / 12 sections = 48; 48 / 4 (# of curves on our outside edge per segment) = 12. Which is evenly divisible by 3.)


is total gibberish. Clearly, math challenged. What I meant is:

360 / 12 sections, 4 curves per section = 48 curves. Which is evenly divided by 3 (16, btw.)

So. I *think* that takes care of the corrections. So sorry if I befuddled you!
Tess
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Old 01-06-2007, 10:24 PM
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Re: Celtic Knots 101 - Working in a Circle

Tess, I have whiplash from following the curves and spirals. I'm getting me a protractor tomorrow (Eyeballing it doesn't work well). - Chris
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Old 01-07-2007, 08:42 AM
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Re: Celtic Knots 101 - Working in a Circle

Quote:
Originally Posted by CPM
Tess, I have whiplash from following the curves and spirals. I'm getting me a protractor tomorrow (Eyeballing it doesn't work well). - Chris

no no no.... no injury! Yep, protractors are your friends...

Tess
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Old 01-08-2007, 06:58 AM
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Re: Celtic Knots 101 - Working in a Circle

this makes my head hurt!
<goes and curls up in the fetal position>
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Old 01-08-2007, 07:37 AM
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Re: Celtic Knots 101 - Working in a Circle

Ok, so far I've given Chris whiplash & mud's head is going to explode...

anyone else going to try?

Post or PM if you need me to, um, put crash pads on this thread...
Tess
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