Being who I am, I began with geometry. A disk takes 6 or 7 increases around because when you increase the radius of a circle by 1 unit (i.e. by one round) the perimeter increases by 2π units, 6.28ish. We have to fudge a little, of course, since an sc doesn’t add exactly the same amount to circumference as to radius and we can only increase by whole stitches, but it works out; we are able to make disks.

For a polygon, there are two distances that could play the role of the circle’s radius: center to corner (radius), and center to edge midpoint (apothem). We have formulas that tell you how much the perimeter increases when the radius or apothem increases by 1, depending only on the kind of polygon you’re expanding.

Shockingly, I’ve decided not to go into the algebra here; you can read all about it Math Open Reference. My previous knowledge says you need 8 extra stitches for a square, and that number should be larger for fewer sides and smaller for more sides (you need more stitches to get around pointier corners). Those both matched the apothem calculation and not the radius calculation.

The apothem numbers leave a lot to be worked out: how to round, what to do when the increases aren’t a multiple of the number of sides, and whether an octagon could even be made when it called for fewer increases per round than corners. I made all six polygons more or less successfully, but they broke out into half easier, half harder.

The easy polygons were the triangle, square, and heptagon.

Triangle: This didn’t go how I expected – I thought I would need to round up to 12 extra stitches per round, but I actually dropped down to 9. I started with 6 sc in a magic ring, and every corner got 4sc. Increases made into previous increases went into the third of the four sc.

Square: As I said, I already knew to put 3sc into the corners to make a square. I started with 6 sc, increased around, and then started making concentrated increases for corners. Increases made into previous increases were made into the middle sc.

Heptagon: Since for me, seven increases is appropriate for making a flat disk, the heptagon was straightforward. YMMV. I started with seven stitches, increased around, and then increased in the second stitch of each previous increase. To improve the point of the corners, in the last round I made 3sc into the second stitch of every previous round increase.

Pentagon, hexagon, and octagon were more difficult, but they did work reasonably well.

Pentagon: The pentagon formula called for 7.3 new stitches per round. Since five 2sc increases would add 5 and five 3sc increases would add 10, I alternated between them: start with 5 sc in a magic ring and make 3sc into each of them. Next round, put 2sc into the center of each 3sc increase; round after that, put 3sc into the second of each 2sc increase. Continue alternating, ending on a 3sc round. I did attempt mixing 2sc and 3sc increases within individual rounds, but it was a mess to keep the side lengths equal.

Hexagon: Like the pentagon, I used a combination of 2sc and 3sc increase rounds. The hexagon’s apothem number was lower and the number of increases per round higher (6 or 12) so I made two 2sc increase rounds for every one 3sc increase round. It perhaps would be even better to make three 2sc rounds per 3sc round, but I worried about maintaining the flatness of the piece. Start with 6 sc in a magic ring, make 3sc into each of them, and then make two rounds of 2sc increasing and one of 3sc. Put your increases into the second stitch of a 2sc predecessor or the middle stitch of a 3sc predecessor, and for best results end on a 3sc round.

Octagon: How can one even make an octagon if even one increase per corner leads to too many stitches around for the piece to stay flat? I suspect the best answer is to make a disk large enough to naturally hit a multiple of 8 stitches around and then do something like (sc, hdc, sc) in each corner on the last round. I wanted to try to stick to the size and methods of the other polygons (though I didn’t quite) and ended up with this: 7 sc in a magic ring; 2sc around; *2sc, sc* around. You’re at 21 stitches. Make a big jump to 32: *2sc, sc* 10 times, 2sc. Last round: sc 2, *(sc, ch, sc), sc 3* 7 times, (sc, ch, sc), sc. The chain in the middle of the last round’s increases gives it a little bit more point without adding even more extra bulk than we already have.

There you have it: all the polygons from 8 sides down rendered in crochet, for your freeform delight. I did these all in spirals and ended with a needle join in the second stitch; the ultimate perimeter would be smoother if you worked in joined rounds.

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Here’s the punch line: stuffing matters more than stitching. At least after a point, of course. All of my spheres were pretty well spherical, and I think the lumps were due as much to my stuffing job as to the shaping.

Some stuffing differences were extremely obvious, as in the spheres above. Those were the same pattern, but the right-hand one is stuffed more fully than the left.

I used four design bases: two from sites that were trying to embed geometry and trigonometry into the design, and two very simple repetitive designs. Those latter two were to increase by either 6 stitches per round or 5, work even for some number of rounds, and then decrease by the same number of stitches as you increased. In both cases I made one adjustment for the larger sphere, which I’ll detail with their patterns, at the end of the post.

The first mathematical approach was the Crochet Sphere Calculator. Its method of calculating the stitch counts is not given, but you tell it the desired circumference in stitches and it gives you a pattern. It may or may not give you a pattern with the exact number of stitches at the largest point that you asked for, I found, but it tells you how to place the increases and decreases. Those placements frequently require significant paying attention to your counting, however. I had one quibble with it, which was that starting with 5 stitches (which is what it means, though it says 0) and then increasing in four and doing a double increase (3 in 1) in the fifth seems less desirable than starting with 6 and then increasing in 5 of them (making a single sc in the sixth). Likewise, but even more so, having a penultimate round of 11 stitches and then decreasing 4 pairs and a trio to get to 5 seems less desirable than decreasing 5 pairs to get to 6 (with or without a single sc in the remaining stitch). Triple decreases, even done by the invisible method, are clumsy and obvious.

The second mathematical approach was the Ideal Crochet Sphere. This is based on viewing each round as a latitude line and calculating its circumference from its angle to the “north pole.” There is a separate pdf with two larger spheres and a blog post with a form that calculates the pattern for a sphere of your desired size, though by telling you only the number of increases or decreases in the round. You must decide on their placement. However, you can certainly place them so that there is as much repetition in the counting as possible, to minimize the need for going back in your round and counting to remember what you’re doing next. The pre-made patterns are arranged to have as much repetition as possible without stacking increases and decreases on top of each other round to round.

Above, left to right: Sphere calculator 20-stitch circumference (this is the one shown in the stuffing picture above), sphere calculator 22-stitch circumference, 10-round ideal sphere, 6-increase sphere, 5-increase sphere (patterns for last two at bottom).

Above, left to right: Sphere calculator 33-stitch circumference, 16-round ideal sphere, 6-increase sphere, 5-increase sphere (patterns for last two below, again; last sphere in different yarn because I ran out).

You can see that they are all more or less equivalently spherical. I didn’t make any effort to offset my increases and decreases in the 5- and 6-increase spheres, and doing so would have made them smoother. The large 5-increase sphere is fairly lumpy, but you should have seen what it looked like before stuffing! A strawberry. It looked pretty much just like a strawberry (at least before too many decrease rounds).

My recommendation if you don’t want to have to work out, say, how many rounds to work even in a 6-increase sphere, is to use the Ideal Crochet Sphere. It gives good results with the least effort.

Patterns for my spheres after the cut…

Smaller 6-increase sphere:
1. Form magic ring, ch 1, and sc 6.
2. 2sc around (12).
3. *2sc, sc* around (18).
4. *2sc, sc 2* around (24).
5-7. Sc around (24 sc; 3 rnds).
8. *Dec, sc 2* around (18).
9. *Dec, sc* around (12).
10. Dec around (6). FO.

Larger 6-increase sphere:
1-4. As for smaller sphere.
5. *2sc, sc 3* around (30).
6. Sc 3, 2sc, *sc 9, 2sc* twice, sc 6 (33). This adjustment was to make the size match better.
7-10. Sc around (33 sc; 4 rnds).
11. Sc 3, dec, *sc 9, dec* twice, sc 6 (30).
12. *Dec, sc 3* around (24).
13-15. As for 8-10 of smaller sphere.

Smaller 5-increase sphere:
1. Form magic ring, ch 1, and sc 5.
2. 2sc around (10).
3. *2sc, sc* around (15).
4. *2sc, sc 2* around (20).
5. *2sc, sc 3* around (25).
6. Sc around.
7. *Dec, sc 3* around (20).
8. *Dec, sc 2* around (15).
9. *Dec, sc* around (10).
10. Dec around (5). FO.

Larger 5-increase sphere:
1-5. As for smaller sphere.
6. *2sc, sc 4* around (30).
7. Sc 3, 2sc, *sc 9, 2sc* twice, sc 6 (33). This adjustment was, again, to make the size match better.
8-9. Sc around (33 sc; 2 rnds).
10. Sc 3, dec, *sc 9, dec* twice, sc 6 (30).
11. *Dec, sc 4* around (25).
12-15. As for 7-10 of smaller sphere.

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The samples were worked for six rounds, and were increased by the same number of stitches in each round. They are in order by number of increases, from the top of each ring around clockwise, 1-4 in the inner ring and 5-10 in the outer. The 1-increase sample was begun with 3 sc in a magic ring and the 2-increase sample was begun with 4 sc in a magic ring; otherwise the beginning number was equal to the number of increases per round. In every case the increases were evenly spaced and the first was in the first stitch of the round.

You can see that increasing by 1 to 4 stitches per round produces a cone (I posted a lot more about cones last winter). As of 5, though, it shifts to a cupped shape. 6 is also somewhat cupped, but 7 is flat. 8 is also flat and not terribly crowded. 9 starts to get crowded and want to ruffle, but real ruffling doesn’t kick in until 10, which refuses to lie flat. Since I used worsted weight yarn with an F/5 hook (3.75mm) the ruffles are rather stiff, but with a larger hook for your yarn they would be soft and malleable. It could be a nice effect for the edge of a doily.

Increasing by six stitches per round is used for both flat disks and cupped ends; the recipe I learned long ago for making a sphere is to increase by six stitches per round until it’s as broad as you want, sc even for some suitable number of rounds, and then decrease by six stitches per round to close off. I wonder looking at these samples whether 5 increases would also make a good sphere (you won’t have to wait too long for that – two sphere samples are already done, though not this one yet). For me, 7 increases per round is a more reliable disk.

Math sidetrack: why 6 and 7? Because of pi. Or tau, I suppose. To be flat, a one-unit increase in the radius of the disk (i.e., an additional round) must be accompanied by a 2π-unit increase in the circumference. If single crochet stitches added exactly the same amount to the circumference as to the radius, you would need 2π additional stitches per round (which would work out pretty well to 7 every fourth round and 6 otherwise). However, for me at least, they add less to the circumference than to the radius, so I need a skosh more. How much of this is due to them being squished together at the base and how much to my particular stitch gauge I can’t say, although it is certainly possible that 7 is overkill for you and your 6 is perfectly flat, or that with a different hook:yarn size ratio I myself might want 6 or even 8.

One last note: You can see slight corners on some of these pieces, and in person they all have a bit of cornering. This is due to stacking the increases on top of each other round to round. An increase produces a slight bump out (more on this in another shaping post as well) and that’s been allowed to shine through. The easiest fix is simply to offset the increases in every other round. When the number of non-increase stitches between increases is odd, begin the round with an increase. When it is even, begin the round with half that number of stitches and then do the first increase. You’ll end the round by making the same number of non-increase stitches as you began with.

The post Single Crochet Shaping 1: cones to ruffles appeared first on ReveDreams.com.

A cone is a basic shape and simple to crochet. The math of it is that the circumference and height of a cone have a linear relationship: for every cone there is a value x so that wherever you are on the cone, the circumference at that point is x times the height at that point. You can use trigonometry to work out stitch counts for cones of specific proportions (here’s a post sort of about that), but more simply it means you add the same number of stitches to each round. That is, at least, on average. You would get a cone by alternating adding 2 and 3 stitches to each round (effectively adding 2.5 per round), or adding 2 stitches to most rounds but only a single stitch to every fourth round (effectively adding 1.75 per round). The fact that increases and decreases in crochet don’t fully take effect until the round after they are made allows this to work smoothly.

By the time you get to seven stitches per round you’re pancake flat, and six is flat enough that it tends to come out as rounded instead of conical. You can get a good wide cone starting with five stitches in a magic ring and adding four stitches per round – even three will come out as fairly wide. For a very narrow cone, adding just one – or even less – is called for. The following cones were made by starting with 3 stitches in a magic ring and adding 3 more each round (gray cone, left side), and by starting with 3 stitches in a magic ring and alternating between adding one stitch and adding no stitches (starting with 1; green cone, right side). They each proceed for 8 rounds, so the gray cone ends with 24 stitches around and the green one with 7.

After the jump: placing increases, making a base, and stitching in back loops only.

The location of your increases makes a difference. The gray cone above wants to have a triangular lip (you can see that in the top picture of this entry) because every round was of the form “*2sc, sc X* three times” – the increases were all made into previous increases. The three cones below were each made by starting with 4 stitches and adding 2 per round. From left to right: blue is most spread out, alternating between rounds that increased in the first stitch and halfway around (patterns of the form “*2sc, sc X* twice”) and rounds that increased 1/4 and 3/4 of the way around (patterns of the form “*sc Y, 2sc, sc Y* twice”). Green is in between; every round is of the form “*2sc, sc X* twice.” And finally, purple is least spread out, with three sc made into the first stitch of each round, and one sc per remaining stitch. I love the spiral shape of the purple one, but it’s really not a cone. The green one has a noticeable puff to its rim in two places (second picture below), and actually a visible slanting line due to the fact that increases tug the previous stitch’s top teardrop up quite a bit. The blue is much smoother.

Now suppose you want to make a stuffed cone. You need a base. For me, a seven-stitch start and increasing by seven stitches per round gives the flattest disk. For the light blue cone I made four rounds, so the last round was *2sc, sc 2* around and had 28 stitches. For the gray disk, I made three full rounds and then did a partial round consisting of three sets of *2sc, sc 2*, for a total of 24 stitches in the round.

The samples here didn’t work out this way, but I have had plenty of examples of cones where the appropriate size disk to cover the bottom opening has fewer stitches around than the base of the cone. In that case, to sew them together, I figure out the overage and space the extra stitches evenly around. When you are sewing this isn’t terribly crucial, but if you want to crochet the edges together, periodically a cone stitch will need to get attached to the same base stitch as the previous cone stitch. I’ve seen a cone that ended with 35 stitches around but had a base disk with only 28 stitches – another round would have made it too wide. Every fifth cone stitch was attached to the same base stitch as its predecessor. Another case was 20 stitches around and a base that was 14 stitches around; every third cone stitch had to be attached to the same base stitch.

Alternatively, you can skip a cone stitch periodically. That might tuck the base of your cone in a bit, though, and make it look like it’s smaller around than it ought to be.

For reasons to be disclosed later, I made some cones stitching in back loops only from round 2 on. Crocheting in the back loop only gives a looser connection between rounds, so the resulting fabric is taller and more malleable than standard crochet fabric. The cones can be pulled up tall or mushed down.

Note that the difference is only in stitch height – the two blue cones fit the same base despite being quite different heights, and the purple twisty cones would too.

Finally, your trailer for next time. This lovely was made in back loops only, starting with 3 stitches in a magic ring, 2 added in the next round, and 1 added in each round after that – and each time I placed the increase in the stitch after the previous increase (so if the first round had an increase in stitch 1, the next had it in stitch 3, after that stitch 5, and so on). I made three blocks like that, which should have resulted in a 40-stitch final round by my calculation, but I had just 36, so I clearly missed some increasing. It’s still a lovely tall Seussian bit of landscape!

The post On cones, shaping, and one-loop stitching appeared first on ReveDreams.com.

The invisible join, which I call the needle join after Serendipity Crochet, is an alternative to slip stitching at the end of a round. I first encountered it on futuregirl’s blog; it’s fairly novel to a lot of crocheters, but it does seem to be a time-worn technique for finishing granny squares and similar items. What I want to add to the conversation is about using the needle join versus the slip stitch to smooth the edge of a crochet piece made in a spiral with single crochet. First let’s review how to close a round of double crochet with a needle join (or you can jump past).

Verbally: finish your last stitch, cut the yarn, and pull it through the stitch as usual. Thread it onto a needle and insert the needle into the first “real” stitch (not the turning chain) the way you would insert your hook to crochet into that stitch. Pull through and then insert the needle down through the top of the last stitch of the round. Tighten the teardrop to match the rest of the stitches (sometimes this requires pulling on parts of the yarn with your fingers) and secure the yarn as usual.

Visually:

Above left: top of turning chain and first “real” stitch marked; the latter (left-hand marker) is what you want to stitch into, as shown above right. Below left: back view of the second step of the needle join: stitching back into the last stitch made. Below right: the finished ring.

Now to compare this with slip stitch for single crochet made in a spiral. The main drawback of spiral crochet is the fact that the end of a round is a full stitch up from the beginning, so pieces with open edges have a significant height jog. Slip stitches are often used to mitigate it, and needle join suggested as an alternative. I have a tweak to the needle join option [spoiler: make a needle join in the second stitch of the last round.] – let’s look.

1. Slip stitch into next stitch (leftmost sample of top picture).

Pro: quick and easy.

Con: teardrop of stitch wants to roll to the front of the work; single strand emerging from top disrupts interlocked teardrop edge; can be problematic if you need a particular stitch count.

Best for: edges that will be invisible, such as those sewn onto another piece of crochet.

2. Needle join in next stitch (middle sample of top picture).

Pro: cleaner than slip stitch.

Con: extra work; makes a steep ramp between the round ends; can be problematic if you need a particular stitch count.

Best for: not much, IMHO.

3. Needle join in the stitch after next (rightmost sample in top picture).

Only two pictures this time because the edge-on view is essentially identical to the previous needle join.

Pro: cleaner than slip stitch; makes a gentle slope between the levels of the round ends; doesn’t add to the stitch count.

Con: extra work.

Best for: exposed round ends, such as on a disk or cup. I believe this gives the smoothest possible edge to spiral crochet.

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