I've continued experimentation with the Y increase and hyperbolic knitting.
In this case, I started with 8 pairs in the round. I alternated one round 1×1 ribbing, one round Y increase in every pair of stitches (thus doubling). I started with 8 pairs and bound off with 512 pairs. The yarn is Lily Sugar 'n Cream kitchen cotton — sturdy, inexpensive, easy-care yarn that comes in a 2½ ounce/120 yard put-up. Some stores also carry it in a 14 ounce cone.
I would love to make a very large hyperbolic poof. I think it would be interesting to be able to fall into one, as if it were some strange hyperbolic version of a bean bag chair. Here is the problem:
Powers of 2
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256
29 = 512
210 = 1,024
211 = 2,048
212 = 4,096
213 = 8,192
214 = 16,384
215 = 32,768
216 = 65,536
217 = 131,072
218 = 262,144
219 = 524,288
220 = 1,048,576
221 = 2,097,152
222 = 4,194,304
223 = 8,388,608
224 = 16,777,216
225 = 33,554,432
226 = 67,108,864
227 = 134,217,728
228 = 268,435,456
229 = 536,870,912
230 = 1,073,741,824
231 = 2,147,483,648
232 = 4,294,967,296
233 = 8,589,934,592
234 = 17,179,869,184
235 = 34,359,738,368
236 = 68,719,476,736
237 = 137,438,953,472
238 = 274,877,906,944
239 = 549,755,813,888
240 = 1,099,511,627,776
This is where imagination bashes up against the laws of physics. You hit the million mark on the 20th increase round, billion mark on the 30th, and the trillionth on the 40th. There are 63,360 inches in a mile. If you got four stitches to the inch, then 253,440 stitches in a mile. That means that at the 20th increase round, you need 4 miles of cables, double-pointed needles, or whatever it is you are using to hold the live stitches. Crocheters do not have this problem. On the other hand, I like the greater drape of the knitted fabric. Crochet is stiffer.
My poof is just under 4 inches radius/8 inches diameter. It is 14 rounds tall: cast on round, 12 rounds pattern, one bind-off round. The next increase round + plain round will take about one skein of yarn. The pair of rounds after that will take about 2 skeins. I might be able to get to around 216 = 65,536. I've made a blanket with 80,000 stitches and a fine-gauge reversible lace scarf with 75,000. So I might be able to make a poof about 16 inches in diameter that weighs roughly 20 pounds (assuming 5 ounces of yarn gets me roughly 2000 stitches). This gives you a sense of why ruffles are a sign of conspicuous consumption. They devour yardage!
Another way of looking at this is that every time you increase, you are committing yourself to using as much yarn as you have already used in the whole rest of the project. I stopped at 512 stitches, which was a little over a full skein. If I had increased again, I would have needed a full skein of yarn just for that increase row and its corresponding plain row. So another approach is to weigh yarn, cast on, and when you have only about half left, bring the project to an end.
If I am understanding this form correctly, the center is the least dense. If I had started with one pair, the form would progress from a point to a cone to a hyperbolic pseudosphere. Since I started at 8 stitches in the round, the center is a circle that becomes a hyperbolic pseudosphere. Although the form ruffles around to fill up three-dimensional space, the edge gets longer and heavier and packs in faster than the radius grows. I am thinking that at some point, the mass of the fabric becomes a well-packed ball. In my example above, can you really crush 20 pounds of kitchen cotton into a 16-inch sphere? It is that packing problem that makes me think to comprehend the form fully, you need to keep knitting.
Of course, another approach is to add even more plain rows between the increase rows. This would allow the diameter to grow more quickly. But if you want one with an 18-inch radius/36 inch diameter, you are still looking at a massive project. (What I would really love is one with a 6- to 8- foot diameter, where I could touch it and interact with it.) Then again, maybe just commit to knitting 50 pounds of kitchen cotton into a hyperbolic beanbag chair and queue up streaming Netflix?
I must admit I've had a fascination with powers of two since I was very young. I can remember learning to multiple at school. Second or third grade, maybe? We had a big green blackboard with yellow chalk in the basement play room at home. I sat on the floor and multiplied by two over and over again until I filled up the blackboard. I was fascinated. And here, decades later, I am still enthralled.
In this case, I started with 8 pairs in the round. I alternated one round 1×1 ribbing, one round Y increase in every pair of stitches (thus doubling). I started with 8 pairs and bound off with 512 pairs. The yarn is Lily Sugar 'n Cream kitchen cotton — sturdy, inexpensive, easy-care yarn that comes in a 2½ ounce/120 yard put-up. Some stores also carry it in a 14 ounce cone.
I would love to make a very large hyperbolic poof. I think it would be interesting to be able to fall into one, as if it were some strange hyperbolic version of a bean bag chair. Here is the problem:
Powers of 2
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256
29 = 512
210 = 1,024
211 = 2,048
212 = 4,096
213 = 8,192
214 = 16,384
215 = 32,768
216 = 65,536
217 = 131,072
218 = 262,144
219 = 524,288
220 = 1,048,576
221 = 2,097,152
222 = 4,194,304
223 = 8,388,608
224 = 16,777,216
225 = 33,554,432
226 = 67,108,864
227 = 134,217,728
228 = 268,435,456
229 = 536,870,912
230 = 1,073,741,824
231 = 2,147,483,648
232 = 4,294,967,296
233 = 8,589,934,592
234 = 17,179,869,184
235 = 34,359,738,368
236 = 68,719,476,736
237 = 137,438,953,472
238 = 274,877,906,944
239 = 549,755,813,888
240 = 1,099,511,627,776
This is where imagination bashes up against the laws of physics. You hit the million mark on the 20th increase round, billion mark on the 30th, and the trillionth on the 40th. There are 63,360 inches in a mile. If you got four stitches to the inch, then 253,440 stitches in a mile. That means that at the 20th increase round, you need 4 miles of cables, double-pointed needles, or whatever it is you are using to hold the live stitches. Crocheters do not have this problem. On the other hand, I like the greater drape of the knitted fabric. Crochet is stiffer.
My poof is just under 4 inches radius/8 inches diameter. It is 14 rounds tall: cast on round, 12 rounds pattern, one bind-off round. The next increase round + plain round will take about one skein of yarn. The pair of rounds after that will take about 2 skeins. I might be able to get to around 216 = 65,536. I've made a blanket with 80,000 stitches and a fine-gauge reversible lace scarf with 75,000. So I might be able to make a poof about 16 inches in diameter that weighs roughly 20 pounds (assuming 5 ounces of yarn gets me roughly 2000 stitches). This gives you a sense of why ruffles are a sign of conspicuous consumption. They devour yardage!
Another way of looking at this is that every time you increase, you are committing yourself to using as much yarn as you have already used in the whole rest of the project. I stopped at 512 stitches, which was a little over a full skein. If I had increased again, I would have needed a full skein of yarn just for that increase row and its corresponding plain row. So another approach is to weigh yarn, cast on, and when you have only about half left, bring the project to an end.
If I am understanding this form correctly, the center is the least dense. If I had started with one pair, the form would progress from a point to a cone to a hyperbolic pseudosphere. Since I started at 8 stitches in the round, the center is a circle that becomes a hyperbolic pseudosphere. Although the form ruffles around to fill up three-dimensional space, the edge gets longer and heavier and packs in faster than the radius grows. I am thinking that at some point, the mass of the fabric becomes a well-packed ball. In my example above, can you really crush 20 pounds of kitchen cotton into a 16-inch sphere? It is that packing problem that makes me think to comprehend the form fully, you need to keep knitting.
Of course, another approach is to add even more plain rows between the increase rows. This would allow the diameter to grow more quickly. But if you want one with an 18-inch radius/36 inch diameter, you are still looking at a massive project. (What I would really love is one with a 6- to 8- foot diameter, where I could touch it and interact with it.) Then again, maybe just commit to knitting 50 pounds of kitchen cotton into a hyperbolic beanbag chair and queue up streaming Netflix?
I must admit I've had a fascination with powers of two since I was very young. I can remember learning to multiple at school. Second or third grade, maybe? We had a big green blackboard with yellow chalk in the basement play room at home. I sat on the floor and multiplied by two over and over again until I filled up the blackboard. I was fascinated. And here, decades later, I am still enthralled.
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