Sunday, October 24, 2010

Knitting the Lorenz Manifold

A few years ago I came across a pattern to crochet the Lorenz Manifold. The crochet pattern was not like any others I had seen, perhaps because it was created by someone who had their mind way into math. Just a sample:

"rnd 4: 60 dtr 0 3 4 7 8 11 12 15 16 19 20 23 24 27 28 31 32 35 36 39 (gd 12.75)"

The short intrepretation: You will end round 3 with 60 dtr (double triple) crochets on the round. you will achieve this be working two dtr in each stitch of the previous row shown in the list. That means that starting with the 0 stitch (the first stitch to a non-mathematician) you will work two dtr in the first stitch, one dtr in each of the 2nd and 3rd stitches, two dtr in each of the 4th and 5th stitches (on the list as 3 and 4) and so on. The gd 12.75 is a measurement of geodesic distance. According to the pattern the gd is given for "orientation and motivation" -- yeah, right, motivation.

One other little note: dtr is double triple crochet. But the pattern is from the UK where double triple crochet means the same as triple crochet in the US.

It took me a while to convert this into a knit pattern. And what I have is still a draft. I am working through it for the second time. The first time I used knitting worsted. I did all the increases as "work two stitches in one stitch." Because of the bulk of the knitting worsted the result was heavy and difficult to mount on a frame. This time I am working all the increases as yarn overs and I am using lace weight yarn

Abbreviations:
K = knit a stitch
YO=yarn over
K2 = knit the next two stitches
Kn = knit the next n stitches
M= make one; knit into the front and back of the stitch. This is an increase of one stitch.
D=decrease one stitch
DM=knit two stitches together and before removing the loops from the needle, K1-b in the same two stitches.
Work even the same as knit around.

Cast on 5 stitches. Join into a ring being careful not to twist the stitches.

Row 1. M in each stitch. (total of 10 stitches)
Row 2. M in each stitch. (total of 20 stitches)
Row 3 work even
Row 4. M in each stitch. (total of 40 stitches)
Row 5 work even
Row 6 work even
Row 7. YO, k1, [K2, (YO, k1) twice] 9 times, K2, YO, k1. (total of 60 stitches)
Row 8 work even
Row 9 work even
Row 10 YO, k1, [K4, (YO, k1) twice] 9 times, K4, YO, k1. (total of 80 stitches
Row 11 work even
Row 12 work even
Row 13. K3, [(YO, k1) twice, K6] 9 times, (YO, k1) twice, K3. (total of 100 stitches)
Row 14 work even
Row 15 work even
Row 16. YO, k1, [K8, (YO, k1) twice] 9 times, K8, YO, k1. (total of 120 stitches)
Row 17 work even
Row 18 work even
Row 19. K111, YO, K5, YO, K4. (total of 122 stitches)
Row 20 work even
Row 21 work even
Row 22. YO, K3, YO, K5, YO, K3, (YO, k1) twice, K2, YO, K5, YO, k1, [K6, YO, K5, YO, k1] 6 times, K2, (YO, k1) twice, K2, YO, K5, YO, K3, (YO, k1) twice, K12, YO, k1. (total of 148 stitches)
Row 23 work even
Row 24 work even
Row 25. K7, (YO, k1) twice, K21, [(YO, k1) twice, K12] 5 times, (YO, k1) twice, K6, YO, K15, (YO, k1) twice, K10, YO, K3, (YO, k1) 4times, K2, YO, k4. (total of 171 stitches)
Row 26 work even
Row 27 work even
Row 28. K3, YO, K3, YO, K19, [(YO, k1) twice, K14] 6 times, YO, K16, YO, K5, YO, K3, YO, k26. (total of 189 stitches)
Row 29 work even
Row 30. K13, YO, K3, YO, k2 132, YO, k40. (total of 192 stitches)
Row 31 work even
Row 32. YO, K25, YO, K3, YO, K5, YO, K3, YO, K10, YO, K5, YO, K67, YO, K5, YO, K10, YO, K3, YO, K5, YO, K3, YO, K23, YO, k2, (YO, k1) twice, K, YO, K3, (YO, k1) twice, K8, YO, K4, YO, k2, YO, k1 (total of 214 stitches).
Row 33 work even
Row 34. K3, YO, k45, YO, K13, (YO, K7, YO, K3, YO, K5, YO, K3) 3 times, YO, K7, YO, K13, YO, K64, (YO, k1) twice, K5, YO, k8. (total of 234 stitches)
Row 35 work even
Row 36. K23, (YO, k1) twice, K152, (YO, k1) twice, K14, YO, K11, YO, K9, YO, k2, YO, K7, YO, k12. (total of 243 stitches)
Row 37 work even
Row 38 K47, (YO, k1) twice, K20, (YO, k1) twice, K64, (YO, k1) twice, K20, (YO, k1) twice, K40, YO, K5, YO, K9, (YO, k1) twice, K6, YO, K7, (YO, k1) twice, K6, (YO, k1) twice, K2, YO, k3. (total of 261 stitches)
Row 39 work even
Row 40 K7, (YO, k1) twice, K86, (YO, k1) twice, K20, (YO, k1) twice, K97, (YO, k1) twice, K43. (total of 269 stitches)
Row 41 work even
Row 42 YO, k2, YO, k196, (YO, K4) twice, (YO, k1) twice, K7, (YO, k1) twice, K13, YO, K5, YO, k2, (YO, k1) twice, K16, YO, K13, YO, k1 (total of 283 stitches)
Row 43 work even
Row 44 K16, (YO, k1) twice, K3, YO, K4, YO, K7, YO, K17, YO, K7, YO, K113, YO, K7, YO, K17, YO, K23, YO, K9, (YO, k1) twice, K2, YO, K5, YO, K3, YO, K9, YO, K12, (YO, k1) twice, K9, YO, k2, YO, K5, (YO, k1) twice, K2, YO, k3. (total of 307 stitches)
Row 45 work even
Row 46 K2, YO, K5, (YO, k1) twice, K71, (YO, K7, YO, K17) 3 times, YO, K7, YO, k1, K108, (YO, k1) twice, K2, YO, k2, YO, K5, YO, K6, YO, K5, YO, k17. (total of 325 stitches)
Row 47 work even
Row 48 K7, YO, K9, YO, K28, YO, k1, K155, YO, K9, YO, K16, YO, K10, YO, K16, YO, K10, (YO, k1) twice, K4, YO, K16, (YO, k1) twice, K2, YO, K10, (YO, k1) twice, K, YO, K14, YO, k11 (total of 343 stitches)
Row 49 work even
Row 50 YO, K6, YO, K24, YO, K26, (YO, K17, YO, K9) 5 times, YO, K23, YO, K25, YO, K9, (YO, K3) times, YO, K11, YO, K4, (YO, k1) twice, K13, (YO, k1) 3 times, K28, YO, K18, (YO, k1) three times, K6, (YO, k1) twice, K3, YO, k1 (total of 375 stitches)
Row 51 work even
Row 52 K54, (YO, k1) twice, K209, YO, K7, YO, K42, YO, K27, YO, k34 (total of 381 stitches)
Row 53 work even
Row 54 K5, YO, k1, YO, k78, (YO, k1) twice, [K26, (YO, k1) twice] 4 times, K26, YO, K8, YO, K7, (YO, k1) twice, K13, YO, K15, YO, K3, (YO, k1) three times, K2, YO, K3, YO, K22, YO, K9, YO, K17, YO, K7, (YO, k1) twice, K15, YO, K26, YO, k3. (total of 411 stitches)
Row 55 work even
Row 56 K2, YO, K21, (YO, k1) twice, K8, YO, K11, (YO, k1) twice, K6, YO, K250, (YO, k1) twice, K6, YO, k2, YO, K5, YO, K25, YO, K5, YO, K7, (YO, K3) twice, YO, K39, YO, K3, YO, K5, (YO, k1) twice, K4. (total of 432 stitches)
Row 57 work even
Row 58 K4, (YO, k1) twice, K12, YO, K54, YO, k150, YO, K15, YO, K39, YO, K37, YO, K16, YO, K26, YO, K7, (YO, k1) twice, K, YO, K3, YO, K4, (YO, k1) twice, K2, (YO, k1) twice, K27, YO, k27 (total of 451 stitches)
Row 59 work even
Row 60 YO, K16, YO, K8, YO, K11, YO, K48, (YO, k1) twice, K6, YO, K15, YO, K7, (YO, k1) twice, K6, YO, K15, YO, K7, (YO, k1) twice, K6, YO, K15, YO, K7, (YO, k1) twice, K6, YO, K15, YO, K7, (YO, k1) twice, K6, YO, K23, (YO, k1) twice, K35, YO, K4, YO, K13, YO, K7, YO, K32, YO, K16, YO, K17, YO, K3, YO, K13, YO, K4, YO, K3, YO, K36, YO, K16, (YO, k1) twice, K13, YO, k1 (total of 491 stitches)
Row 61 work even
Row 62 K13, YO, K38, YO, k1, K237, YO, K11, YO, K14, YO, K18, YO, K8, YO, K21, (YO, k1) twice, K40, (YO, k1) twice, K8, YO, K3, (YO, k1) twice, K2, (YO, k1) twice, (K14, YO) twice, K19, YO, K12, YO, k10 (total of 511 stitches)
Row 63 work even
Row 64 K2, YO, K15, YO, K5, YO, K3, (YO, k1) twice, K21, (YO, k1) twice, K10, YO, K9, YO, K9, YO, k1, K194, YO, K9, YO, K57, YO, K9, YO, K12, YO, K15, YO, K10, YO, K42, YO, K5, (YO, k1) twice, K21, YO, K20, YO, K22, YO, k1, K14 (total of 534 stitches)
Row 65 work even
Row 66 K5, YO, K5, YO, K3, YO, K3, (YO, k1) twice, K4, YO, K24, YO, K67, YO, K136, YO, K9, YO, K81, (YO, K3) 3 times, YO, K7, (YO, k1) 4 times, K10, YO, K3, YO, K43, YO, K24, (YO, k1) twice, K5, YO, K8, YO, K4, YO, K34, YO, K37, YO, k2, YO, k4 (total of 563 stitches)
Row 67 work even
Row 68 YO, K4, YO, K6, YO, K3, YO, K16, YO, K18, (YO, k1) twice, K55, YO, K26, YO, K25, YO, K68, YO, K9, YO, K45, YO, K7, YO, K25, YO, K23, YO, K27, YO, K8, (YO, k1) twice, K12, YO, K2, YO, K31, YO, K11, YO, K41, YO, K5, (YO, k1) 3 times, K14, YO, K76 (total of 591 stitches)
Row 69 work even
Row 70 K9, (YO, k1) twice, K7, YO, K27, YO, K41, YO, K33, YO, K29, YO, K7, YO, K19, YO, K9, YO, K7, YO, K9, YO, K9, YO, K9, YO, K7, YO, K29, YO, K7, YO, K65, YO, K35, YO, K2, (YO, k1) twice, K7, YO, K6, YO, K28, YO, K9, YO, K25, YO, K31, YO, K17, (YO, k1) twice, K3, YO, K9, YO, K3, YO, K3, (YO, k1) 3 times, K2, (YO, k1) 3 times, K8, YO, K12, YO, K41, (YO, k1) twice, K2, YO, K3, YO, K3, YO, K4, YO, K2 (total of 637 stitches)
Row 71 work even
Row 72 K2, YO, K11, YO, K1, K75, YO, K305, YO, K26, YO, K5, YO, K6, YO, K3, YO, K72, YO, K3, YO, K11, YO, K3, YO, K10, (YO, k1) twice, K2, YO, K26, YO, K10, YO, K24, YO, K41 (total of 655 stitches)
Row 73 work even
Row 74 K5, YO, K64, YO, K46, YO, K207, YO, K61, YO, K5, YO, K20, YO, K15, YO, K3, YO, K10, YO, K14, YO, K63, YO, K25, (YO, k1) twice, K2, YO, K114 (total of 670 stitches)
Row 75 work even
Row 76 K4, YO, K6, YO, K17, YO, K92, YO, K16, YO, K189, YO, K56, YO, K18, YO, K18, YO, K7, (YO, k1) twice, K3, YO, K1, K106, YO, K5, (YO, k1) twice, K3, YO, K3, YO, K6, (YO, k1) twice, K2, YO, K7, YO, K10, YO, K71, YO, K12, YO, K9, YO, K4 (total of 695 stitches)
Row 77 work even
Row 78 YO, K8, YO, K13, YO, K17, YO, K100, YO, K23, YO, K17, YO, K97, YO, K17, YO, K21, (YO, k1) twice, K30, YO, K25, YO, K10, YO, K58, YO, K3, YO, K88, YO, K11, (YO, k1) twice, K10, YO, K14, YO, K23, YO, K2, (YO, k1) twice, K87, YO, K16 (total of 720 stitches)
Row 79 work even
Row 80 K12, YO, K10, YO, K28, YO, K61, YO, K57, YO, K15, YO, K23, (YO, k1) twice, K14, (YO, k1) twice, K20, (YO, k1) twice, K14, (YO, k1) twice, K22, YO, K15, YO, K52, YO, K24, (YO, k1) twice, K, YO, K57, YO, K15, (YO, k1) twice, K12, YO, K50, DM, K47, YO, K5, YO, K14, YO, K20, (YO, k1) twice, K4, YO, K11, YO, K41, DM, K5, DM, K, DM, K29, YO, K13, YO, K5, YO, K2 (total of 754 stitches)
Row 81 work even
Row 82 K11, YO, K5, YO, K17, YO, K9, YO, K51, YO, K2, (YO, k1) twice, K23, YO, K270, YO, K18, YO, K45, YO, K10, (YO, k1) twice, K, YO, K18, YO, K5, (YO, k1) twice, K3, YO, K8, YO, K21, DM, K7, DM, K, D, K5, DM, K14, DM, K43, YO, K2, YO, K11, (YO, k1) twice, K6, YO, K3, YO, K7, YO, K8, YO, K62, DM, K8, DM, K26, YO, K6, YO, K7 (total of 782 stitches)
Row 83 work even
Row 84 K2, YO, K2, YO, K4, YO, K13, YO, K8, YO, K2, YO, K33, YO, K3, YO, K5, YO, K21, YO, K12, YO, K1, K273, YO, K7, YO, K48, YO, K24, YO, K14, YO, K6, YO, K19, YO, K3, YO, K63, D, K42, (YO, k1) twice, K4, (YO, k1) twice, K12, YO, K21, YO, K4, YO, K21, YO, K34, D, K, D, K8, D, K11, D, K48 (total of 804 stitches)
Row 85 work even
Row 86 K4, YO, K3, YO, K15, (YO, k1) twice, K2, YO, K19, YO, K2, (YO, k1) twice, K79, YO, K3, YO, K45, YO, K1, K167, YO, K5, YO, K33, YO, K13, YO, K12, YO, K16, YO, K10, YO, K13, YO, K19, YO, K9, YO, K6, YO, K9, YO, K23, YO, K15, YO, K5, YO, K26, DM, K7, D, K3, D, K5, DM, K3, DM, K5, D, K47, YO, K6, YO, K8, YO, K12, (YO, k1) twice, K15, YO, K4, YO, K14, YO, K5, YO, K7, YO, K1, K, YO, K1, K, YO, K12, DM, K21, DM, K48, YO, K4 (total of 840 stitches)
Row 87 work even
Row88 YO, K11, YO, K10, YO, K12, YO, K7, YO, K3, YO, K26, YO, K16, YO, K15, YO, K45, (YO, k1) twice, K20, (YO, k1) twice, K23, YO, K18, YO, K19, YO, K5, YO, K37, YO, K5, YO, K37, YO, K5, YO, K18, (YO, k1) twice, K42, (YO, k1) twice, K17, YO, K60, YO, K22, YO, K42, YO, K8, YO, K6, YO, K7, YO, K10, YO, K1, K90, YO, K8, YO, K13, YO, K15, YO, K8, (YO, k1) twice, K4, (YO, k1) twice, K22, YO, K15, YO, K5, (YO, k1) twice, K11, YO, K19, DM, K56, YO, K12 (total of 887 stitches)
Row 89 work even
Row 90 K11, YO, K5, YO, K18, K34, YO, K13, YO, K5, YO, K25, YO, K8, YO, K12, YO, K23, YO, K24, YO, K3, YO, K38, (YO, k1) twice, K22, YO, K43, (YO, k1) twice, K42, (YO, k1) twice, K66, (YO, k1) twice, K6, YO, K40, YO, K97, YO, K8, YO, K16, YO, K9, YO, K5, YO, K8, YO, K38, D, K9, D, K20, D, K8, D, K41, YO, K16, YO, K7, YO, K12, (YO, k1) twice, K6, YO, K19, YO, K9, (YO, k1) twice, K19, YO, K64, D, K32, YO, K2 (total of 921 stitches)
Row 91 work even
Row 92 K2, YO, K35, YO, K24, YO, K5, YO, K10, YO, K8, YO, K122, YO, K17, YO, K31, YO, K7, (YO, k1) twice, K6, YO, K31, YO, K7, (YO, k1) twice, K6, YO, K31, YO, K7, (YO, k1) twice, K6, YO, K31, YO, K41, YO, K76, YO, K50, YO, K29, (YO, k1) twice, K23, YO, K70, DM, K57, YO, K6, YO, K2, (YO, k1) twice, K10, YO, K16, (YO, k1) twice, K14, YO, K45, YO, K29, YO, K35, YO, K11, (YO, k1) twice, K6 (total of 958 stitches)
Row 93 work even
Row 94 K5, YO, K4, YO, K11, YO, K3, YO, K2, YO, K19, YO, K10, YO, K5, YO, K23, YO, K10, (YO, k1) twice, K41, YO, K54, YO, K15, YO, K23, YO, K200, YO, K12, YO, K89, YO, K23, YO, K10, YO, K43, (YO, k1) twice, K1, YO, K20, YO, K4, (YO, k1) twice, K64, YO, K43, YO, K47, YO, K3, (YO, k1) 4 times, K9, YO, K2, YO, K39, YO, K6, YO, K21, DM, K42, DM, K8, D, K32 (total of 994 stitches)
Row 95 work even
Row 96 K4, YO, K3, YO, K2, YO, K10, YO, K6, YO, K8, YO, K15, YO, K65, YO, K117, YO, K24, YO, K165, YO, K12, YO, K80, YO, K25, YO, K7, YO, K35, YO, K4, YO, K2, YO, K5, YO, K33, YO, K23, YO, K2, YO, K12, YO, K70, D, K9, D, K37, (YO, k1) twice, K2, YO, K8, YO, K12, YO, K8, YO, K3, YO, K15, (YO, k1) twice, K2, YO, K30, YO, K96, D, K34 (total of 1025 stitches)
Row 97 work even
Row 98 YO, K10, YO, K5, YO, K18, YO, K84, YO, K10, YO, K18, YO, K135, YO, K12, YO, K11, YO, K12, YO, K15, YO, K12, YO, K11, YO, K12, YO, K15, YO, K12, YO, K11, YO, K12, YO, K132, YO, K33, (YO, k1) twice, K9, YO, K19, YO, K27, YO, K12, YO, K13, YO, K6, YO, K7, YO, K11, YO, K17, D, K, DM, K4, DM, K2, DM, K53, YO, K9, YO, K34, YO, K27, YO, K5, (YO, k1) 8 times, K2, (YO, k1) twice, K4, YO, K12, YO, K13, YO, K136, YO, K4 (total of 1072 stitches)
Row 99 work even
Row 100 K31, YO, K63, YO, K4, YO, K6, YO, K13, YO, K19, (YO, k1) twice, K39, YO, K358, YO, K73, YO, K45, (YO, k1) twice, K3, YO, K10, YO, K21, YO, K2, YO, K8, YO, K5, (YO, k1) twice, K33, DM, K20, YO, K16, YO, K9, D, K41, YO, K13, YO, K29, YO, K23, (YO, k1) twice, K10, YO, K23, YO, K5, YO, K10, YO, K64, YO, K9, YO, K5, YO, K51 (total of 1104 stitches)
Row 101 work even.

2 comments:

  1. I want to make one, but Now!!! Darn it Christmas time! Unfortunately no mathematicians in the family...thanks for the instructions and all the great info on it.

    ReplyDelete
  2. WOW, This could be a challenge. Thanks for the knit instructions.

    ReplyDelete