Step by step instructions for ages 6 and up

Transcription

1 Step by step instructions for ages 6 and up Daina Taimina 2013

2 How to make a paper model of the hyperbolic plane 1. Cut along the solid lines the sheet with hexagons (6 sides) along the outside edge. 2. Cut out hexagons 1-5 notice where the solid lines are and cut only along them. 3. With Scotch tape glue in extra hexagons in five openings. 4. Cut out five heptagons (seven sides). 5. With Scotch tape glue the heptagons in the holes. 6. You can continue to build your model by adding more heptagons and hexagons along the edges. Make sure that in each vertex come together two hexagons and one heptagon. 2

3 Cut along solid lines only! 3

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5 To crochet a model of a hyperbolic plane we will use only two basic crochet stitches chain (ch) and single crochet (sc). If you know how to crochet, you can skip now to the pattern. If this is the first time you are trying to crochet, let me try to explain what you should do. 1. Take your yarn so that short end (about 2 3 in) is in your front and pull yarn through this first loop. Pay close attention short end is under the log end! If you will take it other way around, then it will unravell once you try to pull the yarn through. 2. This is how first loop will look. Now pull the short end so that the first loop is tightened. 1a. The other way to start the first stitch is to put a yarn across your index finger, then insert your crochet hook under the yarn and rotate it once so that yarn is intertwined under the hook like you can see it in the next picture. 5

6 2a. Now pull the yarn through the first loop and tighten with pulling a short end of the yarn. 3. This is how the first chain stitch looks. 4. Continue this chain as long as you wish. For the first hyperbolic plane model crochet about 10 chain stitches. 5. Once the first row is completed you need to turn around. The second row and all the rest will use single crochet stitch. Insert the crochet hook in second chain (you are using the first chain to turn around ) and pull the yarn through the chain, leaving two loops on the hook. 6

7 6. Take the yarn over and pull through both loops. One single crochet stitch has been completed. To practice you should crochet a little rectangle it means finish the row you just started then at the end of the row turn around, make one chain stitch, and start the next row with single crochet, continue. 7. To crochet hyperbolic plane you will have to make increases. To increase means that you crochet two single crochet stitches in the same loop. When you have completed one stitch, insert the crochet hook in the same loop and crochet another single crochet stitch in the same loop. 7

8 1. Crochet 10 chain stitches. 2. Turn around. Insert the crochet hook into the second chain from the hook. Crochet 3 single crochet one in each chain from the previous row. In the 4th chain do the increase crochet 2 single crochet in the same loop. Crochet again 3 single crochet one in each chain, then do the increase again, finish the row with single crochet. 3. Turn around and crochet one chain, so that you can start a new row on a top of the last stitch in the previous row. 4. Crochet 3 sc, do increase in 4th, and so on. Soon you will notice that your piece will no more be flat. Continue following this pattern as long as you wish. Another option is to crochet hyperbolic plane starting from a circle. I have described in details how to crochet symmetric hyperbolic plane in my book Crocheting Adventures with the Hyperbolic Planes. It involves some calculations. Here I am giving you simpler option. 1. Crochet 5 chain stitches. Join them together with a slip stitch it means insert the crochet hook in the first chain and pull the yarn through both loops. Crochet around this circle single crochet as much as you can. 8

9 2. Once you completed this first row you have to options you can join the end and the beginning of this row with the slip stitch, then one chain to start the next row. In the second and every next row crochet 2 sc, increase in 3rd stitch, 2 sc, increase, etc. 3. The other option is to crochet in spiral pattern once you finished the first row of single crochet, start the pattern without joining the beginning and the end of the row. In both cases soon your piece will no more be flat. In order to crochet model of the hyperbolic plane you should keep the same increase pattern throughout your work. But I encourage you to experiment start another piece and change the place where you do increase. You will notice that if you do increases often your work will have more ruffles. This is the hyperbolic plane when increases are done in every 3rd stitch. If you want to make drawings on the model of the hyperbolic plane, then increases should be made less frequent. For this large model it is crochet 12 sc, then increase, crochet 12 sc, increase, etc. 9

10 If you start with a very litle circle (3 chains joined together) and then crochet in spiral, the result will be so called pseudosphere. Experiment with different ratios, with different colors, yarns! If you want your model to keep its shape, then choose the yarn that will not stretch a lot and choose a smaller number crochet hook than the one that is suggested on the label on a skein of your yarn. Happy crocheting! 10