Sometimes getting crafty means getting mathsy! Try some crochet to learn about curved surfaces.
You will need
- Crochet hook
What to do
Make a chain
- Take the end of your yarn and tie a small loop in it.
- Put your crochet hook through the hole, and hook onto the yarn hanging below the loop.
- Pull your hook back through the loop. You should now have a new loop around your crochet hook.
- Hook onto the yarn again, and pull it through the loop to make another link in your chain.
- Repeat until you have a long enough chain.
Make a stitch
- Once you’ve crocheted a chain, you need to crochet back along the chain. You should already have one loop on your crochet hook
- Put your hook through the second last stitch in the chain
- Hook onto the yarn and pull it through the stitch. You will now have two loops on your hook.
- Hook onto the yarn again, and pull it through both loops. You should end up with one loop on your hook. Congratulations! You’ve just finished one stitch.
- Now, stick your crochet hook through the next loop in the chain and repeat.
- When you get to the end of a row, do one chain stitch (like when you were starting), and then come back the other way.
To make a hyperbolic surface
- Start with a chain of about 15-20 links
- After five stitches, put your hook into the same loop as last time, not a new one. Follow all the other steps to finish this stitch, and then continue with more regular stiches. This is called increasing.
- Increase every fifth stitch as you go along this row, and every row after it. When you think your crochet is big enough, cut the yarn, and feed the end through the last loop you made. You now have a crocheted hyperbolic surface!
Crochet is a traditional way of making fabrics and it has been around for hundreds of years. It is only more recently that people have realised how mathematical it is.
If you just do single stitches back and forward, you end up with a flat sheet of material. You can fold it or roll it, but it will always be able to be laid out flat on the floor.
If you reduce the number of stitches in each row, you end up with a piece of material that won’t lie flat. The edges might lie down, but the middle will always pop up. Mathematicians call this a positively curved surface. Soccer balls are positively curved, as are the toes on socks, and the tops of the shoulders on a shirt.It is also possible to make a negatively curved surface, by adding stitches in each row. A negatively curved surface will also not lie flat, but this time it won’t be the middle that sits up. Instead, there is too much material at the edge, and it will be all ruffled up. This kind of surface is a hyperbolic surface. You can find negative curves in armpits on shirts, and in between pants’ legs.
Geometry on a curved surface is very strange. If you draw a triangle with two small angles on a flat surface, the third angle needs to be big so that the sides meet at a point. But if you draw on a hyperbolic surface, then all three angles can be small. If you draw a triangle on a sphere, all three angles can be big!Although it seems strange, curved geometry is very important. For example, the world is round, so if you want to navigate over long distances, you need to use spherical geometry, not flat geometry.
Scientists also thought that the whole universe could be curved, like a piece of crochet. To test this theory, NASA launched a satellite called WMAP, which looked at microwave energy. In 2008, scientists used the information WMAP had collected, and decided the universe probably wasn’t curved.
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