Straight Line Curves

It’s a really relaxing bit of recreational mathematics (albeit with some serious theory behind it), creating beautiful curves just by drawing straight lines. I can only give a glimpse in this blog, but the possibilities are endless. If you’re tempted to have a go, one word of caution: be as accurate as you can since the final result can be disappointing if you are careless, or work too fast.

My first example is probably about the simplest you can do. I’ve used graph paper (and in later examples) simply to make the accurate measurement easy, but the end result would be better on plain paper since the graph grid lines do distract from the end result.

Simple hyperbolic(ish) curve

Draw x– and  y-axes labelled from 0 to 10. Join (0, 9) to (1, 0), then (0, 8) to (2, 0) and so on. In the diagram I’ve drawn the first three lines:

 

 

 

 

 

 

 

 

You can just see the beginnings of a curve appearing. When you’ve drawn 9 lines, this is what you get:

 

 

 

 

 

 

 

 

By adding in another 9 lines, such as (7.5, 0) to (0, 2.5), the curve is much better delineated:

 

 

 

 

 

 

 

 

 

Four converging spirals

My next example needs great care since each stage depends on the accuracy of the previous stage. First, draw an exact 10cm x 10cm square. Mark a point exactly 1cm along each edge, and join them up to form a new square:

 

 

 

 

 

 

 

 

Now mark points exactly 1cm along the new square, and join them up. In the next diagram I’ve added three more squares:

 

 

 

 

 

 

 

 

 

Keep going until it is impossible to draw any more squares, and you end up with something like this (I’ve accentuated the spirals):

 

 

 

 

 

 

 

 

 

It’s not brilliant, is it? The graph paper, and the rather messy construction spoil it. And it would be better if I’d added a new square every 0.5cm. But here’s a computer generated version of the same thing, and you can see what a difference perfect accuracy makes:

 

 

 

 

 

 

 

 

 

Furthermore, it’s often possible to copy diagrams such as these and then tile them. The two diagrams below show what happens if you rotate and then tile this diagram four times, and then translate the new diagram and tile four times!

 

 

 

 

 

 

 

 

 

Cardioid

My final example is one of my favourites, probably because it’s another pretty simple one to construct but with a spectacular result. Start with a circle marked off at 10 degree intervals, labelled 1 to 36. Join 1 to 2, 2 to 4, 3 to 6 and so on. The first two diagrams show the first 9 lines, and the first 18 lines.

 

 

 

 

 

 

 

 

The curve is being created because each new line passes a bit closer to the centre of the circle, although only the 18 to 36 line actually does pass through the centre.

Now to join 19 to 38, treat 38 as 2; and to join 20 to 40, treat 40 as 4; and so on. Your final line will join 35 to 34, and the shape you have created is a cardioid because it looks like a heart:

 

 

 

 

 

 

 

 

If you’d like more examples there are plenty of websites devoted to the topic. Try searching for ‘straight line curves’ or ‘curve stitching.’ For example, there are some nice ideas on the MathCraft site here.