Tuesday, May 26, 2015

Machinery, Lines and Circles

On Facebook, someone posted an animation of how a sewing machine works. It wasn't enough to help me understand how the top thread manages to get around the bobbin mechanism. I searched on youtube, and nothing helped. This article on math and the sewing machine made me think for a moment that I was getting it, but I still am not. How is that bobbin mechanism held in place in a way that allows the thread to get around it? (Do you see how the top thread moves past the whole back of the bobbin? How is that possible?) They say that the bobbin is held snugly inside its case, but how is the out part attached?

I think I need a transparent sewing machine, so I can really see how this is working.

On thing leads to another (especially online!), and I ended up at this site from a museum for mathematics, called The Garden of Archimedes, in Florence, Italy, where I encountered this very simple statement about the difference between constructing a circle and a line - something I had never thought about before.

The simplest curves are doubtless the line and the circle. To draw circles, one uses a compass. It's sufficient to keep a constant distance between the tracing point and the centre, and one obtains a near-perfect circle, even with a primitive compass. At first sight, one would think that tracing a segment is also a very simple operation: you just need to use a ruler or pull a string taut. In fact, things don't work exactly like that. In order to draw a good straight line with a ruler, one needs the ruler itself to have a "straight" side, but the value of a ruled line depends on the ruler that was used to make it. So, who made the first ruler? To apply the same method to the circle would mean, for example, to take a coin and trace its edge - the circular profile would be "intrinsic" to the instrument itself.

It would be better to apply to the straight line the principle used to draw the circle, rather than vice versa.
Inatead of using a ruler or straightedge, can't you use the "pull a string taut" method, with something a bit less flexible than string? Maybe something that freezes into position? Hmm... Apparently that's not the avenue that was followed. You can find out the fascinating history of the solutions people found for this problem by going to the Garden of Archimedes site.


  1. I was looking up the very same thing not long ago, and had exactly the same question!

  2. One other direction of line vs circle is the comparison between compass+straight edge constructions and origami. I'm not an expert, so take the following with a little caution:

    To a modern way of thinking, origami is more powerful because the set of constructible points is larger. Not larger cardinality, since both are still countable but larger because all C+SE constructible points are origami constructible, but not vice versa. For example, generic angle trisection is possible with origami, but not C+SE.

    On the other hand, origami can't construct a circle. If you consider geometric shapes to be the fundamental objects of study (like the ancient Greek mathematicians seemed to) then origami is less powerful.


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