There used to be one of these in my toolbox:

http://www.machinemart.co.uk/images/library/range/large/0104...

The small remaining bits at the corners were pushed out by the pointed segments by feeding the bits of leftover wood into the rotating bit.

Worked wonderfully, I wonder during which move i managed to lose it, I always thought it to be a pretty ingenious device.

Mathematically not as clever as the one in the article but definitely simpler and it did the job.

Thanks, found it! It's a "mortise chisel and drill": a much more practical solution that I can buy!

While the Reuleaux concept is mathematically cute, it isn't very practical and is only approximate. And the Reuleaux bit requires a special chuck to ensure that the square is "upright" (not tipped at some arbitrary angle): http://upper.us.edu/faculty/smith/reuleaux.htm for details.

There's a joke in this somewhere about mathematicians and carpenters.

I've been pleasantly surprised at how often I get to use my bench-top mortising machine for projects outside of mortise and tenon work.

Here is an animation showing how it works:

http://upload.wikimedia.org/wikipedia/en/2/20/Rouleaux_trian...

That's only the Rouleaux bit. The article details a bit that produces sharp (vs. rounded) squares.

It was confusing b/c the pictures didn't show that the bit's axel had to orbit independently of the bit. This animation cleared that up for me even if it wasn't the exact case.

The link above is an animation of the Rouleaux Triangle and will draw a square with rounded corners. In this article, the author talks about a variant of that triangle that will draw exact squares.

Ok, given my login name, I feel like I have to comment on this article.

Also, here is a video of this in action. http://www.youtube.com/watch?v=9qEhyQfbImY

Rounded corners.. giving problems to engineers for 100+ years. :)

This and other amusing/interesting mathematical diversions are covered in one of Martin Gardner's books:

http://www.amazon.com/dp/0226282562/

Am I missing something? The holes can't be square, because of the rotating motion required to create them they will always have rounded corners.

I'd like to see pictures of the holes created with this bit.

You are right, it's not really square, but the rounded edges can be chiseled away fairly easy, especially if the drill bit has the chisel built right into it as jacquesm's earlier image showed (http://www.machinemart.co.uk/images/library/range/large/0104...). although if it's hard wood, you may need to use a solid chisel and a hammer to pound the heck out if it.

That's what I thought, too, but then I realized that what we were both missing is that the edges of the cutting surface are not the same as the edges of the shape.

The vertices really do trace out a right angled square, so if you just put cutting surfaces at the vertices and inside them, it really will cut a square.

a man in china invented a bike with non-circular wheels

http://technology.timesonline.co.uk/tol/news/tech_and_web/ar...

http://www.china.org.cn/china/photos/2009-05/07/content_1773...

Amusing: "He believes that people will be drawn to the bike because it requires more work to cycle and therefore will provide more exercise for the cyclist than a conventional bike."

I wonder if the unsolved problem of drilling a triangle hole is related to the geometric problem of dividing a line into three equal parts with just a compass.

You can't trisect an arc using only a compass and a straightedge. (You can simulate a straightedge using a compass.) Dividing a line into 3 parts is trivial.

Funny I'd forgotten that that was possible. http://upload.wikimedia.org/wikipedia/en/d/dc/Trisectsegment...

The curve of constant width, square drill thing was in Martin Gardner's Scientific American column 30 years ago....

Nice to see it again.

rotary engines.