Once upon a time there were plenty of square pegs jammed into round holes, but those dark times have passed. Now there is a special shape that allows people to drill square holes.

This special structure was invented long before the Industrial Revolution created a use for machines that could drill uniform holes over and over. Think of this shape as a 'square peg' in the 'round hole' of pre-industrial society. It's called the Reuleaux Triangle, after Franz Reuleaux, who put it to work. It's relatively easy to construct.

Take a point, and draw a circle of radius R around it. Draw a line from the center point of the circle to the edge. It doesn't matter where. Not take the point where the edge intersects the line, and use that as the center point to another circle, with the same radius (R).

You'll now get a structure that looks like overlapping rings. They'll instersect at two different places. Go to one of the intersection points, and use that point as the center point for yet another circle, radius R.

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What you've got now is a venn diagram. The center space, the space in a venn diagram which you would list any objects that shared the characteristics of all three circles, is a Reuleaux triangle. Remember, this isn't any venn diagram. Each of the circles has to have a radius that runs through the other circles' center points.

This 'triangle,' with edges made up of curves, can be moved in a regular motion in a square hole. Reuleaux triangles are often used as models for drill bits. They use the angles of the triangle, spiraling up around a central pole, to create perfectly square, and evenly sized, square holes for square pegs. The geometry of a circle turns into the geometry of a triangle, which turns into the geometry of a square.

Via Wolfram Math World and Whistler Alley.