msb@lsuc.UUCP (Mark Brader) (03/23/85)
Mark Leeper: > Placed inside a square frame, one side of which is the length of the breadth > of the triangle, a Rouloux shaped drill bit can be made to follow the > edges of the square. Norman Diamond: > Doesn't that produce a "square" with rounded corners? Greg Kuperberg: > Nope. Apparantly, the Rouloux drill bit has 90 degree corners. This is > just the right angle for fitting in the corner of a square; ... Me: Hey guys, that's *Reuleaux* triangle. Now remember, the way you construct one is to replace each side of an equilateral triangle ABC with a circular arc centered on the opposite vertex. Consider the arc AB. Since the line segment AC is a radius of this arc, the tangent at A to the arc must be perpendicular to AC, and therefore at 30 degrees to the line segment AB. Similarly the (tangent to the) arc AC will be at 30 degrees to the line segment AC. So the (tangents to the) arcs meet at A at an angle of 120 degrees, and the point A canNOT be fitted into the corner of a square. If there are drill bits that make square holes, they are not shaped like Reuleaux triangles. Perhaps they are a similar shape with arcs centered around a point further away than the opposite vertex, so that they do meet at 90 degrees. Such a shape would not have constant breadth, but perhaps it is close enough that the difference can be handled all right mechanically. Mark Brader