Compass and Straightedge.

As I remember high school Geometry, it was mostly theorems and proofs and was very fascinating.  Bertrand Russell once said, “At the age of eleven, I began Euclid, with my brother as my tutor.  This was one of the great events of my life, as dazzling as first love.  I had not imagined there was anything so delicious in the world.”  (For the most part I agree with that, although I’m reluctant to call it as dazzling as first love.)   In my first Geometry class, there seemed to be a departure from proof of theorems and their corollaries to talk about constructions with a straightedge and compass.  I never really got why there was any emphasis on this at all.  Why was it important?  Nevertheless, that’s what this web site is about and, in particular, what this “musing” is about.

I want to tell you something about any compass and straightedge construction that grabbed my attention when I first learned of it.  In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.  In other words, you don’t need the straightedge.  WHAT you say!  Impossible—the perpendicular bisector of a line segment certainly can’t be drawn with a compass!  Yes, it is clearly obvious that it is impossible to draw a straight line without a straightedge.  But the operative words in the last two sentences are the words “drawn” and “draw.”  Remember, a straight line is determined by two points, and points can be located with a compass alone (no straightedge required); that’s what the Mohr-Mascheroni theorem means. Though the use of a straightedge can make a construction significantly easier, the theorem shows that any set of points that fully defines a constructed figure can be determined with compass alone, and the only reason to use a straightedge is for the aesthetics of seeing straight lines, which for the purposes of construction are functionally unnecessary.

Over the coming months, I am going to show several compass-only constructions.  Currently, one of the animations on the Mathematics as Art page employs a compass-only construction of the curve known as the Astroid.  Compass-only constructions are difficult but extremely pleasing and lend themselves very nicely as the basis for using mathematics to generate abstract art and design.

An interesting little sidebar to this Mohr–Mascheroni theorem is the fact that Georg Mohr was a Danish mathematician and lived in the 17th century while Lorenzo Mascheroni was an Italian mathematician and lived in the 18th century, yet the theorem was not named the Mohr-Mascheroni theorem until 1928.  It turns out that Mohr first proved the theorem in 1672 and published it in an obscure book entitled Euclides Danicus.  Mohr’s work languished in obscurity until 1928 when it was re-discovered.  Meanwhile, Mascheroni came along 125 years after Mohr, hypothesized the theorem, and also found a proof for it.  However, the priority for this result (now known as the Mohr–Mascheroni theorem) belongs to Mohr.

Motivated by Mascheroni's result, in 1822 (now we jump to the 19th century) Jean Victor Poncelet conjectured a variation on the same theme. He proposed that any construction possible by straightedge and compass could be done with straightedge alone. The one stipulation, though, is that a single circle with its center identified must be provided. This result is now known as the Poncelet-Steiner theorem since it was proved by Jakob Steiner 11 years later.

April 2019