Constructions with a compass alone

Back Tuesday I discovered the rather tantalizing Mohr-Masceronic theorem which essentially says that anything you can do with a compass and straight edge you can do with a fixable alone. It’s not exactly that the straight edge is completely redundant as for the compass alone constructions you need a fixable compass, i.e one you can lift or slide along the paper while Euclid makes due with a non-slidable compass. Often called a ‘snap-compass’ imagining that it snaps together if you lift it from the paper.

The equivalence between a fixable compass and a (snap compass+straightedge) as a tool was as one might remember the point of Prop.2 Book 2 ( http://aleph0.clarku.edu/~djoyce/elements/bookI/propI2.html ) and though I’d like everything to be possible using a snap compass alone that’s supposedly not the case.

Anyhow some cut-the-knot apparently has a very nice sequence of compass only construction which lead you through some of the steps to prove the Mohr-Masceronic theorem which are pretty neat as puzzles (http://www.cut-the-knot.org/do_you_know/compass.shtml)

Still I am for whatever insane reason partial to the snap compass over the real compass so I’ve also been playing at finding what problems I can solve with just that tool and I was somewhat delighted that I was able to adapt the construction of a parallel line through a point from my previous post ( https://seriouscephalopod.wordpress.com/2015/04/01/working-with-a-compass-and-straight-edge-takes-time/ ) to a snap compass a unexplained visualization of the construction being shown below:

Compass1

Of course this problem of finding a parallel line through a point (or rather a fourth point which with the off-line point forms a parallel line) is depressingly trivial to solve if you can move the compass as it’s the same as creating a paralellogram:

ParalellUsingFixableCompass

I also tried out making a .gif for the snap-compass construction which is mostly just blurry and annoying but now I’ve tried that for the first time.

AnimatedParaline

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