Thursday, January 12, 2017

Compass only non-collapsing compass (Euclidea Series)

Since I've branched into this topic, I want to include some notes on additional references I've found and some results I've been able to get. First, an admission:

I'm still having trouble finding the intersection points of a circle and a line when the circle's center is on the line!
Another resource
James King (UW home page) has a nice session outline working through compass only constructions. Be warned, there are some spoilers for some of the Euclidea challenges in that material. Unfortunately, I can't tell from write-up when and where this was used. Maybe this NWMI meeting?

In Professor King's notes, he describes the construction I'm struggling with as "important and difficult," so I will have to redouble my efforts.

Non-collapsing compass from collapsing compass
As an aside, perhaps you have noticed that everyone else refers to the two types of compasses as collapsible and non-collapsible. Doesn't that terminology strike you as wrong, too? The point isn't that one type of compass is able to collapse, but rather that it always collapses when not drawing a given circle. Also, a common form of "non-collapsible" compass is able to collapse!

Just squeeze the legs together to collapse

Anyway, I was able to figure out the construction of a non-collapsing compass, just in time for Euclidea 13.2.

I'll give a construction below. For a more mild hint, the key is the ability to reflect points through a line we've already "constructed" (where we have two points on the line).

I've chosen to do this in GeoGebra so I can show a sequence, explain some of the reasoning, and grey out earlier parts of the construction that we don't need anymore. Check my work to make sure I didn't slip in any subtle straightedge moves!

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