I am somewhat embarrassed to admit today was the first time I actually played with a physical compass despite being well into my 20s,having studied university maths for years and worked with the euclidean constructions for a year.
Having run though most of the simple geometry questions I’ve had recently I thought I should return to just playing a bit more with the actual construction parts instead of the proof parts and the puzzle like questions of trying to figure out construction which consume the smallest number of operations. I was originally mesmerized by this sort of question when I looked up some of the constructions for how to trisect a line and finding it kind of surprising how few operations (circles and lines) you can reduce it to. When sketching constructions by hand it’s easy to loose track of how many steps you need to perform an action in practice as you bisect and angle there or draw a perpendicular here while the number of circles and lines being implicitely necessary can be quite staggering. On top of that most of the constructions in the Elements employ the operation of ‘moving a line segment to a point’ which if you’re actually to do it with circles and lines makes you go “HELL NO!. because (as far as I’ve counted) takes 4 circles and 2 lines for that simple thing…
Playing more by hand should probably add the necessary frustration to push me to find or look up some new methods.A for my weekly TikZ practice here is the fastest way to draw a parallel line through a point I’ve come up with so far:
which relies on the similarity of inscribed triangles.I would be genuinely surprised (or embarrassed) and not a little bit impressed if there was a faster one but I suppose eventually I’ll bother looking up whatever theory there is concerning minimality problems.