Tuesday, January 10, 2017

Beginning Restricted Constructions (Euclidea Series)

Note: This is a little bit out of order, but reflects the challenge on which we're currently working.

Choices of Construction Rules
The classic construction problems involve specific, restricted tools:

  1. unmarked straight-edge: this tool is only able to determine a line that contains two existing points (previously given or already constructed). It has no length markings and only one side.
  2. collapsing compass: this tool is only able to determine a circle given the center and a point on the circle. Again, both center and an included point must already be given earlier in the construction. Like the straight-edge, the compass has no way to indicate the length of the radius. (side note: when I was in HS geometry, we didn't require the compasses to be collapsing).
  3. A plane as a writing surface and infinitely fine/precise pens: these allow us to identify the intersection points of the constructed objects (circles and lines).

I, and maybe you, for years considered these rules to be natural. However, there are several other construction rules that are at least as natural:

  • Origami: moves based on folding paper. This is really a fascinating topic. Take a look at this Numberphile video for a taste.
  • Straight-edge only constructions: for those unfortunate to have left their compass at home.
  • Compass-only constructions: easy to imagine how ancients made a good compass, but how would they have gotten a really straight edge anyway?!

Several of the Euclidea challenges impose either straight-edge only or compass-only restrictions, which got me interested in this general topic again.

Examples in Euclidea
Unless I've missed some, the restriction challenges start officially in Theta pack:

  • Drop a perpendicular (8.4): we are only given the straight-edge, but we are also given a circle.
  • Mid-point (8.5): find the midpoint of a segment with the straight-edge and a parallel line.
  • Segment trisection (9.7): same restrictions as 8.5
  • Midpoint (13.1): this time, we've only got the compass, no straight-edge
  • Some I've not yet unlocked: Tangent to Circle (13.5), Drop a Perpendicular (13.7), Line-Circle Intersection (13.8)
In addition, Zeta pack has a some challenges that, while not marked as restrictions, are good warm-ups: Point Reflection (6.1), Line Reflection (6.2), and Translate Segment (6.6).

Humans can forget things!
If you've made it to Zeta pack, you know that the collapsing compass is able to replicate the function of a non-collapsing compass, so that particular restriction doesn't change what is theoretically constructible (but it sure reduces a lot of extra steps and auxiliary objects in real constructions.)

It turns out that the straight-edge is also unnecessary! This result is the Mohr-Mascheroni Theorem. One really fascinating/disturbing fact about this result is that it was first proven, as far as we know, in 1672 by Mohr, but his proof was lost for over 250 years. Mascheroni independently discovered and proved the result about 120 years after Mohr.

I think this is an important lesson that, with search-empowered internet, is easy to forget: human knowledge accumulation isn't always steady or certain.

Compass-only program
Cut-the-Knot (a generally excellent resource) has a good discussion of restricted constructions and outlines a program for proving the sufficiency of compass-only constructions.

I have started to work through this list. Generally, I won't post solutions since Cut the Knot already has solutions. However, I was concerned about whether the compass they were using was collapsing or not. The solution for 6 (given three points, find a fourth point that makes them vertices of a parallelogram) clearly uses a non-collapsing compass. So, that leaves an open challenge: how to build a non-collapsing compass from just a collapsing compass.

I'm currently stuck on this. There are two ways I could break through:
(a) find the point of intersection of two lines. If I could do this, I would use the ability to find perpendicular lines to move distances around.
(b) Find the point of intersection of a line and a circle with the center on the line.

Note, this second one is very similar to Cut the Knot's third challenge, but that assumes the center of the circle is not on the line segment. Amazingly, this slight change makes the challenge much harder, at least for me.

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