0

Questions like this one build a square using only a compass but with variable size. I am wondering if it is possible to construct a square using only a fixed size compass. If so, how to proceed with the construction?

Edit: Please consider Mohr-Mascheroni Theorem and that the square is built once the four vertices are determined

My question was put on hold with the following argument: "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – The Count, Henrik, Paul Frost, Shailesh, Leucippus"

Here is my answer: this question is important to me because as a woodworker and an enthusiast of precision tools and primitive/manual tools, I like to think about ways how one could achieve good levels of precision just using DIY tools e.g suppose you got lost somewhere and need to build some tools from the ground up just using raw materials.

This reasoning leads to my question: if you're lost and without tools, constructing a straight edge is not trivial. Also, a variable-sized compass may also not be so trivial to build. But a fixed compass is trivial: just grab a piece of fallen wood (e.g an irregular wood branch) and use it as a fixed-sized compass. With a fixed compass you can easily build a circle. With this circle, you can build a hexagon and equilateral triangles. You can also get a 90-degree angle from the hexagon. A square is a shape that is common in our society. So, if one is lost and with a piece of fallen wood to use as a fixed-sized compass and need to draw a square, could it do using just this piece of fallen wood?

I don't know if my motivations are valid to this community, but as they were to me and the question has a geometrical background, I gave a shot and decided to ask here.

Hadley Siqueira
  • 1
  • Do you also have a straightedge? – Arthur Jul 31 '19 at 13:48
  • @Arthur No. Only the fixed compass. I am considering Mohr-Mascheroni Theorem and just need the four vertices of the square – Hadley Siqueira Jul 31 '19 at 13:52
  • It's possible. Draw a circle, select a point on the circumference and draw a circle of the same radius. Two intersection points will be two of the vertices. Now you need to do the same with the second circle, I'll leave it to you how to pick center for the third circle of the same radius. – Vasili Jul 31 '19 at 14:12
  • 1
    @Vasya Are you sure Mohr-Mascheroni holds for a compass of fixed size? – Misha Schmalian Jul 31 '19 at 14:19
  • 1
    Mohr-Mascheroni allows/expects compasses to make circles of varying radius. (Just look at the various sample constructions in the Wikipedia page.) Fixing the radius makes for an interesting puzzle, but it doesn't have much to do with M-M (except as concerns the rules of determined figures; eg, the square is considered determined once its vertices have been constructed). ... Or, are you using "fixed size compass" to mean "rigid compass" (vs "collapsible compass")? – Blue Jul 31 '19 at 14:20
  • @MishaSchmalian: I did not mean to say that all circles used during the construction will have the same radius, my suggestion is just a starting point. – Vasili Jul 31 '19 at 14:26
  • @Vasya wouldn't this violate the restriction of using a fixed size compass? – Hadley Siqueira Jul 31 '19 at 14:31

0 Answers0