What's interesting to me is how many of the ancient problems involve using compass and straightedge. Recently I have been trying to draw Islamic geometric patterns (or other tilings, like quasitilings) using compass and ruler, and it can be really difficult!
I can kind of see, though, why considerations of what integer ratios are 'good' for such diagrams and questions like angle bisection or intersections between circles and lines become interesting topics. It can really affect how easy or hard it is to draw such a diagram
A note to the circles: It is not so much about circles, but rather "Some points which share the same distance to another point X"
If you follow a manual of how to construct something with compass and straightedge, the job of the circles is often only to intersect with something else, and these points of intersection are of actual interest (as far as I remember).
Exactly this. There are many patterns that can be constructed by drawing a regular array of circles, then connecting various intersection points with lines, then erasing the original circles.
As it happens, I sometimes find that the same drawing can be achieved by a simpler construction path that involves (say) only midpoints of squares, which makes life a lot easier.
That will work, but you need to have the entire square already marked. (As opposed to e.g. one or two of the sides marked and representing a hypothetical square.)
I wonder, though, is it not a coincidence that they're practical tools to do rather precise multi-step constructions? (E.g. Durer wrote a whole book about type design by those methods.) And with all the geometric algebra in Euclid, did they ever use them for calculations that aren't originally geometrical? Would we know?
I believe that ancient geometry was used for governance and engineering. However the compass and straightedge had an element of abstractness or deliberate simplified impracticality even back then: they had rulers and strings, and could have practically used them as well.
Agreed, I am sure that the drive for axiomisation of geometry drove a lot of this interest.
All I really mean is that actually using these tools for an artistic, constructive purpose gives me a feel for why these problems might of been of interest. Of course, without knowing much about the history of mathematics this far back, I cannot be sure.
I've enjoyed playing around with projective geometry recently: that's even more basic, since it doesn't involve a compass, only a straightedge!
The fundamental objects in projective geometry are points and lines:
- Given two distinct points, a unique line joins them
- Given two distinct lines, they meet at a unique point
We can't do anything with just a single line/point. With a pair we can find their meet/join, but that's it. With three we can find all the meets and joins (forming a triangle). It's only once we have four objects that things get interesting, and we can start joining points, then meeting those lines, then joining those meets, and so on.
Note that lines don't always meet in Euclidean geometry, since parallel lines never meet. Projective geometry avoids this by including "points at infinity". Modelling that with normal 2D diagrams is quite mysterious, especially since opposite directions approach the same point at infinity. Yet it becomes very simple if we switch to 3D:
- We can draw our points on a hemisphere instead of a plane, with great-circles for lines: distinct great-circles will always meet, and the "points at infinity" are simply those on the equator.
- Instead, we can choose an "origin" sitting above our plane, and connect it to our points (forming lines) and lines (forming planes). In that case, the "points at infinity" are just the lines through the origin which are parallel to the plane (thus never meeting it).
There's lots of fascinating results in this framework, from the Greeks to modern times:
I can kind of see, though, why considerations of what integer ratios are 'good' for such diagrams and questions like angle bisection or intersections between circles and lines become interesting topics. It can really affect how easy or hard it is to draw such a diagram