Ancient Greek mathematicians (and many others since) felt that there was something fundamental about straight lines and circles, so they were very interested in what geometrical constructions could be done using those. Solving a problem in plane geometry with "ruler and compass" (it should really be "straightedge and compass" since you aren't supposed to use the markings on the ruler) means solving it using only the following operations:

1. Given two points, draw the straight line passing through both of them.

2. Given two (non-parallel) straight lines, find the point where they intersect.

3. Given two points, draw the circle with one as centre and passing through the other.

4. Given two circles, or a circle and a straight line, find the point or points (if any) where they intersect.

Some problems can be solved using only these operations and some can't. For instance,

These are possible:

  • Given a triangle, find a square of equal area.
  • Construct a regular polygon with 102 sides.

These are not possible:

  • Given a circle, find a square of equal area.
  • Construct a regular polygon with 100 sides.
The Delian Problem is another example of something that's impossible with ruler and compass.

How to distinguish the possible problems from the impossible? That's a task for the (at first glance quite unrelated) field of Galois theory...

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