This post contains images based on the straightedge-and-compass constructions discussed in Part 1.
Plotting points
All of the 5743 points that can be drawn using five or fewer lines/circles in a straightedge-and-ruler construction, starting with just the initial points of (0,0) and (1,0).
If we don't allow use of the straightedge, we get these 1704 points that can be drawn using five or fewer circles and no lines.There's a theorem called the Mohr–Mascheroni theorem that says that every point that can be determined with a rule and compass can be determined with just the compass!
All of the \(5743\) points that can be drawn using five or fewer lines/circles in a straightedge-and-ruler construction, starting with just the initial points of \((0,0)\) and \((1,0)\).All of the \(1704\) points that can be drawn using five or fewer circles (and no lines) in a straightedge-and-ruler construction, starting with just the initial points of \((0,0)\) and \((1,0)\).
All the circles and lines
Just for fun, here are two other images, the first shows all of the distinct lines and circles you can draw using a ruler-and-compass in at most five steps. The second show all of the distinct circles you can draw within five steps using just a compass.
Here are the 1337 circles and 596 lines that can be drawn using five or fewer lines/circles in a straightedge-and-ruler construction, starting with just the initial points of (0,0) and (1,0).The other image is the 480 circles that can be drawn with a compass alone in under five steps.
Sure! What do you have in mind exactly? The point that is closest to the circle centered at the origin of radius sqrt(pi)? The line segment containing two points whose distance is approximately pi?
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