A383082 The number of distinct straightedge-and-compass constructions that can be made with a total of n lines and circles.

1, 3, 3, 16, 205, 5886, 542983

Offset: 0

Author

Peter Kagey, Apr 15 2025

Keywords

• hard
• nonn
• more

Comments

A straightedge-and-compass construction starts with 2 points marked on the plane, traditionally (0,0) and (1,0). One can use a straightedge to draw a line between two marked points or a compass to draw a circle centered at a marked point through another marked point.

New points occur at the intersections of lines or circles with lines or circles.

Constructions are considered distinct if the location of the lines, circles, and points is different. In particular, a translation of a construction is considered distinct. - Peter Kagey, May 07 2025

Examples

			For n = 0, the a(0) = 1 diagram is the one consisting of two points.
For n = 1, there are a(1) = 3 possible constructions:
1) a line between (0,0), and (1,0),
2) a circle of radius 1 centered at (0,0),
3) a circle of radius 1 centered at (1,0).
For n = 2, there are a(2) = 3 possible constructions:
1) a line between (0,0), and (1,0) and a circle of radius 1 centered at (0,0), which marks the point (-1,0);
2) a line between (0,0), and (1,0) and a circle of radius 1 centered at (1,0), which marks the point (2,0);
3) two circles, both of radius 1, centered at (0,0) and (1,0), which marks the points (1/2,sqrt(3/4)) and (1/2,-sqrt(3/4)).
For n=3, see the a(3)=16 diagrams in the link.
		

Links

• Peter Kagey, Illustration of a(3)=16, with the origin marked as a red point.
• Peter Kagey, Illustration of a(4)=205, with the origin marked as a red point.
• Wikipedia, Straightedge and compass construction

Crossrefs

Cf. A333944, A383083.