A383087 The number of distinct distances between points in the Euclidean plane where the points are constructed via a straightedge-and-compass construction using n lines and circles.

1, 1, 3, 5, 73, 6628

Offset: 0

Author

Peter Kagey, Apr 16 2025

Keywords

• nonn
• more
• hard

Comments

We say that a real number is a constructible number if it is the distance between two points that can be determined from a straightedge-and-compass construction.

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 points.

Examples

			For n = 0 and n = 1, the only number that is constructible is 1, the distance between the two initial points.
For n = 2, we additionally can construct sqrt(3) and 2.
To construct sqrt(3), draw two unit circles, centered at each of the two starting points. These unit circles intersect in two places, which are a distance of sqrt(3) apart.
To construct 2, draw a unit circle along with the line connecting the starting points. The line marks two points that are opposite of each other on the unit circle.
For n = 3, we additionally can construct 3 and 4.
		

Crossrefs

Cf. A383082, A383084, A383086.