A383086 The number of distinct distances between points in the Euclidean plane where the points are constructed via a straightedge-and-compass construction using n circles and no lines.

1, 1, 2, 4, 35, 2480

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). In the constructions counted by this sequence, only the compass is used. Circles can be drawn at any marked point through any other marked point, and new points are marked where circles intersect.

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): 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.
For n = 3, we additionally can construct 2, and 3.
		

Links

• Wikipedia, Constructible number
• Wikipedia, Mohr-Mascheroni theorem

Crossrefs

Cf. A383083, A383085, A383087.