A383085 -id:A383085 - cp's OEIS frontend

A383084 The number of points in the Euclidean plane that can be determined via a straightedge-and-compass construction using n or fewer lines and circles.

2, 2, 6, 14, 147, 5743, 900487

Offset: 0

Peter Kagey, Apr 16 2025

			For both n = 0 and n = 1, the a(0) = a(1) = 2 points are the initial points: (0,0) and (1,0).
For n = 2, the a(2) = 6 points are the two above points along with (-1,0), (2,0), (1/2,-sqrt(3)/2), and (1/2,sqrt(3)/2):
  (-1,0): formed by intersecting the line between (0,0) and (1,0) with the circle of radius 1 centered at (0,0).
  (2,0): formed by intersecting the line between (0,0) and (1,0) with the circle of radius 1 centered at (1,0).
  (1/2,-sqrt(3)/2), and (1/2,sqrt(3)/2): formed by intersecting the circle of radius 1 centered at (0,0) with the circle of radius 1 centered at (1,0).

Peter Kagey, Illustration of the a(5)=5743 points.

Cf. A333944, A383082, A383085.

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.

Original entry on oeis.org

1, 1, 2, 4, 35, 2480

Offset: 0

Peter Kagey, Apr 16 2025

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.

			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.

Wikipedia, Constructible number
Wikipedia, Mohr-Mascheroni theorem

Cf. A383083, A383085, A383087.

cp's OEIS Frontend

A383084 The number of points in the Euclidean plane that can be determined via a straightedge-and-compass construction using n or fewer lines and circles.

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