N-Curve

An N-curve, in the construction of Whale (2026), is a plane curve obtained from an integer sequence by joining circular arcs of equal arc length tangentially. Each term determines the signed change in tangential angle along the corresponding circular arc. The construction gives a geometric representation of the sequence and its residues. It is related to turtle geometry, in which successive instructions specify line segment lengths and turns (Abelson and diSessa 1981, Colestock 2017).

For a sequence , , ..., a positive integer modulus , and an integer offset , define

(1)

where is the least nonnegative residue and is measured in radians. Thus is the turn in degrees. The choice , gives signed turns from through 179 degrees.

Identify the plane with the complex plane, set and , and normalize the arc length of each piece to 1. The displacement of a piece whose initial tangential angle is zero is

(2)

with given by the continuous limit. The sequence of endpoints and directions satisfies

			(3)
			(4)

For , the corresponding piece is

(5)

Its unit tangent vector is , so the join has a continuous tangent vector. A zero turn gives a line segment.

The examples show N-curves for the triangular numbers, square numbers, Fibonacci numbers, and prime gaps (OEIS A000217, OEIS A000290, OEIS A000045, and OEIS A001223). The first three sequences begin at index 0. The prime gaps begin with , where is the th prime number. The orange dots mark the initial points. The modulus, offset, and number of pieces are shown for each example. The prime gaps panel shows a finite initial segment without asserting closure.

The closure question for a periodic sequence of turns reduces to a rigid motion. Suppose and write . Then

(6)

where . When , this rigid motion is a rotation about . If is a root of unity of order , then . Consequently, after blocks both the point and its tangent vector return to their initial values. The completed trace has rotational symmetry of order at least . Additional symmetries of the trace can make the full order larger. A return of the point alone can also occur before the tangent vector returns.

For integer turns in degrees, let be their sum over a block. If , then

(7)

where denotes the greatest common divisor. In the illustrated examples, the triangular numbers have a residue period of 720 and , the square numbers have a residue period of 180 and , and the Fibonacci numbers modulo 36 have a residue period of 24 and . Their traces therefore close after 2160, 2160, and 240 pieces, respectively.

When , the block gives a translation. The curve closes with matching tangent vector after a block iff , and otherwise repeats along an unbounded chain. For example, the two signed turns , give

(8)

so returning to the initial direction does not imply closure. The three panels illustrate a block with turns 120 and 0 degrees, a block with turns 180 and degrees, and a block with two turns of 180 degrees. These produce a rotation of order 3, a nonzero translation, and the identity transformation, respectively.