Unibike Spiral

A unibike spiral is a conjectured spiral-shaped unibike curve investigated by Wagon (2025, 2026). His construction begins with the scaled positive branch of Fermat's spiral

(1)

for . Given a front track , the rear track satisfies the differential equation

(2)

with initially, where denotes the dot product (Wagon 2025, p. 8). Each successive rear track starts at , with its initial front-wheel point one unit away on the preceding track's second turn. A reparameterization makes the polar angle.

The illustration uses an extrapolated approximation obtained from a modified starting track (Wagon 2025, pp. 18-19). The right panel magnifies the distance between the first front-wheel position and the rear track by .

For the starting track, Wagon (2025, p. 6) proves the asymptotic relation

(3)

as . An improved starting track is

(4)

for . Wagon (2025, p. 18, Theorem 2) proves that its unibike error satisfies

(5)

for all sufficiently large .

The error comparison shows numerical samples over for Fermat's spiral, a modified rear-track iterate, and the extrapolated approximation.

For the iterates starting from , numerical experiments suggest the error model

(6)

as (Wagon 2025, pp. 12-13, 16-17). This dependence on iteration count has not been proved.

Numerical experiments suggest uniform convergence of the iterates on to a smooth curve with increasing polar radius whose unibike error vanishes everywhere (Wagon 2025, p. 16, Conjecture 1). Modified starting tracks give reported numerical errors below over computed domains. The paper does not prove this convergence or the existence of an exact spiral unibike curve, and the computed errors are not certified bounds on the entire infinite track (Wagon 2025, pp. 18-20).