ABSTRACT (original article): A unibike curve is a track that can be made by either a bicycle or a unicycle. More precisely, the end of a unit tangent vector at any point on a unibike curve lies on the curve (so the bike’s front wheel always lies on the track made by the rear wheel). David Finn found such a curve in 2002, but it loops around itself in an extremely complicated way with many twists and self-intersections. Starting with the polar square root curve
r=
t/(2π)
and iterating a simple construction involving a differential equation apparently leads in the limit to a unibike curve having a spiral shape. The iteration gets each curve as a rear track of its predecessor. Solving hundreds of differential equations numerically, where each depends on the preceding one, leads to error buildup, but with some care one can get a curve having unibike error less than 10^-7. The evidence is strong for the conjecture that the limit of the iteration exists and is a unibike curve. CITATION (original article): Stan Wagon (2025), A Spiral Bicycle Track that Can Be Traced by a Unicycle, arXiv:2503.11847. https://doi.org/10.48550/arXiv.2503.11847
τ
A spiral curve that can be traced by both wheels of a bicycle, with error less than 1 part in ten million.

Introduction

An intriguing open question in the world of bicycle mathematics is whether there is a spiral track that a bicycle can follow. That is, we want both wheels of the bike to follow the same track. A curve that can be traced by both wheels of a bicycle is called a unibike track, as it can be traced by either a unicycle or a bicycle (we exclude a straight line). Here I will describe an iterative procedure that appears to generate a spiral unibike track in the limit. The image above shows the 128th step of the process along with the motion of a bike: both wheels stay on the curve, at least to our eyes. In fact, the front point is not exactly on the curve: the error is less than 1 part in 14,000,000.
Basic properties of bicycle tracks are at [4]. That post discusses how the tangent line to the rear track generates the front track and how one can get a rear track from a front track. All bikes in this discussion have unit length between the two points where the wheels touch the ground.
In 2002 David Finn [2] found a unibike track (Fig. 1; an animation is at [4]). His construction is clever, but the resulting track is excessively complicated and impossible to follow by a real bike. Consequently, many have wondered if the problem admits a spiral solution. A spiral is a polar curve
r(θ)
with monotonically increasing radius.
Figure 1. David Finn’s unibike curve, with white arrows showing some bicycle positions.
I have found a relatively simple method that appears to give a perfect spiral unibike curve. Proof is lacking but the evidence is very strong. My preprint is at [5].

The Polar Square Root

The starting point is the polar square root spiral (known as a Fermat spiral), which here we take as
F
1
(t)=
t
2π
(cost,sint)
(Fig. 2). Figure 3 shows how the front track for
F
1
almost coincides with
F
1
. Figure 3 shows
F
1
and how close it comes to being a unibike track.
In[]:=
PolarPlot
t
2π
,{t,2π,20π},Frame->True,PlotStyle->Blue
Out[]=
Figure 2. The polar square root
F
1
(t)
.
Figure 3. The polar square root
F
1
(t)
is shown in blue ; the corresponding front track in yellow comes tantalizingly close to matching the rear.
An inappropriate use of this spiral is as follows. Look at the curve over
[n,∞)
for large
n
. This will indeed have vanishingly small unibike error as
n
gets large. But this process will not have a limit as
n->∞
, so it cannot lead to a perfect unibike curve. Similarly, one can consider
r=
1/n
θ
; the error approaches 0 as
n->∞
, but there is no limiting curve. So these approaches are useless.
The basic idea of a method that works is simple: Start with
F
1
. Get
F
2
, the corresponding rear track obtained by solving a differential equation, let
F
3
be the same from
F
2
, and continue this iteration. Full details and many variants and extensions are discussed at [5]. The motivation for this idea is the observation that going to a front track seems to make the unibike error larger. So one hopes that the front-to-rear transformation would make the error smaller. The figure at the top of this post shows
F
256
. The evidence is overwhelming that this iterative process converges and that the limit is a spiral unibike track.
For a near-unibike track, we need a way to quantify the error. The unibike error of
G(t)
at
t=
t
0
is the minimum distance from
H(
t
0
)
, the front point corresponding to
G(
t
0
)
, to the curve
G(t)
. At the closest point
G(
t
1
)
on the rear track, the derivative vector
′
G
(
t
1
)
is perpendicular to the line joining
G(
t
1
) and
H(
t
0
)
. So
FindRoot
can be used on the appropriate vanishing dot product to quickly locate the closest point. High precision is useful. The unibike error for
F
1
is shown in log-log form in Figure 4. That plot illustrates a theorem from [5]: the unibike error for
F(t)
is asymptotic to
π
3
-2
t
.

The Rear Track

Towards a Perfect Unibike Track

Conclusion

References

CITE THIS NOTEBOOK

A spiral bicycle track that can be made by a unicycle​
by Stan Wagon​
Wolfram Community, STAFF PICKS, May 14, 2025
​https://community.wolfram.com/groups/-/m/t/3461376