Treebeard's Stumper Answer 9 March 2001

Bicycle Tracks

We sure have had wild weather in Central California lately! We had 13+ inches of rain in one storm at my home in the mountains last week, enough to interfere with school and make us postpone our annual DMS Piece of Cake bike ride that should have been this weekend. Of course it's irresistible to ride through the mud after the heavy rains, and bikes leave such interesting tracks in the mud! Look at the pictures of my bike tracks below. Which track is my front wheel, and which is the rear? What direction was I riding? Hint: this is a geometry problem! Remember that the distance between my bike wheels is fixed, and only the front wheel can turn.

Here are my bike tracks in the mud at school. Which track is my
front wheel, and which is the rear? What direction was I riding?

Look close at the top photo, and you'll see that bike track 2 passes over track 1, so track 2 must be the rear wheel. Also the front wheel turns and the rear follows, so track 1 with the wider turns must be the front. The rear wheel always points toward the front wheel at the wheelbase distance ahead, but not vice-versa. So pick a few points along the rear wheel track and extend a bike-length tangent line in each direction to see which way consistently matches the front wheel's path. I must be riding from right to left. Knowing the secret, it's fun to play detective with real bike tracks!

Notes:

Track 2 must be my rear wheel because the front wheel turns and the rear wheel follows, so track 1 with the wider (and more wobbly) turns must be in front. Another clue is that straight tangent lines drawn along track 1 (like line X) don't intersect track 2 at all. But the rear wheel on a bike must always point towards the front wheel at a fixed distance ahead, equal to the distance between my bike wheels. That also tells me that I'm going from right to left. Pick a few points along the rear wheel track 2, and extend the lines in each direction. It's clear that the lines going away from the right are all about the same length, but they are different lengths in the other direction. This would be more obvious if the photo were taken straight down. Remember that the background is farther away, and further distances look shorter. A-A', B-B', C-C', D-D' are all the same real length. In the other direction, they are all different. Therefore I'm going from right to left. It's geometry!

Graybear's answer was so good that I stole some of his words for my answer:

Track 1 (blue) is from the front wheel and track 2 (red) is from the rear. This is easy since the rear wheel moderates the action of the front wheel. When going around a curve, the front wheel forms a larger circle, even on a four-wheeled vehicle like a car. You can also see in the foreground of the picture that track 2 crossed over track 1 and therefore was made later. The way to determine the direction is by knowing that the rear wheel is always aimed towards the front wheel (though the front wheel is not always aimed toward the rear wheel) and that the wheelbase is constant. If we estimate the wheelbase and, for several points along the rear wheel's path, we extend a wheelbase-length line segment in both directions to see which consistently connects to the front wheel's path. (I hope you follow what I'm trying to say.) This method would be easier if the photo was taken from directly above the tracks, but my guess is that the tracks start from the lower right.

Here's a couple more photos that help explain the answer:

We put my bike back on the tracks near the B-B' position. In the other direction, it doesn't fit. Note how the rear wheel points towards the front wheel along a tangent line at the wheelbase distance ahead. Imagine this tangent line for every point on the path and visualize it changing as my bike moves along. That's calculus!	This close shot of the crossing tracks shows that track 2 must be the rear wheel that follows track 1. But that's not enough to determine the direction! Many mountain bike wheels show their direction in the shape of the tread, but not mine. It does appear that track 2 is deeper than track 1, and the left side of both tracks is deeper than the right side. That's another hint.

I put this stumper to some of the kids at Dunn Middle School with no advance discussion to see what they think:

Track #1 (blue) is front wheel	18	correct
Track #2 (red) is front wheel	16
Bike is going right to left	14	correct
Bike is going left to right	21

I'm not surprised by the results. I asked the kids to write their reasons, and many admitted they were just guessing. I was surprised how many kids had never noticed the cool intersecting curves that bike wheels leave. *Sigh* It's interesting that the kids were evenly divided on which wheel was the front, which is compatible with guessing, but they were wrong by a 3:2 margin on which way the bike was going. What makes this so misleading? Here are a few comments from the kids, right and wrong:

• The rear tire takes the shortest distance to each new curve the front tire makes.
• You can see that the #1 track is swerving a lot more than track #2, and since the front wheel is the one that turns, it looks more curvy,
• The front wheel makes wider turns.
• I think the bike is going right to left because on the left the swerve is bigger, and you need to have speed to do that.
• The front tire leads and the back tire follows, so it doesn't have as far to go.
• Line 2 is the back tire because it overlaps line number 1.
• The front wheel always turns wider that the back.
• When you turn rapidly, the back wheel is still going one way when the front wheel is making it's next move.
• The bike is going to the right because the diamonds on the wheels are facing that way.

I figured this was a geometry question, but I got this interesting email from "xcks-line" (name?) who did it another way with dynamics, taking the hint in the right picture above:

It took not quite a minute to figure it out, as you stated the rear tire couldn't move... It was obvious the front wheel was #1 because of the in & out motion. As far as which way you were going, I imagined a person on a bicycle and turning a corner like that you would have probably stood up to make a turn/curve so I examined the depth of the marks in the sand and at the take off point.

Length x Force = Torque. A lot more energy or force would be needed to speed up, and a normal person after a "burn out or start up" would relax and let "perpetual motion" do it's thing, since torque was no longer needed... The imprint of the later part of the curve you were making would not be as deeply imprinted as the take off point. Look closely at the relaxed ending of the curve to the right in the middle of the picture and you'll see what I mean. You are gliding as opposed to enforcing motion.

Actually I'm a master auto tech, and if you've been to a drag race before, at startup the ass-end of a car squats in an attempt to apply the energy in a forward motion. After it has sling-shotted forward, catching up to the forced energy, the ass-end rises and weight is attempting to balance itself, so the "black marks" are no longer present because the weight has shifted. As I said, look at your tire mark at the relaxed part of your turn to the right, middle of picture, and the front tire's weight print in the sand.

Interesting answer, completely different than I expected. Before I did my ride in the mud, I tried riding through a puddle on the school tennis court and photographed the tracks:

I thought it was too obvious that I was going from right to left because of the thicker tracks on the right (and my footprints), so I went to the mud. I figured my wheels were just shedding water continuously as I went along. But maybe you're right, I'm also peddling harder on the turns, which sheds more water. You can see the difference on the left side of the photo where the tracks are faint, but still thicker on the turns when I peddle and thinner when I coast. But why is the straight section in the middle-right so thick? Ah, that's when I'm turning the front wheel sharply, so I am pumping on the cranks! There's another question here about why we accelerate on curves when riding or driving, but I'll save it for another stumper.

Even Sherlock Holmes got this one wrong, or at least not quite right, in the Sir Arthur Conan Doyle story The Priory School, one of the stories in The Return of Sherlock Holmes. Here's the relevant section:

We had come on a small black ribbon of pathway. In the middle of it, clearly marked on the sodden soil, was the track of a bicycle.

"Hurrah!" I cried. "We have it."

But Holmes was shaking his head, and his face was puzzled and expectant rather than joyous.

"A bicycle, certainly, but not the bicycle " said he. "I am familiar with forty-two different impressions left by tyres. This as you perceive, is a Dunlop, with a patch upon the outer cover. Heidegger's tyres were Palmer's, leaving longitudinal stripes. Aveling, the mathematical master, was sure upon the point. Therefore, it is not Heidegger's track."

"The boy's then?"

"Possibly, if we could prove a bicycle to have been in his possession. But this we have utterly failed to do. This track, as you perceive, was made by a rider who was going from the direction of the school."

"Or towards it?"

"No, no, my dear Watson. The more deeply sunk impression is, of course, the hind wheel, upon which the weight rests. You perceive several places where it has passed across and obliterated the more shallow mark of the front one. It was undoubtedly heading away from the school. It may or may not be connected with our inquiry, but we will follow it backwards before we go any farther..."

"Do you observe," said Holmes, "that the rider is now undoubtedly forcing the pace? There can be no doubt of it. Look at this impression, where you get both tyres clear. The one is as deep as the other. That can only mean that the rider is throwing his weight on to the handle-bar, as a man does when he is sprinting. By Jove! he has had a fall."

Hmmm, looking at the close shot of my tracks crossing, it does appear that the rear track 2 is deeper than the front track 1. The problem is that knowing which track is the front and rear wheel is not enough to determine the direction the bike was traveling, since the rear wheel would obliterate the front wheel track in either direction.

Here are some links for further research:

• This great problem with bike tracks was originally posed in a course at Princeton (and elsewhere) on Geometry and the Imagination by John Conway, Peter Doyle, Jane Gilman, and Bill Thurston. The lecture notes are a treasure trove of interesting problems, including several more about bicycles. There's an account of John Conway's teaching style in giving these lectures.

• This problem also appears as the title problem in Which Way Did the Bicycle Go? by Joseph D.E. Konhauser, Dan Velleman, and Stan Wagon. This fine book is a collection of problems from the Macalester College Problem of the Week archive. Stan Wagon, who now runs the Problem of the Week, lists it as one of his dozen favorite puzzles. The problem is also discussed here and here.

• Suppose the rear bike wheel was the same, but I was riding in the other direction. Can you reconstruct the path of the front wheel? Stan Wagon in his book Which Way Did the Bicycle Go? solves that problem "in Mathematica by using a Bézier curve to get the back-wheel path, and then symbolic differentiation to get the corresponding front-wheel paths in the two directions." I want that program!

• Bike tracks appear to be examples of a class of mathematical curves called cubic splines or Bézier curves based on cubic polynomials. These smooth twisting curves are the basis for Postscript fonts and have many other applications in math and graphic design. Don Lancaster's Cubic Spline Library has info and links. There are live JAVA demos here, here, and here.

• The original Sherlock Holmes stories, including The Priory School, are all available on the Web at Sherlockian.Net, as well as The Online Books Page. There is an entertaining site In the Bicycle Tracks of Sherlock Holmes that describes a bike trip through the south of England complete with Sherlockian references.

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