Squaring the Wheel – a Perfectly Rolling Car With Square Wheels

Have you ever felt that a road is so bumpy you might as well put square wheels on your car? In this instructable I will show you how to construct a small car with square wheels! Oh, and also a track on which the car will run completely smoothly.

There are several ways to do this. I’ve used plywood and a laser cutter, because I have access to one, but cardboard and a knife is also an option.

Step 1: Some Background

Feel free to skip this bit if you’re anxious to get going. If you’re the type of person who gets sweaty palms and a high pulse from seeing mathematical equation then close your eyes, scroll past and proceed to the next step. You don’t need to read this to be able to build the car and track. However, it might prove interesting, and useful. You’ll get to understand why this works, and you will be able to make a car with other shaped wheels if you want to take it a step further.

When something rolls smoothly it’s because the axle is always at the same "altitude". That is, it is always the same distance from a (possibly imaginary) horizontal line. If a wheel is not round, then this can still be accomplished by having the ground not be flat (in exactly right way).

Imagine a straight, horizontal line, l (or even better, look at the one in the figure above); this will be our altitude reference. If the vertical distance, r, from the axle to the point of contact, P, on the ground directly underneath it and the height, h, of the ground above the reference line always sums to the same number, then the car will be rolling smoothly.

The details of this is beautifully presented in an article written by Leon Hall and Stan Wagon. I will therefor call our square wheeled car a Hall Wagon.

For a wheel that is shaped like any straight sided shape (mathematicians call these polygons), like a square, a triangle or a pentagon, the ground shape that does the trick is a curve known as the catenary curve. That is the curve a chain will make when it hangs from two points. One way you could make this curve is to simply hang a chain between two points and copy the curve.

Let’s say the shape you want your wheel is what we call a regular polygon. That means that all sides and angles in the shape are the same. To make your track make several copies of the catenary curve and place them so that the angle in the corner between two curves is exactly the same as the (interior) angle of your regular polygon. (Look at the second picture above to get an idea of what I mean)

The
mathematical function that graphs a catenary curve is

y = cosh(x)

You don’t have to worry to much about what that means, but if it looks a bit familiar it might be because you’ve met its cousin, the cosine function in trigonometry in school. This one is called the hyperbolic cosine. If you think that sounds interesting, I suggest you google it and read a bit about it.

I’ve used a free graphing program called GeoGebra to graph my cosh(x), however you might notice that it’s upside down to the way we want it to be (third pic above). When this happens mathematicians have a neat trick to fix it: stick a minus sign in front of it.

We might want more bumps in our track. This can be done by printing several of these graphs and using ye olde scissors and tape, but we can also use a bit more maths. If we put one more cosh(x) in the graph it will just sit right on top of the old one, so we need a trick to move it over. Instead of having the top at x = 0, we want that point at about 1.1752. (That is so that the angle between the two arcs just at the corner is 90°. The exact value is actually 2×arcsinh(1)). So for the graph to be the same shape, only shifted over by some other value, a, you want a graph that if you subtracted a from it you get your original graph. In other words, we want to graph cosh(x-1.1752). Now we just repeat the trick with cosh(x-2×1.1752) and cosh(x-3×1.1752) and so on.

In the next step we'll ponder some potentially profound consequences of the mathematics we've just discussed...

Step 2: Taking the Maths to Extreme Limits

By using a bit of imagination you can probably see that if you were to make a wheel with really many edges - infinitely many - the shape of the wheel approaches the shape of a circle, and the track that will make it roll smoothly actually approaches a straight line.

Though they may seem like strange, hypothetical, mathematical artefacts, rumours have it that these types of wheels are actually in use some places in the real world.

Now that we know a bit about what’s happening behind the scenes, let’s get to it.

Step 3: Gather Your Materials

What materials you need depends a bit on how you want to make your curve. You could use a chain and draw the curve by hand, like I mentioned earlier (I’ll call this method 1), but a printer (method 2) or laser cutter (method 3) and some free software (or a rummage around google images) is a bit easier. The way I’ve done it is to make catenary curves in a free program called GeoGebra. This software is completely free, and you can use the file I’ve provided or you can read the background material in step 1 and make your own.

Materials for method 1

• A piece of chain or thin, but relatively heavy rope
• Pen or marker
• Paper
• Scissors
• Cardboard
• Ruler
• Square
• Pen
• Heavy stock paper (e.g. 200g/m2)
• Hot glue

For this method you'll need to draw the curve using a chain as a guide. Clever use of masons chalk or other methods may make this easier. Get creative!

Materials for method 2

• GeoGebra or a picture of a catenary curve (you could for instance use the one from step 1)
• Printer with paper
• Scissors
• Cardboard
• Ruler
• Square
• Pen
• Heavy stock paper (e.g. 200g/m2)
• Hot glue

If you print catenary curves you can glue them to cardboard. Use a square to figure out whene the angle of the corner between two curves are 90 degrees.

If you cut the side panels out of cardboard I suggest cutting another piece of cardboard to glue them on to. This will serve the same function as the clips we use in the next step.

Materials for method 3

• GeoGebra (optional)
• Vector graphics program (optional. I’ve used Inkscape)
• Laser cutter
• Plywood
• Heavy stock paper (e.g. 200g/m2)

The next steps will detail how I've used method 3. If you want to use one of the other methods I suggest you read on and use the next steps as inspiration.

Step 4: Preparing and Cutting

You might want to grab installation files for GeoGebra and Inkscape and install them on your computer before you continue.

From GeoGebra you can export your track shape as an eps file. This can then be opened in Inkscape to draw the rest of the side boards for the track.

Instructables are having some problems with non-text or image file formats at the moment, but here are links to the files I have used:

GeoGebra file

Inkscape file

In Inkscape you import the eps file (File – Import) and make sure it’s a path by selecting it and clicking Path – Object to path, or pressing Shift+Ctrl+C. Now you can use the edit nodes tool to draw and edit the shape of the side panels of your track.

The file I’ve provided is made for 4 mm plywood. If you use something else then you will definitely want to edit the sizes of the taps and holes so they match the thickness of your material. Also when a laser cutter makes a cut it removes a bit of material. The width of this cut is called the kerf. The file provided has a kerf of 0.23 mm, so the holes for the 4×4mm taps are 3.54 mm wide (= 4 – 2×0.23, as there are two cuts, one on each side). Take a closer look at the fourth picture above and you'll see where the size of the selected rectangle is given in Inkscape.

You might want to clean up this vector file and make your own improvements and ajustments.

When the file is ready you need to send it to your laser cutter. Depending on what type of machine you have there will be different procedures for this, and you probably already know them for your laser cutter. If not, please ask someone to assist you haven’t used it before or you are uncertain how to operate it. If you are using a laser cutter at a makerspace, or some similar place, then there will be someone there who can teach you.

Step 5: Assembling the Bottom Part of the Track

First you'll want to build the two beams that hold the two side panels together. Look at the second picture above to see how they should look.

The taps and holes are pressure fit together. If it's a tight fit you can use a pair of pliers to squeeze the tip of the tap a bit.

These two beams have three taps on either side. These go into the three holes at the bottom of the lower part of the side panel.

Repeat on the other side panel.

Step 6: Assembling the Top Part of the Track

Now you'll want to make your track from heavy stock paper. If you are using methods 1 or 2 then you’ll want to start by measuring the length of your curves. Then use a ruler and a pen to score parallel folding lines in the paper that distance apart. If you are using method 3 then you can use your laser cutter to cut perforations in the paper. Use the provided file or make your own. When cutting the paper I set my laser cutter to cut 2 mm dashes with 1 mm gaps around the edges and 1 mm dashes with 2 mm gaps along the scoring lines.

Again: Instructables are having problems with file handling, so here is a link to the file I use to cut the paper.

Tear of the side parts of the paper. Before you fold the paper along the score lines, roll it up into a tight roll. This is to curve the paper. Then valley fold along all the score lines so the track gets bumps, vaguely like on the finished track.(See picture number 4 above)

Now place it on top of the assembled side panels and use the top part of the side panels to force the paper in place and into the correct curve. I like to wiggle the upper part of the panel a bit back and forth to force the folds down into the corners. Use the small clips to fasten the top and bottom parts of the side panels together.

Turn around the track and repeat on the other side.

Your track is now complete!

Step 7: Driving On

You may have noticed you have one more part you haven’t used yet. That’s for joining two or more tracks together to make one long track.

Step 8: The Hall Wagon

Just in case you skipped step 1: the reason we call the square wheeled car a Hall Wagon, is because of an article explaining the mathematics beind square wheels, written by Leon Hall and Stan Wagon.

To make your Hall Wagon you need:

• a bit of cardboard
• a drinking straw
• a 3mm wooden dowel rod (for the axel)

Feel free to make a better looking car. This one is more a proof of consept car.

The straw is to allow the axes to rotate. Cut the straw in two pieces short enough to get both wheels between the side panels. Tape them to the cardboard parallel to each other and far enough apart that the wheels don’t interfere with each other.

If the axel is too thin you can put bit of tape around it to make it slightly thicker, for a snugger fit. If it’s too thick use a pencil sharpener to make it a bit thinner. Remember to stick the axle through the straw before fastening both wheels.

Step 9: Your're Done!

Play around with your new toy

Pop it on to your track and feel the satisfaction of a square wheeled Hall Wagon driving smoothly across a perfectly bumpy road.