What the heck is a square wheel??!? Why reinvent the wheel at all? Well, because we can. The inspiration for this Intstructable came from the San Francisco Exploratorium, which features a square wheel display, explaining the basic science behind it, which can be found here. The basic idea behind any wheel, is to keep the load or center of mass at a __constant height__ to make for a smooth ride. This is why wheels are round...until now. By exploring the math behind the square wheel, we can actually design a roadway that allows a square wheel to ride smoothy along.

P.S. Although I will go into some of the math behind it, you don't need to know or understand it to make this project

## Step 1: The Square

Ok so to start we have to assign some values to sides and such. For this project, I made the length of a side simply "l" (disregard the P, that was from an earlier sketch), the difference in length from the center to a side and the center to a corner "A", and the center itself "C". Because we assign variables instead of specific values to these units, we can solve for a square of ANY size, not just the specific square we have this time. Also, from trigonometry, we know the relationship of the length from the center to a corner (which will be referred to as "r") and the length from the center to a side to be a 1: √2 ratio. From this relashionship, we can later derive equations we'll need.

## Step 2: How Do We Get the Center to Stay at a Constant Height?

While this may seem tricky or confuzzeling at first, the truth is much more simple. The square wheel is still a wheel, and so it travels in a uniform rotation right? So the track it runs on and the pattern of it will be uniform too. And because its rotating in such a repeating constant way, the track itself is part of a circle! As we'll explore in the next step, getting the shape of the track is fairly simple, as long as we know the length of a side of the square and understand some basic algebra and trigonometry. Notice how in the pictures, as the square rotates along the track, __the center stays at a constant height__. Again, the actual curvature of the track is related to the length of a side of the square.

## Step 3: Drawing the Track

To understand exactly how to make the track, we need to understand where we're getting our lengths and curves from. Consider this; as the square rotates, the distance from the center to the bottom (directly below it) is constantly changing, but in a uniform way. Now, because the height of the center needs to be constant, we can use that for our radius of the uniform curvature of the track. I know what you're thinking, "Now wait, you said the radius is the height of the center, but that the diagonal is the radius in the picture.." Great observation.

Because the corner stays on the ground when the square wheel is "rolled" on a flat surface, __the diagonal from the center to the corner can be used to chart the overall motion of the square as it travels, giving us a pattern for the motion of the center__, showing us how to construct a track to keep it at a constant height. Simple!! Now where does the math come in?

## Step 4: Math

There's no easy way around it. Math. But it does't have to be difficult! Ok, say you're on a stranded island with only a compass (the math kind of course), a protractor, a blank piece of paper, a pencil, and the length of one side of your square wheel. No way to trace it onto the paper, because you need it to stay intact. Using math functions, we can derive what the chord length, or the distance between curves in our case, will be. Using math, we can also find the angle of the sector for ALL square sizes.

We know that the arc length of the curve will be equal to the length of one side of the square, and that the equation for arc length is S=2πr(ø/360), where ø is the angle of the arc, and r is the radius. We also know chord length is d=2r*sin(ø/2). Additionally, by using the relationship we found between l and r the beginning with a trigonometric relationship for right triangles (r=((l*√2)/2)), we can substitute this in for r in the above equations. By solving these, as shown in the pictures, for first theta and then for d, the chord length, we find that the length of the side cancels out, meaning that the length of a side of a square has no effect on the angle associated with the arc. This meant that the angle, theta, is the same for ALL sizes of squares, at roughly 81 degrees. Furthermore, we find that because this angle remains constant, the chord length, or distance between curves, is directly proportional to the length of a side of the square, being about l*(.4535).

Ok ok slow down, what the heck does this all mean? Well, this information helps us to be able to construct an exact model for any square wheel of any size! It also tells us, using mathematical models, that the period and amplitude of the curves on the track is directly proportional to the size of our square. Nice. Does this mean you have to be an algebra and trigonometry whiz to do this project? Of course not!!

## Step 5: Trace and Trace and Trace...

Ok. So now we have our radius, and an understanding of how this whole thing should work. Maybe not a complete understanding, but enough. Let's draw. To begin, cut out a square slightly larger than that of your wheel. Draw a square of equal size to that of your wheel on it, leaving room in one corner, as shown in the picture. Punch holes in the center and corner using a pencil or pen, and then use them to trace the shape of the curve. Notice how during the tracing, the square rotates as in normally would while rolling on the ground. Shift the corner over between curves to keep it at a constant height. Note that there is a pencil required in the middle, I just left it out to provide a clearer picture. This bumpy shape becomes our track. Additionally, a compass (again, the math kind) could be used to make a more accurate curve for the track, by setting the radius as the distance from the center to a corner of the square.

## Step 6: Construction

I traced this onto a blank piece of paper and then cut it out, then using that to trace onto a piece of cardboard. This step could be simplified by tracing straight onto the cardboard. After tracing the pattern onto two pieces of cardboard and cutting them out, all that was left was to construct the wheel and put it all together. To construct the wheel, I took two squares of equal size (obviously), and connected them with toothpicks and hot glue. I put some lead bb's in the middle to give it some mass, but this step isn't necessary. After constructing the wheel, I connected the two sides of the track the same way, at the distance of the wheel, so everything would line up correctly and the wheel wouldn't fall off.

## Step 7: Roll It!

Alright, it's all led up to this! Roll your wheel! Ok, ok, it may not be the Mercedes smooth ride expected, but it's pretty dang good right?? Remember, this model is made out of cardboard, and the precision associated with that will be reflected in the results, but it is really neat to try it out and roll it along, seeing how the height of the center really is what determines whether a wheel gives a smooth ride or not. The concept behind this is really interesting and can be applied in a ton of different ways, but you, genius tinkerer, have just physically demonstrated it. By yourself! Congrats!!

## Step 8: Final Remarks

In the end, it's really cool to see the wheel, a freakin square, roll smoothly along a track you've constructed yourself. The concepts, ideas, and mathematical models that show this are neat, but whats even more satisfying is to see and hold them in the form of a contraption you've created. Not only have you learned, explored, and discovered how this works, you now know something that most people don't. Not to over exaggerate, but how cool will you be when you pull that rabbit out of the hat next time you have friends over. Or cousins. Or nephews. Or grandparents. Or neighbors. Maybe you can even win a few bets on it. The possibilities are limitless. Use it for good. Please comment any questions or comments you have!! Even critiques. Lay it on me. I want to hear them! Plus it will help me edit this Instructable for clarity. Thanks!