Circles and Squares

How pretty is that!?

This was achieved by tracing the outline of a 3×3 inch square while rotating this square along the center line.
Of course, this was done by hand- and thus, not perfect.

BUT ITS STILL AWESOME. To show you how awesome, I’m gonna point out these super duper cool archs.

HOW COOL IS THAT. When you rotate a square- you get circles!

Could a square really roll? If it could, what would the ground look like? Obviously a square couldn’t roll on a flat ground.

For a square to roll smoothly it’s center must remain a steady distance from the ground. But the edges of a square aren’t equal from the center at all points, so somehow the ground must change so as to support and make up for this difference.

Let P represent any point on the edge of the square. Let X represent the distance from the center of the square to P. Let A represent the radius of the circle that the square would inscribe. In other words, A is the distance from the center to a corner of the square, the largest possible value of X.

So, if X is the length from the center to the edge at any given point and A is the largest possible distance from the center to the edge, then the height of the ground at any given point is given by A-X.