hand) any of several provided gearwheels (or rotors)—each having holes for a ballpoint pen—could be spun around the ring to draw geometric shapes. Later, the Super-Spirograph introduced additional shapes such as rings, triangles, and straight bars. All edges of each piece have teeth to engage any other piece; smaller gears fit inside the larger rings, but they also can rotate along the rings' outside edge or even around each other. Gears can be combined in many different arrangements. Sets often included variously colored pens, which could enhance a design by switching colors, as seen in the examples shown here. Beginners often slip the gears, especially when using the holes near the edge of the larger wheels, resulting in broken or irregular lines. Experienced users may learn to move several pieces in relation to each other (say, the triangle around the ring, with a circle "climbing" from the ring onto the triangle). Mathematical basis Consider a fixed outer circle of radius centered at the origin. A smaller inner circle of radius is rolling inside and is continuously tangent to it. will be assumed never to slip on (in a real Spirograph, teeth on both circles prevent such slippage). Now assume that a point lying somewhere inside is located a distance from 's center. This point corresponds to the pen-hole in the inner disk of a real Spirograph. Without loss of generality it can be assumed that at the initial moment the point was on the axis. In order to find the trajectory created by a Spirograph, follow point as the inner circle is set in motion. Now mark two points on and on . The point always indicates the location where the two circles are tangent. Point , however, will travel on , and its initial location coincides with . After setting in motion counterclockwise around , has a clockwise rotation with respect to its center. The distance that point traverses on is the same as that traversed by the tangent point on , due to the absence of slipping. Now define the new (relative) system of coordinates with its origin at the center of and its axes parallel to and . Let the parameter be the angle by which the tangent point rotates on , and be the angle by which rotates (i.e. by which travels) in the relative system of coordinates. Because there is no slipping, the distances traveled