Moving a disk in edge contact to another disk of the same size, how many rotations?

Two disks of the same size are in edge-to-edge contact. One is held stationary. The other and the one being asked about is moved around the stationary disk while maintaining the edge to edge contact without slippage . How many rotations around its own center does the moving disk make while making one complete movement around the stationary disk?

Coins of the same size with milled, ridged or serrated edges are excellent to model this problem, a diameter of 25mm or 1 inch is suggested; but, the size is not important is long as they are the same diameter . You specifically are invited to use this method to solve this problem.

Here are the two disks after a small rotation. Initially, the two disks were side-by-side horizontally and the two images both vertical. The two images were derived from the Wolfram Mathematica popular curves of Ada Lovelace (often credited as being the first programmer) and al Khwarizmi (the person from whose name the word "algorithm" was derived). This image is one frame out of an animated GIF that illustrates the answer.