Composition isn't formulaic. And pictures can't be boiled down to rules or turds, repetitive patterns, leading lines, or face in the dirt perspectives.

If there is anything that can begin to classify what makes a good composition, it would need to start with the quality of the relationships, and your ability to recognize or forge them out of the chaos of the environment. 

I think of Photographs as being made up of physical forces—mass, gravity, objects in motion, tension and resolution. Every element in the picture affects every other element in the frame, much like objects in space have their own mass and gravitational pull.

Take the above photo made on an old farm in the Texas Panhandle. In addition to the way we naturally "read" a photograph (due to the way western language is written and read), all of the force and movement is moving the viewers' eye to the right because tree in the upper right and the direction the shadows are pointing. To balance the picture, the tires in the lower left, and darker grasses on the left just below the horizon, become important compositional elements pulling some unconscious attention back into the picture. Their presence has visual mass and can pull you back across the frame and "read" it all over again. Each time you might notice another detail, or experience other feelings. Making those dynamic compositional relationships can keep the viewer more engaged and interested in what else is going on conceptually and allow them to bring more of themselves to the picture too. 

The thing about composition as it relates to art (rather than pure graphic design or predefined commercial requirements), is that when you are out making the picture and then experiencing the resulting picture—either your own or those in museums and galleries—those graphical elements are more often "felt" than "seen."

Since pictures, like humor, can suffer the same death by analysis, any attempt to understand or critique the picture is best left for educational purposes, and certainly until after the picture is made. You have to be able to trust your intuition first and then later you are able to consider the reasons why you think it is or isn't a good picture and let that inform your intuition the next time you pick up the camera.