The Rule of Thirds is a mathmatical expression of a three principles: balance, contrast and movement. By breaking a photo into three or nine pieces, the illusion of movement and a sense of proportion can be established.
A number of techniques to balance a photo can be used. Diagonals can be used to balance the photo. Light and dark halves work very nicely. Another method is use an s type of curve such as a winding road to divide the photo and to lead the viewer's eyes. Leading lines off the photo page is also a popular device.
In Photo 1, a diagonal is used to break the photo into sections of movement and non-movement. The darker half is moving while the lighter half is suspended in time, but still in a frozen action state.
In Photo 2, a Yosemite National Park waterfall is broken into nine pieces by superimposing what looks like a tic-tac-toe board. Also, superimposed is a serpentine dotted line, in the shape of an 'S'. A third line drawn is a diagonal red line. The eye follows the curving serpantine line and breaks the photo into halves along the diagonal.
Moving counter-clockwise, the eye also can focus on numbers 1, 2, 3. These areas are the fore, middle, and background of the frame. Finally, as you look at the thirds, see that there is movement of light to medium to dark from the top to the bottom, another movement line.
When speaking of the Rule of Thirds, inevitably, the conversation will turn to the Golden Mean/Phi of 1:1.618. The Phi Ratio is often found in nature. A DNA helix and plant growth patterns are examples. It is hypothesized that buildings such as the Parthenon were built using the Golden Proportion.