Data visualization became very popular in 2011, as evidenced by NYT pieces like this one and the release of Nathan Yau's book Visualize This. It seems to me that the upper limit of the amount of information a dataviz/infographic/pick-your-term can convey is bounded by three things: the creativity of the designer, technology available to him/her, and the perceptibility of the viewer. But is there an optimal point where simplicity of design and information conveyed are both maximized?
For one answer to this question, consider multiplication. In most (all?) American schools that I know of, multiplication is taught in terms of area (two terms) or volume (three terms). Harvard's Stats 110 begins by teaching probability as area. This concept is simple enough, and is particularly handy because often what we care about in practical terms can be expressed as an area/volume: how much wallpaper do I need? how much water will fit in that bucket?
But in terms of just manipulating the numbers, the area/volume interpretation can be a bit clunky--it doesn't really save any steps, and once you have more digits than you can hold in your head, most people will reach straight for a calculator. There's nothing wrong with that, except that there are many applications of multiplication beyond area or volume (take total cost of large order as an example). The Japanese have a different method, as the video below shows.
Two characteristics readily recommend this method. First, it is a very basic visualization that, if practiced, seems like it could make multiplication of large numbers in your head simpler. Second, there is no strict interpretive paradigm imposed on the answer. The practical meaning of the answer need not be an area, or anything geometric at all. However, it is not clear how this would extend beyond three terms either.
It may be that readers more experienced than myself with linear algebra or geometry will sense intuitively how this works. If you have a straightforward explanation, please share it in the comments. (h/t @brainpicker)