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Visual Logic

This expression

$\begin{displaymath}2+x\end{displaymath}$

and this expression

$\begin{displaymath}x+2\end{displaymath}$

are the same. You probably learned this by memorizing a rule, called the commutative property, which states that the sum of several terms is the same regardless of their order. Because the rule is so simple, most people instantly know that the two expressions are equivalent, and can swap one for the other without much thought.

There is a way this process can be modelled visually. Start with this expression:

$\includegraphics{figures/movsym/fig_movsym_ga.eps}$

Visualize it, so that you see it mentally. In your image, chunk the x and the 2 separately.

$\includegraphics{figures/movsym/fig_movsym_gc.eps}$

Now, rotate them around the + sign.

$\includegraphics{figures/movsym/fig_movsym_gd.eps}$ $\includegraphics{figures/movsym/fig_movsym_ge.eps}$ $\includegraphics{figures/movsym/fig_movsym_gb.eps}$

While we rotated it counter-clockwise around the +, there are other possibilities. You can rotate it the other way, or in the third dimension by having one symbol pass in front and the other pass behind. They can even 'teleport', immediately swapping places with each other. Try these out right now. Notice which one seems more natural for you.

The conversion above is a hint of something: we can take an image that has mathematical meaning, and by doing certain graphic manipulations on the image itself, end with the expression in a different, equivalent form. Done correctly, the graphic manipulations have the same effect as normal mathematical methods. Visual logic has to do with the abstract pattern of the change in the image. If the expression above was y + 3, we would transform it to 3 + y, by shifting things around in a similar way to what we did above. Even though the symbols in that expression are different, the visual logic of the transformation would be the same.

When we do a transformation on an equation's image like this, there are two things going on at once. There is the visual logic, the pattern of change in the image itself. There is also the mathematical logic; x + 2 really does equal 2 + x. A pattern of visual logic is really only useful to us if it completely honors the mathematical logic that is represented. As techniques are presented in this and the following chapters, keep in mind that this connection between the visual and mathematical logic has to be maintained in everything we do.

Next: Movement As Arithmetic Up: Symbol Motion Previous: Symbol Motion