### Elegance and Complexity

Chaitin and others refer to the minimal representation issue in AIT as "elegance"--the situation where we have an expression with the property that no smaller expressions yield the same result.

In cognitive science we're often interested in the complexity of a stimulus or system. Complexity in this particular case is roughly equivalent to the inverse of elegance, in an Occam's razor fashion. The (very, very rough) idea is that as something becomes more "complex," it becomes more difficult to do / harder to remember / harder to interpret / insert your limitological position here. It caught on quite well in the fields of attention, perception, and, oddly enough, aesthetics. Mathematician George Birkhoff believed that the aesthetic value of an object could be determined as a simple ratio of where represents the amount of order in the object and represents the amount of complexity [10]. The rather obvious problem with this ratio is that its two constituents are quite difficult to define in a truly operational and meaningful way. Complexity in this sense has been defined in an almost countless number of ways. One example from the world of shape is the number of vertices in a polygon.

Two shapes, four vertices each. Is one more "complex" than the other? Depends how you look at it. Is one more aesthetically pleasing than the other? Again, perspective is everything. (See Komar and Melamid's Scientific Guide to Art for further details on [11].) I'm sure you'd agree that this probably isn't quite an adequate description of the situation. There have been coding-based approximations. In the above case, you may have a better chance of the measure working out if you described the rectangle in a raster-encoding sort of way, sort of like the following.

`Thickness 1Draw a black line from 0 to .`

The economy is clear when you look at the metacode you'd need to encode the polygon on the right.

`Draw a black line from .2 to .3.Draw a black line from .21 to .32.Draw a black line from .22 to .35....Draw a black line from 0 to 0.01.`