Nuts, Fish, Babies, and Nuts: Modeling with Cubic and Nearly-Cubic Functions

Abstract

In this chapter, we reflect on an example from Engel (2010) in which he models how the weight of hexnuts depends on their size. Based on our understanding of scaling, we expect the relationship to be cubic, and it is. We then explore examples that are nearly cubic, using data on snook (a kind of fish), where the power-law exponent is greater than 3, and human children, where the exponent is less than 3. On our return to the hexnuts, we find that the exponent is not 3 after all. Further analysis shows that in fact the relationship is not even a power law. Along the way we make observations about the role of data in generating insight, and the meaning of fit parameters such as the non-integer exponents. Finally, we reflect on how these problems illuminate our understanding of the nature of modeling as a whole.