Bean-Shaped Bodies

Mathematics is not usually considered an experimental science. However, there are many analogies between the approach to mathematical research and the more familiar experimental scientific process. While mathematicians do not usually go out and directly observe and experiment with real-world phenomena, research is by no means entirely deductive. Attempting to solve a problem can entail much trial and error, guesswork, and experimentation. When trying to prove or disprove a theorem, a good starting point is looking for counter-examples. When looking for relationships and patterns within a specific class of mathematical object, it is often useful to examine a great number of such objects. For example, if you are trying to better understand lines of curvature on different surfaces, it is helpful to investigate a host of different surfaces, looking for patterns.

Two of our models, X 11a and X 11b, were designed for this sort of purpose. They are each described by Schilling as "bean-shaped bodies", and were built to experimentally determine lines of curvature, asymptotic curves, and parabolic curves on the surface. Like many of our models, these are likely to have been used in lectures to give insight into these types of curves. Their unusual shape means that it is not so easy to guess the behaviour of their lines of curvature, making it interesting to study them and explore their properties. 

 

Model X 11a

X 11aModel X 11bX 11a

 

Model X 11b 

Model X 11bModel X 11bModel X 11b