But sometimes we take too much for granted in the universality of our use of mathematics. Many science papers become unreadable to a large fraction of their intended audience because the author incorrectly assumes "everyone uses the same symbols that I do" and fails to define terms clearly.

A particularly dangerous example of this is units. Occasionaly, scientists or engineers will fail to specify the unit system they are working with because they incorrectly assume "we're all using the same system." Tragically, NASA lost a $125 Million mars probe because two groups of engineers were using different units and each assumed that everyone was doing what they were so they didn't need to specify.

When we fit a length, the number of times our unit length, λ, fits in depends linearly on the length we choose. When we fit square boxes of unit area, λ^{2}, the number of boxes we fit in depends quadratically on the length we choose.

To see what this means, imagine that instead of a length λ, we choose a different length, μ = λ/2. Our new measurements can be found algebraically in a straightforward way.

L = 12 λ = 12 (2 x λ/2) = 24 μ

A = 24 λ^{2} = 24 (2 x λ/2)^{2} = 24 x 4 (λ/2)^{2} = 96 μ^{2}.

A = 24 λ

Note that we have to change λ __inside__ the square when we are working with the area. This shows that if we change our unit scale for length, the length and the area change in different ways.

This clarifies the reason why we cannot equate a length and an area. Even if the number assigned to them agrees in one system, it will not in any other.

How things change when we change some of the input descriptors is a general and powerful method in advanced research physics. The general technique is associated with group theory: the mathematical tools for studying how things change. For example, general relativity was developed from Einstein's idea that the equations of physics should remain the same under *any* change of coordinates -- even complicated non-linear ones. Modern particle theories are constructed by choosing the descriptors and choosing groups associated with possible changes of the description of the system.

A fascinating modern research example of scaling arises from the observation that biological systems show scaling laws when they change their size. If you plot the log of the average number of heartbeats a minute for a given species vs. the log of the average mass of that species you get a straight line. That line has been known for almost 100 years, but there has been no explanation of it (on the order of our understanding of how the area changes as we change the length scale) until recently. The answer involves fractals. For an introduction to these ideas, check out the talk by Geoffrey West, one of the discoverers of the explanation.

University of Maryland | Physics Department | Physics 374 Home |
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This page prepared by Edward F. Redish

Department of Physics

University of Maryland

College Park, MD 20742

Phone: (301) 405-6120

Email: redish@umd.edu

Last revision 11. September, 2005.