Conformal Deformation of Conductors
I’d like to describe an observation so simple that it would not be worth mentioning if not for its consequences. \
Consider an electrically conducting lamina with constant resistivity
where the resistance is measured between opposite sides of the square. Figure 1b explains why: putting

This observation has an immediate implication for conformal maps, since they map infinitesimal squares to infinitesimal squares. Thus,
as Figure 2 illustrates. Indeed, let us divide the domain into infinitesimal squares (to make this rigorous, one has to use equipotential lines and the lines of current). The conformal image is then partitioned into infinitesimal squares as well. The map clearly preserves the ratio
Observation

The conventional name for the resistance is modulus, although “resistance” might be a better term. Another interpretation of the modulus is capacitance, in which case we must replace the current in the conductive lamina with the electrostatic field in the non-conductive plane.
The same electric interpretation of the modulus applies to deformed annuli, i.e., to doubly connected regions. Two such regions are conformally equivalent only if their resistances—measured between the inner and outer boundaries—are equal. And the combinatorial meaning of modulus
The figures in this article were provided by the author.
About the Author
Mark Levi
Professor, Pennsylvania State University
Mark Levi (levi@math.psu.edu) is a professor of mathematics at the Pennsylvania State University.
Stay Up-to-Date with Email Alerts
Sign up for our monthly newsletter and emails about other topics of your choosing.