The Theorem that Won the War: Activities for Part 1.1 (Rotors)

Author(s):
Jeff Suzuki (Brooklyn College)

1. Suppose a rotor has the wiring shown:

1. Describe the permutation if the current is running from left to right.
2. Describe the permutation if the current is running from right to left.
3. What do you notice about the two permutations?
2. Draw the side view of a rotor whose right-to-left permutation is given. Then give the left-to-right permutation for the rotor.
1. $\left(\begin{array}{cccccc} a & b&c &d &e &f \\ d&f&b&a&c &e \end{array}\right)$
2. $\left(\begin{array}{cccccc} a & b&c &d &e &f \\ c&e&d&a&f &b \end{array}\right)$
3. Consider a permutation described as $\left(\begin{array}{cccccc} a & b&c &d &e &f \\ b&e&d&c&f &a \end{array}\right)$.
1. Explain why it is reasonable to say $\left(\begin{array}{cccccc} a & b&c &d &e &f \\ b&e&d&c&f &a \end{array}\right)=\left(\begin{array}{cccccc} a & f&d &b &e &c \\ b&a&e&c&d &f \end{array}\right)$.

Suggestion: “Things that do the same thing are the same thing.”
We can view a permutation as a function, so the permutation on the left maps a → b, d → e, and so on;
what does the permutation on the right do?

2. Rewrite the permutation shown by completing the second row: $\left(\begin{array}{cccccc} a & b&c &d &e &f \\ b&e&d&c&f &a \end{array}\right)=\left(\begin{array}{cccccc} & & & & & \\ a&b&c&d&e &f \end{array}\right)$
3. Rewrite the permutation shown by completing the first row: $\left(\begin{array}{cccccc} a & d&e &b &c &f \\ b&e&d&c&f &a \end{array}\right)=\left(\begin{array}{cccccc} & & & & & \\ a&b&c&d&e &f \end{array}\right)$
4. Consider the permutation $\left(\begin{array}{cccccc} a & b&c &d &e &f \\ b&a&e&c&d&f \end{array}\right)$.
1. Draw a rotor corresponding to this permutation. (Assume this permutation is for a left-to-right flow of current.)
2. Find the right-to-left permutation.
3. Find the first row of the permutation on the right: $\left(\begin{array}{cccccc} a & b&c &d &e &f \\ b&a&e&c&d &f \end{array}\right)=\left(\begin{array}{cccccc} & & & & & \\ a&b&c&d&e &f \end{array}\right)$
4. Compare the permutations you found in Activity 4b with Activity 4c. Does this suggest an easy way to find the right-to-left permutation from the left-to-right permutation?

Jeff Suzuki (Brooklyn College), "The Theorem that Won the War: Activities for Part 1.1 (Rotors)," Convergence (October 2023)