A Math Forum Web Unit
Mutsumi Suzuki's

Transforming Stars

First let's look at how to apply the transformation (or exchange) rule to yield stars with the number 1 not at the top, but at an inner corner.

Consider this star:

There are three exchange rules:

1. A<-->L   B<-->C   D<-->E   H<-->I   J<-->K

   

2. B<-->K   H<-->F   A<-->C   L<-->J   D<-->G

   Rotate the star 60° to the right, use rule 1, and rotate 60° back to the left.

3. E<-->H   A<-->D   B<-->F   G<-->K  I<-->L

   Rotate the star 60° to the left, use rule 1, and rotate 60° back to the right.

1. Let's apply Exchange Rule 1:
      

2. Now let's rotate Derived Star 1 180°:

      

   Derived Star 2 is not identical to the original but the number 1 is still on top!

3. Let's apply Exchange Rule 3 to the Original Star and rotate Derived Star 3 60° to the right:

   

In the case of Derived Star 3, the number 1 is at an inner corner.

Here Exchange Rule 3 is the same as (1) rotating 60° to the left, (2) applying Exchange Rule 1, and (3) rotating 60° back to the right: after the rotation, the number 1 is at B; with the transformation the number 1 moves to the inner corner.