Larry Riddle, Agnes Scott College

#### Description

The translations used in the IFS are determined by the coordinates of the starting endpoints P1, P2, P3, P4, and P5 of the line segments as shown in the figure below. Each line segment is obtained by scaling the unit horizontal line by r, rotating by the appropriate angle, then translating the line so that the origin is moved to P1, P2, P3, P4, or P5.

The coordinates of the five points can be determined using trigonometry. Recall that the scaling factor is $$r = \sqrt{\frac{{6 - \sqrt 5 }}{31}} = 0.34845$$ and A = 11.81858573°.

$$P1 = \left[ {\begin{array}{*{20}{c}} {r\cos (A)} \\ {r\sin (A)} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {0.34106} \\ {0.07137} \\ \end{array}} \right]$$

$$P2 = P1 + \left[ {\begin{array}{*{20}{c}} {r\cos (72^ \circ + A)} \\ {r\sin (72^ \circ + A)} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {0.37858} \\ {0.41779} \\ \end{array}} \right]$$

$$P3 = P2 + \left[ {\begin{array}{*{20}{c}} {r\cos (A)} \\ {r\sin (A)} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {0.71965} \\ {0.48916} \\ \end{array}} \right]$$

$$P4 = P3 + \left[ {\begin{array}{*{20}{c}} { - r\cos (36^ \circ + A)} \\ { - r\sin (36^ \circ + A)} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {0.48567} \\ {0.23095} \\ \end{array}} \right]$$

$$P5 = P4 + \left[ {\begin{array}{*{20}{c}} {r\cos (72^ \circ - A)} \\ { - r\sin (72^ \circ - A)} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {0.65894} \\ { - 0.07137} \\ \end{array}} \right]$$