Details for McWorter's Pentigree IFS
Description
The coordinates of the five points can be determined using trigonometry by starting at (0,0) and moving along the red line segment paths from point to point. See the "Trig Value Details" page for the exact values of sine and cosine at 36° and 72°.
\(
P1 = \left( {\begin{array}{c}
{r\cos {{36}^ \circ }} \\
{r\sin {{36}^ \circ }} \\
\end{array}} \right) = \left( {\begin{array}{c}
{ \frac{1}{4}\sqrt 5 - \frac{1}{4}} \\
{\frac{3}{8}\sqrt {10 - 2\sqrt 5 } - \frac{1}{8}\sqrt {50 - 10\sqrt 5 } } \\
\end{array}} \right) = \left( {\begin{array}{c}
{0.309017} \\
{0.224514} \\
\end{array}} \right) \)
\(
P2 = P1 + \left( {\begin{array}{c}
{ - r\cos {{72}^ \circ }} \\
{r\sin {{72}^ \circ }} \\
\end{array}} \right) = \left( {\begin{array}{c}
{ - \frac{1}{4}\sqrt 5 + \frac{3}{4}} \\
{\frac{1}{4}\sqrt {10 - 2\sqrt 5 } } \\
\end{array}} \right) = \left( {\begin{array}{c}
{0.190983} \\
{0.587785} \\
\end{array}} \right) \)
\(
P3 = P2 + \left( {\begin{array}{c}
{r\cos {{36}^ \circ }} \\
{ - r\sin {{36}^ \circ }} \\
\end{array}} \right) = \left( {\begin{array}{c}
{\frac{1}{2}} \\
{ - \frac{1}{8}\sqrt {10 - 2\sqrt 5 } + \frac{1}{8}\sqrt {50 - 10\sqrt 5 } } \\
\end{array}} \right) = \left( {\begin{array}{c}
{0.5} \\
{0.363271} \\
\end{array}} \right) \)
\(
P4 = P3 + \left( {\begin{array}{c}
{ - r\cos {{72}^ \circ }} \\
{ - r\sin {{72}^ \circ }} \\
\end{array}} \right) = \left( {\begin{array}{c}
{\frac{3}{2} - \frac{1}{2}\sqrt 5 } \\
0 \\
\end{array}} \right) = \left( {\begin{array}{c}
{0.381967} \\
0 \\
\end{array}} \right) \)
\( P5 = P4 + \left( {\begin{array}{c} {r\cos {{36}^ \circ }} \\ { - r\sin {{36}^ \circ }} \\ \end{array}} \right) = \left( {\begin{array}{c} { - \frac{1}{4}\sqrt 5 + \frac{5}{4}} \\ { - \frac{3}{8}\sqrt {10 - 2\sqrt 5 } + \frac{1}{8}\sqrt {50 - 10\sqrt 5 } } \\ \end{array}} \right) = \left( {\begin{array}{r} {0.690983} \\ { - 0.224514} \\ \end{array}} \right) \)
