Larry Riddle, Agnes Scott College

## Durer's Pentagons

#### Description

"Tile Patterns Formed by Pentagons" by Albrecht Durer in The Painter's Manual, 1525.
"Fifth, you can combine pentagons in the following manner: First draw a pentagon and place pentagons of the same size on each side. Then place five pentagons on their sides, particularly along the two sides. This will result in the formation of five narrow lozenges between them. Then add pentagons in the angles which will have formed, so that these will touch the narrow lozenges with their corners. You can continue in this manner as long as you desire."

To convert this into an iterated function system, image starting with the initial polygon and scaling by the same factor r = 0.382 used for the Sierpinski pentagon. This time, however, translate six copies so that they would fit exactly inside the initial polygon. The copy in the middle must first be rotated by 180°. Now repeat the construction on each of the six polygons to get the second iteration. Continue indefinitely.

Construction
Animation
 first iteration second iteration third iteration fourth iteration

#### IteratedFunctionSystem

The IFS for this construction is the same as the one for the Sierpinski pentagon with the addition of a sixth function for the scaled polygon in the middle. The scale factor is $$r = \frac{{3 - \sqrt 5 }}{2}$$ = 0.381966. The sixth function includes a rotation by 180°.

\begin{align} {f_1}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} \\ {f_2}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.618} \\ 0 \\ \end{array}} \right] \\ {f_3}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.809} \\ {0.588} \\ \end{array}} \right] \\ {f_4}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.309} \\ {0.951} \\ \end{array}} \right] \\ {f_5}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {0.382} & 0 \\ 0 & {0.382} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} { - 0.191} \\ {0.588} \\ \end{array}} \right] \\ {f_6}({\bf{x}}) &= \left[ {\begin{array}{*{20}{c}} {-0.382} & 0 \\ 0 & {-0.382} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} { 0.691} \\ {0.951} \\ \end{array}} \right] \\ \end{align}

Details are the same as for the Sierpinski pentagon.

#### Dimension

The Durer fractal is self-similar with 6 non-overlapping copies of itself, each scaled by the factor r < 1. Therefore the similarity dimension, d, of the attractor of the IFS is the solution to

$\sum\limits_{k = 1}^6 {{r^d}} = 1 \quad \Rightarrow \quad d = \frac{{\log (1/6)}}{{\log (r)}} = \frac{{\log (6)}}{{\log (2/(3 - \sqrt 5 )}} = 1.86172$

#### References

1. Dixon, R. Mathographics, Dover Publications.
2. Durer, Albrecht. The Painter's Manual, translated by Walter L. Strauss, Abaris Books, Inc., N.Y., 1977.
3. Jones, Juw. "Fractals Before Mandelbrot-A Selective History," in Fractals and Chaos, Crilly, Earnshaw, and Jones, Editors, Springer-Verlag 1991.
4. Weisstein, Eric W. "Pentaflake." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Pentaflake.html