Pentaflake
المؤلف:
Aigner, M.; Pein, J.; and Stechmüller, T. T
المصدر:
Math. Semesterber. 38
الجزء والصفحة:
...
24-9-2021
2121
Pentaflake
The pentaflake is a fractal with 5-fold symmetry. As illustrated above, five pentagons can be arranged around an identical pentagon to form the first iteration of the pentaflake. This cluster of six pentagons has the shape of a pentagon with five triangular wedges removed. This construction was first noticed by Albrecht Dürer (Dixon 1991).
For a pentagon of side length 1, the first ring of pentagons has centers at radius
 |
(1)
|
where 
is the golden ratio. The inradius 
and circumradius 
are related by
 |
(2)
|
and these are related to the side length 
by
 |
(3)
|
The height 
is
 |
(4)
|
giving a radius of the second ring as
 |
(5)
|
Continuing, the 
th pentagon ring is located at
 |
(6)
|
Now, the length of the side of the first pentagon compound is given by
 |
(7)
|
so the ratio of side lengths of the original pentagon to that of the compound is
 |
(8)
|
We can now calculate the dimension of the pentaflake fractal. Let 
be the number of black pentagons and 
the length of side of a pentagon after the 
iteration,
The capacity dimension is therefore
(OEIS A113212).
An attractive variation obtained by recursive construction of pentagons is illustrated above (Aigner et al. 1991; Zeitler 2002; Trott 2004, pp. 21-22).
REFERENCES:
Aigner, M.; Pein, J.; and Stechmüller, T. T. Math. Semesterber. 38, 242, 1991.
Ding, R.; Schattschneider, D.; and Zamfirescu, T. "Tiling the Pentagon." Discr. Math. 221, 113-124, 2000.
Dixon, R. Mathographics. New York: Dover, pp. 186-188, 1991.
Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, pp. 76 and 109, 2002.
Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, pp. 64-65, 2002.
Lück, R. Mat. Sci. Eng. A 263, 194-296, 2000.
Sloane, N. J. A. Sequence A113212 in "The On-Line Encyclopedia of Integer Sequences."
Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 60 and 88, 1999.
Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, pp. 40-42, 2004. http://www.mathematicaguidebooks.org/.
Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, p. 19, 2004. http://www.mathematicaguidebooks.org/.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 104, 1991.
Zeitler, H. Math. Semesterber. 49, 185, 2002.
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