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Pentaflake

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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).

PentaflakeDistances

For a pentagon of side length 1, the first ring of pentagons has centers at radius

d_1=2r=1/2(1+sqrt(5))R=phiR,
(1)

where phi is the golden ratio. The inradius r and circumradius R are related by

r=Rcos(1/5pi)=1/4(sqrt(5)+1)R,
(2)

and these are related to the side length s by

s=2sqrt(R^2-r^2)=1/2Rsqrt(10-2sqrt(5)).
(3)

The height h is

h=ssin(2/5pi)=1/4ssqrt(10+2sqrt(5))=1/2sqrt(5)R,
(4)

giving a radius of the second ring as

d_2=2(R+h)=(2+sqrt(5))R=phi^3R.
(5)

Continuing, the nth pentagon ring is located at

d_n=phi^(2n-1).
(6)

Now, the length of the side of the first pentagon compound is given by

s_2=2sqrt((2r+R)^2-(h+R)^2)=Rsqrt(5+2sqrt(5)),
(7)

so the ratio of side lengths of the original pentagon to that of the compound is

(s_2)/s=(Rsqrt(5+2sqrt(5)))/(1/2Rsqrt(10-2sqrt(5)))=1+phi.
(8)

We can now calculate the dimension of the pentaflake fractal. Let N_n be the number of black pentagons and L_n the length of side of a pentagon after the n iteration,

N_n=6^n
(9)
L_n=(1+phi)^(-n).
(10)

The capacity dimension is therefore

d_(cap)=-lim_(n->infty)(lnN_n)/(lnL_n)
(11)
=(ln6)/(ln(1+phi))
(12)
=1.861715...
(13)

(OEIS A113212).

PentaflakeRecursiveGrowth

An attractive variation obtained by recursive construction of pentagons is illustrated above (Aigner et al. 1991; Zeitler 2002; Trott 2004, pp. 21-22).

See also

Pentagon

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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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Pentaflake

Cite this as:

Weisstein, Eric W. "Pentaflake." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Pentaflake.html

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