Fractals are geometric objects used to describe irregular shapes that have a fractional dimension and commonly appear in nature. Although several proposals for the study of perfect fractals at the basic level are present in the literature, only few proposals for the study of real fractals exist, which does not seem reasonable considering the wealth of the theme and its potentiality as an interdisciplinary topic. In this text, we present a simple and easily assimilable procedure to calculate the fractal dimension of any two-dimensional object. It only demands a computer with the free software ImageJ installed. This procedure is easily understandable, considering the basic school curriculum, and easily usable in a variety of school settings.
References
1.
J.
Sanudo
et al., “Fractal dimension of lightning discharge
,” Nonlinear Processes Geophys.
2
(2
), 101
–106
(1995
).2.
T.
Gallo
et al., “Thermal collapse of snowflake fractals
,” Chem. Phys. Lett.
543
, 82
–85
(2012
).3.
R. D.
Campbell
, “Describing the shapes of fern leaves: A fractal geometrical approach
,” Acta Biotheoretica
44
(2
), 119
–142
(1996
).4.
I.
Rian
and M.
Sassone
, “Tree-inspired dendriforms and fractal-like branching structures in architecture: A brief historical overview
,” Front. Archit. Res.
3
(3
), 298
–323
(2014
).5.
H.
Takayasu
, Fractals in the Physical Sciences
(Manchester University Press
, Manchester
, 1990
).6.
B.
Mandelbrot
, D.
Passoja
, and A.
Paullay
, “Fractal character of fracture surfaces of metals
,” Nat.
308
(5961
), 721
(1984
).7.
H-O.
Peitgen
, H.
Jurgens
, and D.
Saupe
, Chaos and Fractals: New Frontiers of Science
(Springer-Verlag
, New York
, 1992
).8.
R. H.
Ko
and C. P.
Bean
, “A simple experiment that demonstrates fractal behavior
,” Phys. Teach.
29
, 78
–79
(Feb.
1991
).9.
D. H.
Esbenshade
, “Fractal bread
,” Phys. Teach.
29
, 236
(Oct
. 1991
).10.
M.
Amaku
, L. B.
Horodynski-Matsushigue
, and P. R.
Pascholati
, “The fractal dimension of breads
,” Phys. Teach.
37
, 480
–481
(Nov.
1999
).11.
M.
Zanoni
, “Measurement of the fractal dimension of a cauliflower
,” Phys. Teach.
40
, 18
–20
(Jan.
2002
).12.
K.
Zembrowska
and M.
Kuzma
, “Some exercises on fractals for high school students
,” Phys. Teach.
40
, 470
–473
(Nov.
2002
).13.
P.
Knutson
and E. D.
Dahlberg
, “Fractals in the classroom
,” Phys. Teach.
41
, 387
(Oct.
2003
).14.
View the supplementary material at TPT Online, http://dx.doi.org/10.1119/1.5126825 under the Supplemental tab.
16.
A.
Karperien
, “Fraclac for imageJ
.” Available at https://imagej.nih.gov/ij/plugins/fraclac/FLHelp/Introduction.htm, accessed Jan. 17, 2018.17.
“
Eclipse
,” The Landscape Photography Podcast
, https://www.landscapephotographypodcast.com/podcast/2017/ 8/6/landscape-photography-podcast-ep-8, accessed Jan. 17, 2018.18.
It is important to mention that it would not be necessary to binarize the figure before calculating its size through FracLac once the package itself already does this automatically. Our choice to binarize the figures before calculating their dimension has a didactic objective, namely, to allow students to understand the process of binarization that precedes the estimation of the fractal dimension.
© 2019 American Association of Physics Teachers.
2019
American Association of Physics Teachers
AAPT members receive access to The Physics Teacher and the American Journal of Physics as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.