1 | Mathematics In the Modern World
Ramos, Ivan Henry A.
GUIDE QUESTIONS:
1. What are fractals?
Fractals may be defined as patterns, quantities, or objects that are indefinitely
intricate and recurring across various scales—self-similar. Although common examples
of fractals are graphs or visual representations of iterated function systems (e.g.,
Sierpinski triangle), fractal-like forms are prevalent in nature. As discussed in the attached
video, the principle or idea of fractals may be seen in trees—a mother branch diverging
to daughter branches, snowflakes, blood vessels, and in myriad more natural
phenomena.
2. What does “self-similarity” mean?
In terse terms, self-similarity refers to a quality of an object to have its prevalent
structure resemble the pattern of its fragments or portions. One may call an object selfsimilar if it has its substructure practically similar to itself. In simple terms, self-similar
structures are things, objects, or patterns that appear notably identical when viewed
normally or magnified.
3. Name TWO (2) mathematicians who are known for their contribution to the field
of fractal geometry. What did they do to earn recognition?
On various accounts, (1) Benoît Mandelbrot is recognized as the Father of Fractal
Geometry. He is regarded as the title for his discovery of the Mandelbrot set in 1979
through the help of computers. This discovery has engendered and forged the
fundamental principles of fractal geometry. Another esteemed mathematician in fractal
geometry is (2) Helge von Koch for introducing the Koch Curve and, eventually, the Koch
Snowflake. The Koch Snowflake is a fractal that may be visualized by creating an
equilateral triangle, repeatedly taking out each side’s inner third, and filling it with another
equilateral triangle (Weisstein, n.d.).
4. Cite THREE (3) applications of fractals that illustrate solutions to real-life
problems.
Based on its definition (i.e., “… patterns that are infinitely intricate…”), fractals are
physically impossible. Even so, the principles and concepts from the ideal fractal may be
employed to create assumptions. (1) For instance, truncated fractals—a limited fractal,
imitating only its geometric form—are used to construct more efficient and functional
antennas, circuit filters, filters and microwave substrates, and other metamaterials
(Krzysztofik, 2017). (2) In the medical field, fractal analysis helps in histopathology and
diagnosis, mainly cancers and tumors (Bose et al., 2015; Fractal Foundation, n.d.). (3)
Fractals are used in creating guided and informed inferences and decisions in the field of
business and economics (Liberto, 2021; Takayasu & Takayasu, 2009).
2 | Mathematics In the Modern World
Ramos, Ivan Henry A.
5. Provide a link to a web source (video or website) that gives a piece of interesting
information about fractals. Write at least three (3) concepts you have learned from
this web source.
Nowell, W. (2013). Fractal Geometry. Resources for College Mathematics.
http://www.cmath.info/html/fractal_geo.html
5.1.
Iteration or the repetition of a certain process—commonly applied to geometric
entities or functions—is a fundamental part of constructing and visualizing all
forms of fractals.
5.2.
The material expounds on six major classifications of fractals. (1) Classic
fractals which are fractals on geometric elements (e.g., segments, planes, or
even three-dimensional shapes); (2) Complex polynomial fractals are
fractals that involve the application of the iteration process on complex
polynomial functions (e.g., Mandelbrot Set); (3) Iterated Function Systems
such as the Barnsley fern (4) Strange attractors which are fractals involving
differential equations; (5) Lindenmayer systems that model botanical
development; and (6) Plasma fractals that are used in computer-aided design
programs.
5.3.
Fractal dimension is a property that serves as a parameter for the complexity
of patterns—particularly fractal patterns. Often, fractals have dimensions that
are non-integer (or fractions). The formula in calculating for fractal dimensions
is:
𝐷=
log(𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑒𝑙𝑓 − 𝑠𝑖𝑚𝑖𝑙𝑎𝑟 𝑝𝑖𝑒𝑐𝑒𝑠)
log(𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟)
3 | Mathematics In the Modern World
Ramos, Ivan Henry A.
OTHER NOTES: Fractals the Hidden Dimension
“There is order in chaos.”
Fractal Geometry – a branch of Geometry
introduced by Mandelbrot that focuses on the
study of fractals. It bridges the gap between
mathematics, nature, and art.
Georg Cantor was a German mathematician
who presented the first-ever monster
(Cantor Set).
“Think not on what you see, but on what it
took to produce what you see.”
The quote was referring to iteration – the
endless repetition (of applying functions
repeatedly) to visualize fractals.
Fractals – infinitely intricate self-similar
structures.
Fractals may be found in nature; some
examples include trees, organ systems, and
even galaxies of stars.
Figure 2. The Cantor Set
The said monster was created from a line
that was broken into three equal pieces,
whose middle segment was removed. The
process was repeated to the smaller lines.
Helge von Koch was a Swedish
mathematician who presented a fractal curve
that was later known as the Koch Snowflake.
“The whole of the fractal looks just like the
part.” — referring to self-similarity
Self-similarity – the trait of an object or
pattern (or, in this case, a fractal) to have the
resemblance between its substructures and
its overall structure.
Figure 3. Koch Snowflake
The Koch Snowflake was formed from an
equilateral triangle. A portion of each side of
the triangle was be replaced with two
segments, and this process was repeated to
create the snowflake.
Figure 1. The Sierpinski Gasket
In the 19th Century, early attempts to formally
describe or define (fractal) curves had led to
the discovery of what was called back-then
as monsters or things beyond the realm.
Monsters – were patterns that satisfy the
conditions of being a fractal but are
unimaginable and could not be physically or
visually depicted.
Fractal curves such as these were labeled as
pathological curves as it does not conform to
the principles of measurement and Euclidian
geometry.
A problem presented by Gaston Julia
regarding iteration of complex polynomial
functions
piqued
the
interest
of
Mandelbrot—who, through the help of
computers, devised the Mandelbrot Set.
4 | Mathematics In the Modern World
Ramos, Ivan Henry A.
References:
Bose, P., Brockton, N. T., Guggisberg, K., Nakoneshny, S. C., Kornaga, E., Klimowicz, A. C.,
Tambasco, M., & Dort, J. C. (2015). Fractal analysis of nuclear histology integrates
tumor and stromal features into a single prognostic factor of the oral cancer
microenvironment. BMC Cancer, 15, 409. https://doi.org/10.1186/s12885-015-1380-0
Fractal Foundation. (n.d.). Fractal Applications. http://fractalfoundation.org/OFC/OFC-12-4.html
Krzysztofik, W. C. (2017). Fractals in Antennas and Metamaterials Applications. Fractal Analysis
- Applications in Physics, Engineering, and Technology. 10.5772/intechopen.68188
Liberto, D. (2021, August 30). Fractal market hypothesis (FMH).
https://www.investopedia.com/terms/f/fractal-markets-hypothesis-fmh.asp
Takayasu, M., & Takayasu, H. (2009). Fractals and Economics. Complex Systems in Finance
and Econometrics, 444–463. doi:10.1007/978-1-4419-7701-4_25
Weisstein, E. W. (n.d.) Koch Snowflake. MathWorld: A Wolfram Web Resource.
https://mathworld.wolfram.com/KochSnowflake.html