Biomimicry and Product Design (Part 3) - Mathematics Relations in Nature
The most of nature creation or optimisation
processes are supported by mathematics and algorithms representations, and
possible through this understanding, these relations can be applied to human
products and services.
Was in 1202 when a 32-year-old Italian,
Leonardo of Pisa, referenced as the greatest mathematical of middle age, and
known as Fibonacci (is a shortening of the Latin filius Bonacci, that means the
son of Bonaccio), finished one of the most influential books of all time,
introducing the Hindu-Arabic numerical system to Western Europe. Before that,
Europeans used Roman numerals in arithmetic. The Fibonacci Sequence (FS), or
Fibonacci Numbers, is an integer sequence where the next number is the result
of the sum of the two previous numbers, starting with 0 and 1 (Table
1).
Fibonacci Sequence rule:
xn = x(n-1) + x(n-2), where n is the number
position on sequence
Fibonacci presented the sequence due to an
observation of an old 6th-century Indian mathematical problem about rabbit
reproduction starting with one male and one female, and for one year. The
premises of this problem are: the initial couple have just been born. The
rabbits´ sexual reproduction occurs at each month, and the gestation period is
for one month. Each couple gives birth to one more couple. No one rabbit dies
for one year.
Figure 1 - Mathematical problem of
rabbit reproduction that inspired the Fibonacci Sequence.
There are numerous curiosities and mathematical
relations about the FS that has been discovered along
the centuries see (Garland,
1987; Posamentier & Lehmann, 2007).
The Fibonacci Sequence is also found in
numerous natural phenomena. But, before discussing this, is necessary first to
introduce the concept of the Golden Ratio (φ), also called the Golden Mean or
the Divine Proportion. The Golden Ratio has been used since ancient times by
Egyptians and Greeks, and has attracted the attention of many talented minds
over centuries such as mathematicians, physicists, biologists, artists etc. The
Greek symbol φ (Phi) was adopted in 1900´s in honour of Phidias (500 BC – 432
BC), a Greek sculptor and mathematician, that applied it to the design for the
Parthenon and its arts (Meisner, 2012) (Figure
4). The
number φ is equal to 1,6180339…, and it starts to appear when two sequential
Fibonacci Number are divided by each other. Initial divisions have results far
from φ, but with each successive division, the result comes closer and closer
to 1.6180339… (Figure 2).
Figure 2 - The Golden Ratio (φ) starts
to appear when two sequential Fibonacci Number are divided by each other.
Most of the Golden Ratio manifestation is
found in geometry patterns. The golden rectangle is formed with any proportion
of edges with 1 x 1,6180339… If this rectangle has its bigger edge separated in
1 and 0,618… (creating one square and one rectangle), another golden rectangle
will appear, and doing it again to the new rectangle, another rectangle will
appear again, and again and again (Figure
3 (a)).
If in this division, is draw a 1/4 arc in
square vertex, and again for the other smaller square, again and again, will be
created the golden spiral (Figure 3 (b)).
If all squares are measured, their dimensions will follow the Fibonacci
Sequence (Figure 3 (c)).
These geometries relationships are found in many natural events, alive or
inanimate (Figure
5). There
are many other geometries that show the Golden Ratio, found in literature. The
Golden Angle (ψ) is the complementary angle found by the division of a complete
circumference with φ.
Fibonacci Sequences and Golden Ratio
relationships are often present in plants (flowers, leaves, fruits, branches
etc). Many flowers use Fibonacci Numbers as the number of petals, and seed
heads are organised following the golden spiral, the golden angle, and
Fibonacci Numbers quantities (Figure
6).
This very efficient arrangement to catch solar energy gave inspiration to
positioning mirrors to the creation of a thermal-solar energy generator (Figure
7).
Figure 7 – Sunflower seed heads
arrangement is composed of golden relationships. That arrangement inspired to
the creation of a thermal-solar energy generator. FONT: Solar Group.
Is common also to find golden relationships
in animals and human (Figure 8, Figure 9). The
human body has numerous divisions that respect the Golden Ratio (Figure 10).
Figure 10 - Examples of divisions in
human anatomy that is found the Golden Ratio. FONT: Newhacks.info
An explanation of why the Fibonacci
Sequence often appears in patterns of growth in nature is that the growth and
self-renewal cells process induce hierarchical patterns generation. This hierarchical
pattern, in different scales, is the same as the problem of rabbit population
growth. Thus, mathematical laws involving temporal and spatial rules for cell
division and growth patterns, end up being the Fibonacci Sequence (Raymond & Schleiniger, 2017). However, is essential to emphasise that the Golden Ratio only
appears in living beings if they are healthy. Any mutation or interruption in
the natural growing process will create anomalies that will not respect the
proportion. For this reason, it is possible to suppose that the human brain
unconscious associates figures that have the Golden Ration as harmony and
beauty, independent of culture or time, doing this proportion be considered a
universal beauty standard. Due to this peculiar propriety, the Golden Ratio has
been intensively explored by artists, architects, designers, marketers etc (Figure
12, Figure 13, Figure 14, Figure 15, Figure 16). One
of the most famous was Leonardo Da Vinci (Figure
11).
The maxillofacial surgeon Dr Stephen
Marquardt, based on the Golden Ratio and studies of beauty patterns throughout
history and hundreds of faces, created the beauty mask (Figure
17, Figure 18). How
more a person´s face fits with the mask, more is the probability to be
considered beautiful. Even a person not considered beauty, through digital
manipulation of the picture using the mask as the reference, is possible to
become considered more attractive (Figure
19).
Another frequent pattern that appears in
nature is the fractals. Fractals are decomposition (or growth) geometric, on
its edge, of a similar geometry (exact, approximate or statistical), but in
different scales, that can repeat infinitely. Different from the Fibonacci
Sequence, fractals have irregular proprieties that cannot be described by
Euclidean Geometry[1] (Meakin, 1990), although,
that does not mean that the Fibonacci Sequence and fractals cannot be found
together in naturals phenomena, that in fact is possible (Figure
20).
All these mathematical manifestations in
nature, the Fibonacci Sequence, the Golden Ratio and fractals, are an affluent
source of inspiration to the product design and development. The incorporation
of these in the creative process with Topology Optimisation, can give more
options and unexpected solutions, and even control beauty and intangible
aspects in the final form.
Bibliography:
Garland, T. H. (1987). Fascinating
Fibonaccis: Mystery and magic in numbers. Palo Alto: Dale Seymour.
Meakin, P. (1990). Fractal
structures. Progress in Solid State Chemistry, 20(3), 135–233.
https://doi.org/10.1016/0079-6786(90)90001-V
Meisner, G. (2012). History
of the Golden Ratio. Retrieved April 20, 2018, from
https://www.goldennumber.net/golden-ratio-history/
Posamentier, A. S., &
Lehmann, I. (2007). The fabulous Fibonacci numbers. Amherst, NY:
Prometheus Books.
Raymond, C., & Schleiniger,
G. (2017). Why do fibonacci numbers appear in patterns of growth in nature?, 55(5),
30–41.
[1]
Euclidean Geometry is the study of lines, planes and solids figures based upon
intuitive and deductible postulates, attributed to the Greek mathematician
Euclid (c. 300 BCE).
Comments
Post a Comment