Geometry of water
geometry La geometría of del water agua
- Page 3: The geometry of water Laura Zamudio
- Page 7: why fractals? Nature, architecture
- Page 11: The theory of fractals is relativel
- Page 15: what are fractals? From the above a
- Page 19: ability to perform and process a la
- Page 23: that each of the parts that compose
- Page 27: Photo recovered from: https://flic.
- Page 31: Fractals in medicine Fractals, bein
- Page 35: should be noted that Pollock did no
- Page 39: Analysis / conclusions After unders
- Page 43: Si bien la naturaleza y los seres q
- Page 47: pentagonal formations. The former a
- Page 51: Masaru Emoto and the memory of wate
geometry La geometría<br />
<strong>of</strong> del <strong>water</strong> agua
The geometry <strong>of</strong> <strong>water</strong><br />
Laura Zamudio Robles<br />
Degree project<br />
2020-1
index<br />
INTRODUCTION<br />
1. Introduction<br />
2. Why fractals?<br />
3. Understanding fractals<br />
a. The father <strong>of</strong> fractals<br />
b. What are the fractals?<br />
c. Supporting theories<br />
d. Types <strong>of</strong> fractals<br />
4. Application <strong>of</strong> fractals<br />
a. Fractals in medicine<br />
b. Fractals in art<br />
c. Fractals in nature<br />
d. Analysis / conclusions<br />
5. Searching <strong>of</strong> the fractal for the proyect<br />
a. Water as a research path<br />
b. Dr. Mu Shik Jhon and the geomety <strong>of</strong> <strong>water</strong><br />
c. Cluster - the <strong>water</strong> alpabeth<br />
d. Masaru Emoto and the memory <strong>water</strong><br />
e. Analysis / conclusions<br />
4<br />
5<br />
7<br />
27<br />
39<br />
6. Value proposal<br />
a. Initial approach<br />
b. Moodboards analysis<br />
c. Moodboards and references<br />
d. Geometries<br />
e. Analysis for design<br />
7. Final pieces<br />
a. Prototypes<br />
b. Design proposal<br />
c. Pieces<br />
d. Brand<br />
8. Bibliography<br />
55<br />
83<br />
121<br />
This paper reports the conceptual and material<br />
development <strong>of</strong> a design project inspired by<br />
the fractals <strong>of</strong> <strong>water</strong>. The project begins with<br />
a research on fractals, in order to understand<br />
them from their mathematical approach and<br />
their representations in different fields <strong>of</strong><br />
knowledge. From this research, two main<br />
categories <strong>of</strong> fractal classification are<br />
determined: mathematical and natural fractals.<br />
For the development <strong>of</strong> the project, natural<br />
fractals are taken as the object <strong>of</strong> research and<br />
it is focused on the study <strong>of</strong> <strong>water</strong> as an element<br />
analyzed from fractality. From this, Dr. Mu Shik<br />
Jhon’s research on the molecular structure <strong>of</strong><br />
<strong>water</strong> is used, which results in the formation <strong>of</strong><br />
geometric structures called clusters.<br />
From these structures, the development <strong>of</strong> the<br />
proposal begins, in which the product design<br />
seeks to represent the geometry generated<br />
by the cluster in the <strong>water</strong>, making use <strong>of</strong> four<br />
principles that arise from research on fractals<br />
and <strong>water</strong> and that guide the development <strong>of</strong><br />
prototypes and final pieces. These principles<br />
are self-similarity, proportionality, dynamic<br />
systems that reflect the vibrations that modify<br />
the structure <strong>of</strong> <strong>water</strong> and the cluster as its<br />
constitutive pattern.<br />
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why fractals?<br />
Nature, architecture and art are just some<br />
<strong>of</strong> the multiple contexts in which fractals<br />
can be found and although we <strong>of</strong>ten do<br />
not recognize them as such, they manifest<br />
themselves in organized forms that produce<br />
an inexplicable beauty. In my case, these forms<br />
have attracted my attention since I was very<br />
young and involuntarily I have approached<br />
them throughout my life through drawings<br />
<strong>of</strong> mandalas, observing the rosettes or the<br />
characteristic forms <strong>of</strong> Islamic art, and more<br />
recently in the contemplation <strong>of</strong> the forms <strong>of</strong><br />
nature. I do not remember the moment when I<br />
discovered that all these forms were fractal<br />
representations, but my interest in these forms<br />
has remained and has been complemented and<br />
developed throughout my career as a designer,<br />
so I decided to go into this research for the<br />
development <strong>of</strong> my degree project.<br />
Photo recovered from : https://flic.kr/p/9AbWTS<br />
6
Entendiendo<br />
understanding<br />
los fracals fractales<br />
Foto recuperada de: https://flic.kr/p/pvgfEQ
The theory <strong>of</strong> fractals is relatively recent, if<br />
compared to other scientific theories that have<br />
generated important changes in the way <strong>of</strong><br />
understanding the world. Fractal geometry has<br />
become the tool for studying irregular shapes,<br />
especially the apparently random shapes <strong>of</strong><br />
nature. Due to the change that fractals represent<br />
for the study <strong>of</strong> the world, it has awakened a great<br />
interest in studying them from different branches<br />
<strong>of</strong> knowledge, not only from the sciences.<br />
the father <strong>of</strong> fractals<br />
The first to use the term “fractal” was the<br />
Polish mathematician Benoit Mandelbrot.<br />
However, before understanding fractals, it<br />
is important to know a little more about his<br />
life and the questions that led him to the<br />
development <strong>of</strong> the fractal theory and for<br />
which he is called the father <strong>of</strong> fractals, even<br />
though he was not the first to talk about them.<br />
Benoit Mandelbrot was born in Warsaw in 1924<br />
into a Jewish family, which forced him to migrate<br />
to France during the First World War and to go into<br />
Picture 1. Benoit Mandelbrot<br />
Photo recovered from : https://flic.kr/p/5Z5v4U<br />
hiding during the Second World War. Mandelbrot<br />
migrated with his family in 1936 to France where his<br />
uncle took charge <strong>of</strong> his education. Subsequently,<br />
he entered Paris to study, but had to withdraw due<br />
to the outbreak <strong>of</strong> the Second World War, which<br />
motivated him to become self-taught. Due to his<br />
unconventional education, he develops a great<br />
interest in geometry and the observation <strong>of</strong> nature<br />
(Sanz, 2019).<br />
For Mandelbrot, nature was a complex system that<br />
could not be understood from the traditional laws<br />
Photo recovered from : https://unsplash.com/photos/9x-7p0fvKRM<br />
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<strong>of</strong> Euclidean geometry. In nature there were<br />
irregular and chaotic forms, which apparently<br />
had no order or way to be understood. However,<br />
for Mandelbrot there were patterns, so he<br />
dedicated his studies to analyze the irregular<br />
forms <strong>of</strong> nature and to find a mathematical<br />
formula that could explain what he observed.<br />
During his observations he was able to<br />
determine that shapes in nature such as<br />
clouds, coastlines, rivers, trees and so on,<br />
had self-similar shapes, meaning that when<br />
looking at a small portion <strong>of</strong> the shapes,<br />
it is similar to the whole <strong>of</strong> what is being<br />
observed. An example <strong>of</strong> this is tree branches,<br />
which when taking a single one gives the<br />
impression <strong>of</strong> looking at the tree on a smaller<br />
scale, as seen in image 2 (Ventura, 2019).<br />
This principle <strong>of</strong> self-similarity would<br />
allow Mandelbrot to establish that his<br />
observations corresponded to a new<br />
type <strong>of</strong> geometry, which explained the<br />
irregular and chaotic forms <strong>of</strong> nature.<br />
The first studies on this new geometry were<br />
presented in his first article published in<br />
Science magazine in 1967, entitled “How long is<br />
the coast <strong>of</strong> Great Britain?” (Gaussianos, 2010).<br />
In this article he posed the situation in which<br />
someone would want to measure the coast <strong>of</strong><br />
Great Britain, in which case he would encounter<br />
the problem that, being an irregular surface, the<br />
final measurement will depend on the unit <strong>of</strong><br />
measurement used, so that the forms <strong>of</strong> nature<br />
cannot be studied from classical mathematics.<br />
Image 2. Branching <strong>of</strong> a tree showing its fractal formation.<br />
Photo recovered from : https://flic.kr/p/dNY6uW<br />
Photo recovered from: https://flic.kr/p/4qmmzf<br />
Image 3. Simulation <strong>of</strong> the Mandelbrot ensemble<br />
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what are fractals?<br />
From the above and in the same article<br />
Mandelbrot uses for the first time the term<br />
fractal, which comes from the Latin Fractus<br />
and refers to the self-similar property that<br />
these possess, i.e. it is a whole that is formed<br />
<strong>of</strong> several similar parts at different scales. It<br />
can also be understood from the example <strong>of</strong><br />
the measurement <strong>of</strong> the coast, in which it is<br />
stated that while normally the measures used<br />
are whole numbers such as 1 or 2, for the coast<br />
<strong>of</strong> Great Britain it will be more or less 1.25.<br />
Finally, it is necessary to mention that<br />
Mandelbrot relies on previous studies that<br />
dealt with the subject, such as the studies<br />
carried out by the French mathematician<br />
Gaston Julia. However, none <strong>of</strong> them<br />
was able to make a demonstration that<br />
would support the approaches <strong>of</strong> fractal<br />
mathematics. Mandelbrot was able to<br />
make the first simulation <strong>of</strong> fractal theory,<br />
because he worked for IBM and there he had<br />
total freedom to develop his theory and the<br />
technological tools (computers) from which<br />
he was able to arrive at the equation Zn=Z2+C,<br />
with which he generated the “Mandelbrot<br />
Set” (see image 3). Because it was he who<br />
gave it its name and managed to make the<br />
first simulation <strong>of</strong> a fractal, he is known as<br />
the father <strong>of</strong> fractals and is credited with<br />
the discovery <strong>of</strong> a new geometry capable<br />
<strong>of</strong> mathematically explaining nature.<br />
As already mentioned, Mandelbrot was not<br />
the first to study the irregularity <strong>of</strong> nature,<br />
but by proposing fractal geometry he opened<br />
the door for new researchers from different<br />
branches <strong>of</strong> knowledge to become interested<br />
in studying and understanding this new world<br />
<strong>of</strong> mathematics. Therefore, one would think<br />
that the first thing necessary to enter into such<br />
research would be to ask oneself what are<br />
fractals, which is paradoxical is that there is no<br />
precise definition, not even Benoit Mandelbrot<br />
managed to propose a satisfactory definition.<br />
However, it is possible to reach an<br />
understanding <strong>of</strong> them by understanding the<br />
characteristics that they possess and that<br />
differentiate them from traditional geometry.<br />
To begin with, fractals are geometric figures<br />
fractioned in different scales and self-similar,<br />
this means that when observing a fractal and<br />
regardless <strong>of</strong> the scale in which it is seen, it<br />
will always be similar since it is composed <strong>of</strong><br />
several copies similar to the original figure,<br />
but in smaller scales. This property is <strong>of</strong> great<br />
importance, since in nature there are no<br />
perfect shapes, understood from Euclidean<br />
geometry, since being shapes composed <strong>of</strong><br />
other smaller ones generates the irregularity <strong>of</strong><br />
nature (What are fractals? - Silicon News, n.d.).<br />
Another characteristic found in fractals is<br />
that they have finite areas but their length<br />
or perimeter is infinite, to better understand<br />
this property we will take as an example the<br />
article about the coast <strong>of</strong> Great Britain. In<br />
this example we can say that Great Britain<br />
has a defined land area, however, when trying<br />
to measure the perimeter <strong>of</strong> the coast, the<br />
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esulting measurement will be different<br />
depending on the unit <strong>of</strong> measurement used<br />
given the irregularity <strong>of</strong> the surface, therefore,<br />
it is assumed that by taking a smaller scale<br />
the measurement will increase, but you can<br />
always take a smaller scale making the<br />
length increase to infinity (see image 4).<br />
Another way in which fractals can be understood<br />
is from the mathematical principles that<br />
support them.<br />
“In mathematical terms a fractal is a shape that<br />
begins with an object that is constantly altered<br />
by infinite application <strong>of</strong> a certain rule. This<br />
can be described by means <strong>of</strong> a mathematical<br />
formula or by means <strong>of</strong> words.” (Cruz Gomez,<br />
Perez Abad and Gomez Garcia, n.d.)<br />
The previous quote can be understood from<br />
the Mandelbrot Fractal (image 3), which starts<br />
from the complex equation Zn=Z2+C, where C<br />
corresponds to the points <strong>of</strong> the plane that are<br />
part <strong>of</strong> the set from the sequence 0, f(0), f(f(f(0)),<br />
f(f(f(f(0))), ... taking into account that this<br />
sequence must be performed for each <strong>of</strong> the<br />
points <strong>of</strong> the plane and thus determine whether<br />
or not they belong to the set. It is because <strong>of</strong> the<br />
latter that these equations are called dynamical<br />
systems, which is also the reason why before<br />
Mandelbrot and more specifically before having<br />
access to computers it was impossible to<br />
perform a simulation that would prove fractal<br />
geometry from a mathematical explanation<br />
(What is the Mandelbrot fractal?, 2010).<br />
It is important to emphasize that the<br />
contribution <strong>of</strong> computers corresponds to their<br />
Photo recovered from: https://vonneumannmachine.files.wordpress.com/2012/11/costagranbretac3b1a.jpg<br />
Simulation <strong>of</strong> the measurement <strong>of</strong> the coast <strong>of</strong> Great Britain with different scales.<br />
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ability to perform and process a large number <strong>of</strong><br />
equations, which allows generating figures such<br />
as fractals that can be observed at different<br />
scales, making these forms infinite. Given the<br />
infinite characteristic <strong>of</strong> fractals, it can be said<br />
that they are figures in movement, in constant<br />
development and that they are observed at a<br />
specific time and scale but not in their totality.<br />
Although the mathematical explanation <strong>of</strong><br />
fractals is complex to understand, there are<br />
examples such as the Koch curve (image 5),<br />
Sierpinski’s triangle (image 6), which show<br />
the principle <strong>of</strong> self-similarity from Euclidean<br />
geometric figures, as for example in the<br />
Sierpinski triangle, in which we start from an<br />
initial equilateral triangle and create three new<br />
triangles from the corners <strong>of</strong> the first one, this<br />
process can be repeated infinitely maintaining<br />
always the principles <strong>of</strong> self-similarity and<br />
infinity <strong>of</strong> the fractal. In the case <strong>of</strong> the Koch<br />
curve it can be observed how the total figure<br />
generates a shape similar to that <strong>of</strong> snowflakes,<br />
which allows us to understand the relationship<br />
between fractal geometry and the shapes <strong>of</strong><br />
nature (Poizat, Sauter and Spodarev, 2014).<br />
It should be clarified that the review <strong>of</strong> the<br />
mathematical principles <strong>of</strong> fractal theory is<br />
part <strong>of</strong> the research, as an element to reach a<br />
theoretical understanding <strong>of</strong> fractals and their<br />
characteristics, but the formulas and theories<br />
exposed above will not be taken as guidelines for<br />
the development <strong>of</strong> the project.<br />
Photo recovered from: http://images.treccani.it/enc/media/share/images/orig/<br />
system/galleries/Enciclopedia_della_Matematica/fig_lettk_00530_001.jpg<br />
Image 5. Representation <strong>of</strong> Koch’s curve<br />
Photo recovered from: http://scpdptomatematicas.blogspot.<br />
com/2017/11/construccion-de-nuestro-fractal.html<br />
Image 6. Representation <strong>of</strong> the Sierpimski triangle.<br />
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supporting theories<br />
As mentioned above, Benoit Mandelbrot<br />
studied and complemented his studies for<br />
the approach <strong>of</strong> fractal geometry, based<br />
on several theories previously proposed,<br />
some <strong>of</strong> these approaches were close to<br />
fractal theory directly and others were not.<br />
The first theory <strong>of</strong> which it is necessary to<br />
speak is the Julia set, proposed by the French<br />
mathematician Gaston Julia. This theory<br />
corresponds in general terms to the theory<br />
proposed by Mandelbrot, since the latter based<br />
his investigations on those carried out by Julia,<br />
with the difference that in this theory the term<br />
C <strong>of</strong> the equation corresponds to a complex<br />
number, which may or may not be part <strong>of</strong> the set,<br />
making the required calculations even greater.<br />
Another difference, and perhaps the most<br />
important, is that Gaston Julia was never able<br />
to see a simulation <strong>of</strong> a fractal, so his theory<br />
was left aside in mathematical research and<br />
only taken up again by Benoit Mandelbrot.<br />
Another <strong>of</strong> the theories that greatly<br />
influenced the approach <strong>of</strong> fractal geometry<br />
was what is known as chaos theory or the<br />
butterfly effect, since, as mentioned, fractals<br />
are the geometry <strong>of</strong> the chaotic forms <strong>of</strong><br />
nature. Chaos theory was proposed by the<br />
meteorologist Edward Lorenz in 1960, while he<br />
was conducting mathematical experiments<br />
to predict the weather, using 12 equations.<br />
According to the knowledge <strong>of</strong> the time, it was<br />
believed that the results obtained from the<br />
same origin (a number with at least 2 known<br />
decimal places) would result in a pattern<br />
similar to the previous one, however, being<br />
dynamic systems, even the smallest change<br />
in the beginning would generate a change in<br />
the subsequent development. To explain the<br />
above, we take as an example the movement <strong>of</strong><br />
a butterfly flapping its wings, which represents<br />
a small disturbance in the air but can generate<br />
a tsunami on the other side <strong>of</strong> the world.<br />
From this discovery Lorenz focused his studies<br />
on the development <strong>of</strong> 3 equations that were<br />
derived from the 12 previously mentioned,<br />
but that had the same effect, thus arriving at<br />
the equations that would explain the chaos<br />
theory and that later would be proven to<br />
describe the movement <strong>of</strong> a <strong>water</strong> whirlpool<br />
(image 7), being a fractal form <strong>of</strong> nature (Cruz<br />
Gómez, Pérez Abad and Gómez García, s.f.).<br />
One <strong>of</strong> the main characteristics <strong>of</strong> fractals is<br />
their self-similarity, however, it is worth noting<br />
Photo recovered from: https://www.elcompositorhabla.com/corps/<br />
elcompositorhabla/data/resources/image/Ruth/Lorentz%20Attractor1.png<br />
Image 7. Simulation <strong>of</strong> the Lorenz equations<br />
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that each <strong>of</strong> the parts that compose it complies<br />
with the ideal or harmonic proportions. This idea<br />
<strong>of</strong> harmonic proportions comes from Pythagoras’<br />
theory <strong>of</strong> numbers and proportions, so it can be<br />
thought that there is a relationship between<br />
Pythagorean theories and fractal theory.<br />
For the fractal theory the proportions are <strong>of</strong><br />
great importance and are appreciated between<br />
the different elements that compose a fractal,<br />
so a relationship with the theory <strong>of</strong> numbers<br />
and proportions <strong>of</strong> Pythagoras is appreciated.<br />
This theory proposes a relationship between<br />
music and mathematics, since musical tones<br />
and intervals can be expressed in numerical<br />
ratios, being the comparison <strong>of</strong> one quantity<br />
with another, making the intervals concordant<br />
or discordant (harmonic or disharmonic) with<br />
each other.<br />
To better understand the relationship between<br />
music and mathematics, an instrument called<br />
Monochord (image 8) was used, which consists<br />
<strong>of</strong> a single string supported on a wooden<br />
base (like a guitar), which is stretched until<br />
it reaches a fundamental sound determined<br />
as a tone. Subsequently, it was divided<br />
into 12 equal parts, in such a way that the<br />
proportion between the string fragments<br />
was maintained, producing different sounds<br />
depending on where the string was stepped<br />
on. Using this instrument, it was possible to<br />
define the octave, the fourth and the fifth,<br />
which are the concordant sounds. From these<br />
sounds the general property <strong>of</strong> the harmonic<br />
arithmetic mean ab=mh was generated, where<br />
there is a proportionality between ab equal<br />
to that between mh. (Correa Pabón, 2006).<br />
Proportions and harmony are fundamental<br />
factors in the world <strong>of</strong> fractals, since they are<br />
intrinsic characteristics <strong>of</strong> these, making them<br />
beautiful to the eye and being the manifestation<br />
<strong>of</strong> an order and a reason for being in the apparent<br />
chaos <strong>of</strong> nature. It is worth noting that these<br />
harmonic patterns <strong>of</strong> fractals and their beauty<br />
are one <strong>of</strong> the main reasons why they arouse so<br />
much interest in different disciplines.<br />
Photo recovered from: https://docplayer.com.br/<br />
docs-images/70/63756304/images/166-0.jpg<br />
Image 8. Pythagorean Monochord<br />
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types <strong>of</strong> fractals<br />
Photo recovered from:: https://i2.wp.com/upload.wikimedia.<br />
org/wikipedia/commons/4/4b/Fractal_fern_explained.png<br />
Image 9. Accurate self-similarity<br />
Since fractal theory is very recent and under<br />
constant study, it is not completely defined<br />
and classified, so different forms and<br />
parameters have been proposed to classify<br />
the known fractals, highlighting mainly their<br />
properties <strong>of</strong> self-similarity and linearity.<br />
Regarding the classification by self-similarity,<br />
fractals are divided into three groups; the<br />
first are those with exact self-similarity<br />
(image 9), which means that they appear<br />
identical in their different scales, these<br />
are fractals generated by iterated function<br />
systems, which is a program specialized<br />
in the generation <strong>of</strong> fractals. The second<br />
group corresponds to fractals with quasisimilarity<br />
(image 10), which means that in<br />
their different scales they are spread out but<br />
not identical. Finally, there are fractals with<br />
statistical self-similarity (image 11), which<br />
Photo recovered from: https: https://img.<br />
culturacolectiva.com/content/2013/03/fractales.jpg<br />
Image 10. Quasi-likelihood<br />
Photo recovered from: https://flic.kr/p/75Rior<br />
is the weakest because the only condition<br />
is that it complies with the numerical or<br />
statistical measures in its different scales.<br />
For the classification by linearity the division<br />
is given in two groups; the first are the linear<br />
fractals (image 12) that are constructed<br />
from a change in the variation <strong>of</strong> its scales,<br />
but without losing the equality in all its<br />
scales. On the other hand, the nonlinear<br />
fractals (image 13) are created from<br />
complex distortions generating a similar<br />
but not identical structure for the different<br />
scales <strong>of</strong> the fractal (Fractal Classes, 2015).<br />
Finally, we can speak <strong>of</strong> a third classification<br />
that corresponds to the mathematical<br />
fractals or fractals created on computer<br />
(image 14) and natural fractals (image 15).<br />
Photo recovered from: https://flic.kr/p/7LF3Wb<br />
Image 12. Linear fractal<br />
Photo recovered from: https://flic.kr/p/5sd2XB<br />
Image 13. Non-linear fractal<br />
Statistical self-similarity<br />
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Photo recovered from: https://flic.kr/p/5bMnJa<br />
Image 14. Mathematical fractal<br />
Mathematical fractals are called fractal<br />
sets and are defined from mathematical<br />
formulations and simulated in computers,<br />
some examples are the Mandelbrot set, or the<br />
Koch curve. On the other hand, natural fractals<br />
are those found in nature so they are not so<br />
precise in their different scales, however,<br />
these are the result <strong>of</strong> evolution processes<br />
making them more complex (Gómez Cumaco,<br />
2009). Additionally, natural fractals do not<br />
have infinite scales, but they do comply with<br />
the self-similar characteristic.<br />
Photo recovered from: https://i.pinimg.com/<br />
originals/95/28/3f/95283fdc7c216f0fc7e90bc5bade968c.jpg<br />
Image 15. Natural fractal - Aloe polyphylla<br />
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aplicaciones applicationde<br />
<strong>of</strong> fractals<br />
los fractales<br />
Foto recuperada de: https://flic.kr/p/25PSiWK
Fractals in medicine<br />
Fractals, being a relatively new theory, but<br />
mainly for having the ability to explain a large<br />
number <strong>of</strong> aspects <strong>of</strong> nature, which so far<br />
seemed not to follow any kind <strong>of</strong> principle,<br />
have aroused an interest and fascination<br />
for different branches <strong>of</strong> knowledge. In<br />
the following, we will present some <strong>of</strong> the<br />
applications that have been given to fractals,<br />
which in my opinion are contrasting because<br />
they are seen from apparently very different<br />
disciplines, but which allow us to see the<br />
versatility <strong>of</strong> fractal theory.<br />
In the human body a great variety <strong>of</strong> fractal<br />
structures can be found, so the fractal<br />
theory is very useful in the field <strong>of</strong> medicine<br />
especially to understand the functioning<br />
<strong>of</strong> these systems, some examples are the<br />
nervous system (image 16), circulatory<br />
or pulmonary system. In the case <strong>of</strong> the<br />
circulatory system, fractal formation is what<br />
ensures that the blood pressure is constant<br />
throughout the body, so that all cells can<br />
receive the necessary supplies carried by<br />
the blood. Another case is the prediction <strong>of</strong><br />
diseases such as osteoporosis from the first<br />
changes in the bones. (Cruz Gomez, Perez<br />
Abad and Gomez Garcia, n.d.)<br />
Image 16. Nervous system<br />
Photo recovered from: https://1.bp.blogspot.com/-pDeP9-LYbeQ/Vm2BXU8fRHI/<br />
AAAAAAAAUiY/Rx46lYJsD6E/s1600/Sistema-nervioso-neuronas-biologia.jpg<br />
30 31
Fractals in art<br />
Fractals arouse great interest and attraction<br />
for those who see them, due to the beauty<br />
produced by the creation <strong>of</strong> harmonic and<br />
proportional patterns. The first fractals<br />
proposed in the world <strong>of</strong> art emerged since<br />
the appearance <strong>of</strong> fractal theory in 1980 from<br />
different computer experiments, however,<br />
these were initially very similar to each<br />
other, so they were not considered pieces<br />
<strong>of</strong> art. As the study in programming and the<br />
advances in computers became possible to<br />
create new fractal patterns by experimenting<br />
with new mathematical formulas or by<br />
making variations in the existing ones.<br />
From 1995 onwards, the development <strong>of</strong> fractal<br />
art began to focus on color patterns, since<br />
the experimentations based on mathematical<br />
formulas had been almost entirely developed.<br />
The experimentations from color consisted <strong>of</strong><br />
developing algorithms that allowed coloring<br />
the different points <strong>of</strong> the plane in different<br />
ways, depending on given conditions, such as<br />
whether or not they belonged to the fractal<br />
set, or the oldest one which is by escape<br />
time, which consists <strong>of</strong> determining whether<br />
the point <strong>of</strong> the plane tends to infinity or not,<br />
when subjected to the fractal algorithm. The<br />
most important thing in the development<br />
<strong>of</strong> color algorithms is that it allows to<br />
generate very different compositions from<br />
the same fractal base. (Calonge, 2013).<br />
One <strong>of</strong> the main discussions about fractal art<br />
has to do with whether the images generated<br />
by computers can be considered works <strong>of</strong> art<br />
or not and where is the creativity <strong>of</strong> the artist<br />
during the simulation <strong>of</strong> an algorithm. Faced<br />
with these questions, some fractal artists such<br />
as Jean-Paul Agosti (image 17) or Susan Conde<br />
decided to dedicate themselves solely to<br />
making works based on their manual skills with<br />
painting, highlighting the fractality <strong>of</strong> nature.<br />
However, it should be noted that the use <strong>of</strong><br />
fractal patterns as inspiration for art has<br />
been known long before fractal art. Cultures<br />
such as the Arabian, used patterns <strong>of</strong> nature,<br />
to make the paintings <strong>of</strong> their mosques, as<br />
seen in the Asfahan Mosque or in the Indian<br />
culture with the development <strong>of</strong> mandalas<br />
known as Kolam (Poizat, Sauter and Spodarev,<br />
2014). Another example that has been found<br />
between fractal theory and art is the work <strong>of</strong><br />
the painter Jackson Pollock, who belonged to<br />
abstract expressionism and who developed the<br />
techniques <strong>of</strong> Action-painting and Dripping. It<br />
Photo recovered from: https://biblioteca.acropolis.org/wpcontent/uploads/2014/10/Jean_Paul_Agosti-Harmonie.jpg<br />
Image 17. Jean-Paul Agosti, Garden <strong>of</strong> Metamorphosis.<br />
32 33
should be noted that Pollock did not create<br />
his works thinking in the fractality <strong>of</strong> nature,<br />
but inspired by the rhythms <strong>of</strong> nature and it<br />
was the physicist Richard Taylor, who during<br />
an artistic experiment in the Yorkshire Park,<br />
in which they mounted a structure and let<br />
the paint spread without any control during<br />
a storm, obtained as a result a canvas that<br />
reminded him <strong>of</strong> Pollock’s work. After the<br />
experiment Taylor decided to investigate<br />
Pollock’s work and found that his painting<br />
techniques consisted <strong>of</strong> spilling paint and<br />
other materials using his whole body, while<br />
following the rhythms <strong>of</strong> nature or the chants<br />
<strong>of</strong> the Navajo Indians. During the analysis <strong>of</strong><br />
the works Taylor used the technique <strong>of</strong> Boxcounting<br />
in which he divides the work into<br />
boxes and to study each part until completing<br />
the work, with this technique he discovered<br />
the fractal character that Pollock’s works<br />
have (image 18) and that was perfected<br />
throughout his work. From this discovery it<br />
was found that the rhythms <strong>of</strong> nature, being<br />
chaotic, are fractal. (Paissan, 2015).<br />
Imagen 18. Jacson Pollock, Number 1<br />
Photo recovered from: https://flic.kr/p/otURgt<br />
34 35
Fractals in nature<br />
Although it has been said from the beginning<br />
that the fractal theory comes from observation<br />
and the desire to understand nature, the most<br />
explored examples are usually mathematical<br />
and those made on computer, so I consider it<br />
important to explore the fractals that exist in<br />
nature and that as Pollock said are the result<br />
<strong>of</strong> life itself. Since the beginning <strong>of</strong> humanity,<br />
nature has 1 always been attributed a divine<br />
character, given its capacity to surprise by<br />
being chaotic and incomprehensible on many<br />
occasions, this divine character <strong>of</strong> nature is due<br />
to the harmonic and proportional forms that<br />
are created and which Mandelbrot describes as<br />
fractals.<br />
1. Throughout human history, within diverse cultures around the world,<br />
one <strong>of</strong> the main characteristics attributed to divinity is aesthetics. With<br />
geometry, rhythm and chromatics as three <strong>of</strong> the favorite resources <strong>of</strong><br />
this divine discourse, nature reaches the most spectacular and at the<br />
same time the most discreet divine manifestation as a hyper-aesthetic<br />
paradox”, (Villar, J., 2010).<br />
Nature is characterized by evolving and<br />
developing while maintaining order, harmony<br />
and balance among all beings that exist and in<br />
themselves, so harmony is a feature that stands<br />
out. One <strong>of</strong> the differences that exist between<br />
natural and digital fractals has to do with the<br />
infinite characteristic, since in nature they do<br />
not have this property and generally they come<br />
to present a specific amount <strong>of</strong> self-similar<br />
scales. However, natural fractals retain the<br />
property <strong>of</strong> being changing or dynamic systems,<br />
as they are constantly evolving which in turn<br />
makes them more complex, from this principle<br />
Mandelbrot presents two fundamental variables<br />
for their understanding; irregularity at the level<br />
<strong>of</strong> form and pattern at the level <strong>of</strong> rhythm.<br />
(Villar, 2010).<br />
Finalmente, cabe resaltar que los fractales<br />
se encuentran en una gran cantidad de<br />
fenómenos naturales tales como, hongos,<br />
plantas, truenos e incluso en el universo, lo<br />
que nos indica que desde el elemento más<br />
pequeño hasta lo más grande se rige por los<br />
principios fractales.<br />
Photo recovered from: https://flic.kr/p/aUFgL<br />
36 37
Analysis / conclusions<br />
After understanding the fractal theory and the<br />
different examples, both from simulations,<br />
as well as in art and in nature in general, it is<br />
possible to determine that fractals are divided<br />
into two large groups, mathematical and<br />
natural, for the development <strong>of</strong> this project I<br />
will take as a point <strong>of</strong> interest the natural ones.<br />
Fractals are dynamic systems with<br />
figures in constant motion, which can be<br />
observed in time. In the case <strong>of</strong> natural<br />
fractals we can determine that they are<br />
systems with two main characteristics,<br />
proportionality understood as irregular<br />
patterns that determine the shape <strong>of</strong> the<br />
fractal, and self-similarity, which generates<br />
rhythms, resulting in fractal composition.<br />
Proportionality is a characteristic that<br />
is understood from the irregularity that<br />
characterizes nature and that is governed<br />
under the chaos theory, or more specifically<br />
it is called chaos geometry. This theory<br />
states that a change, no matter how<br />
small, in a dynamic system will generate<br />
a major reaction in the development <strong>of</strong><br />
the system, due to the fact that these<br />
systems are highly sensitive to change and<br />
are in constant development. In addition,<br />
it should be noted that these variations<br />
preserve the proportional relationships<br />
between all the elements <strong>of</strong> the system,<br />
since the changes occur progressively.<br />
In the case <strong>of</strong> natural fractals, changes come<br />
from the context that surrounds them and<br />
from the fortuitous events that affect them.<br />
An example <strong>of</strong> this can be the growth <strong>of</strong> a<br />
tree branch that, when exposed to a great<br />
force produced by the wind, can bend or even<br />
break and give rise to a new branch, which<br />
integrates harmoniously with the whole.<br />
On the other hand, self-similarity is the<br />
main characteristic <strong>of</strong> fractals and is<br />
directly related to the chaotic development<br />
<strong>of</strong> dynamic systems, since, as I mentioned<br />
before, changes in the system occur<br />
progressively, which indicates that there are<br />
rhythms in the development <strong>of</strong> the system<br />
that give rise to fractal compositions. These<br />
compositions are the result <strong>of</strong> the union<br />
<strong>of</strong> multiple parts similar to each other, but<br />
which are in different scales, or in different<br />
degrees <strong>of</strong> development, taking into account<br />
that fractals are observed in a certain time<br />
and not in their totality.<br />
38 39
Searching <strong>of</strong> the fractal<br />
En busca del fractal<br />
for the proyect<br />
para el proyecto<br />
Foto recuperada de: https://flic.kr/p/aasMrK
Si bien la naturaleza y los seres que<br />
pertenecemos a ella somos sistemas<br />
dinámicos y en constante cambio, estos<br />
tienen un ritmo o una secuencia en la que<br />
se van dando. Estos ritmos son diferentes y<br />
esenciales para el desarrollo de cada sistema,<br />
son los responsables de que la naturaleza<br />
fluya y en el caso de los fractales, que tengan<br />
un orden interno perceptible y repetible, a<br />
esto se le conoce como ritmo dinámico. (“El<br />
Ritmo de la naturaleza en nosotros – Mindalia<br />
Noticias y Artículos”, s.f.)<br />
leaves following the theory <strong>of</strong> the golden<br />
ratio; the roots <strong>of</strong> some plants such as<br />
mangroves or the Guacari tree, which have<br />
adapted to their contexts by making their<br />
roots grow out <strong>of</strong> the ground, evidencing their<br />
fractal growth present both in the branches<br />
and in the roots and among themselves; and<br />
the third case corresponds to the movements<br />
or forms generated by <strong>water</strong> in nature, rivers,<br />
<strong>water</strong>falls, sea, etc. , in which their fractal<br />
character is not always completely evident,<br />
but which when observed in detail reveal a<br />
large number <strong>of</strong> fractal patterns.<br />
Taking into account the characteristics<br />
defined by proportionality and self-similarity<br />
in fractals, select three possible avenues<br />
<strong>of</strong> investigation; plants such as succulents,<br />
which have a fractal arrangement <strong>of</strong> their<br />
Photo recovered from: https://flic.kr/p/a3fD1E<br />
42 43
<strong>water</strong> as a research path<br />
Dr. Mu Shik Jhon and the geometry <strong>of</strong> <strong>water</strong><br />
As previously mentioned, natural fractals are<br />
dynamic systems, which must be understood<br />
from their capacity to flow and evolve. For<br />
this reason I decided to investigate the<br />
movements <strong>of</strong> <strong>water</strong>, since it is an element<br />
where these characteristics are evident<br />
and constant. However, this makes it appear<br />
to be a stable system when viewed on a<br />
large scale, which only shows its fractal<br />
character when observed on smaller scales,<br />
so I find it intriguing to delve into its different<br />
scales both from the contexts in which<br />
it is found, as well as from its states and<br />
the characteristics it has in each <strong>of</strong> them.<br />
Additionally, during the investigation <strong>of</strong> the<br />
three possible paths I found that <strong>water</strong>,<br />
although it is mentioned on several occasions<br />
as an element where the principles <strong>of</strong><br />
fractality can be observed, has not been<br />
studied or at least there is not as much<br />
information as in the case <strong>of</strong> plants. Due to the<br />
lack <strong>of</strong> information that I could find, I found it<br />
intriguing to go into the investigation <strong>of</strong> <strong>water</strong><br />
to understand it under the fractal principles.<br />
In the investigation <strong>of</strong> the fractal character<br />
<strong>of</strong> <strong>water</strong> I found two scientists who have<br />
focused on the study <strong>of</strong> this element,<br />
finding two points <strong>of</strong> view from which <strong>water</strong><br />
is studied at different scales and states,<br />
but resulting in a fractal understanding <strong>of</strong><br />
<strong>water</strong> from its molecular composition to its<br />
different manifestations in nature.<br />
Water is an element that can be in three<br />
states, solid, liquid and gaseous, so it can be<br />
transformed and become part <strong>of</strong> everything<br />
that exists in the world from living beings<br />
to the atmosphere, but more importantly<br />
it is constantly transforming to transport<br />
energy. It is precisely these changes and<br />
the characteristics it has in each <strong>of</strong> its<br />
states, which arouses the interest <strong>of</strong> the<br />
Korean chemist and physicist Mu Shik Jhon,<br />
who states that <strong>water</strong> can have a hexagonal<br />
or pentagonal molecular structure, with<br />
important differences between the two.<br />
Water is the fundamental element for the<br />
life <strong>of</strong> all living beings, it is responsible for<br />
transporting vital energy, however, to fulfill<br />
this function it must have a good quality. The<br />
good quality <strong>of</strong> <strong>water</strong> is due to its molecular<br />
structure, which is explained by Mu Shik Jhon.<br />
Dr. Mu Shik Jhon states that <strong>water</strong> needs<br />
to move and flow to be charged with<br />
energy, this conclusion is obtained from<br />
studying <strong>water</strong> in its natural state, where<br />
movements persist in the form <strong>of</strong> vortices,<br />
helicoids, whirlpools and in general chaotic<br />
movements <strong>of</strong> <strong>water</strong>. Mu Shik Jhon explains<br />
that these movements are fundamental for<br />
the purification <strong>of</strong> <strong>water</strong>, since they allow<br />
<strong>water</strong> molecules (H2O) to attract other<br />
disorganized <strong>water</strong> molecules and structure<br />
them into organized molecules with a<br />
large amount <strong>of</strong> trapped energy. From the<br />
crystalline <strong>water</strong> approach and its relation<br />
to the structural order <strong>of</strong> <strong>water</strong> molecules<br />
Mu Shik Jhon proposes that in nature one<br />
can find molecules with hexagonal or<br />
44 45
pentagonal formations. The former are those<br />
found in natural, crystalline <strong>water</strong>, especially<br />
in what is known as super-frozen <strong>water</strong> such<br />
as that <strong>of</strong> glaciers. This structure provides the<br />
greatest benefits for living organisms, since<br />
it generates strong bonds that allow large<br />
amounts <strong>of</strong> energy to be contained in addition<br />
to containing more oxygen atoms. In terms <strong>of</strong><br />
its behavior, this structure allows <strong>water</strong> to<br />
flow naturally and facilitates the formation<br />
<strong>of</strong> harmonic crystals; these hexagonal<br />
formations are known as clusters (image 21).<br />
On the other hand, pentagonal structures are<br />
commonly found in <strong>water</strong> when it has been<br />
exposed to ions <strong>of</strong> elements such as fluorine<br />
or magnesium or others that are used to<br />
decontaminate <strong>water</strong> for human consumption,<br />
in other words, the <strong>water</strong> that comes out <strong>of</strong><br />
the taps in cities. This structure breaks<br />
down easily, so it contains low amounts <strong>of</strong><br />
energy and oxygen. (“The <strong>Geometry</strong> <strong>of</strong> Water<br />
(I). Biological aspects.”, 2015)<br />
Photo recovered from: https://www.researchgate.net/pr<strong>of</strong>ile/Pei_Zhong_Feng/publication/301933759/figure/fig7/<br />
AS:360516078850052@1462965139883/a-molecular-structure-<strong>of</strong>-<strong>water</strong>-b-molecular-structure-<strong>of</strong>-ice-crystals.png<br />
Cluster, hexagonal formation <strong>of</strong> <strong>water</strong> molecules.<br />
46 47
Cluster - the <strong>water</strong> alphabet<br />
The cluster is the hexagonal structure<br />
that is formed from the union <strong>of</strong> multiple<br />
<strong>water</strong> molecules (image 21), these<br />
unions occur through hydrogen bonds<br />
which are weaker than the bonds <strong>of</strong> the<br />
<strong>water</strong> molecule (H2O), so they allow it to<br />
be malleable and take multiple forms.<br />
It should be noted that this cluster<br />
composition occurs in the liquid state <strong>of</strong><br />
<strong>water</strong>, and it is also important to bear in<br />
mind that this structure is changeable<br />
and adapts to the vibrations that pass<br />
through the <strong>water</strong> and influence it.<br />
This last characteristic <strong>of</strong> the clusters<br />
is the one that allows us to understand<br />
the idea <strong>of</strong> the memory <strong>of</strong> <strong>water</strong>,<br />
since as it happens with the crystal<br />
formations (solid state <strong>of</strong> <strong>water</strong>), where<br />
each crystal starts from a hexagonal<br />
structure, none is equal to another since<br />
the hexagonal formations that they reflect,<br />
vary depending on the vibration to which<br />
the <strong>water</strong> has been exposed. The same<br />
happens with the clusters, where although<br />
they are all formations between tetrahedra<br />
(hydrogen bridge junctions), which together<br />
form hexagons, these are not the same in<br />
all cases, since they depend on the vibration<br />
that has altered them, so if we could<br />
understand these hexagonal formations<br />
we could know the vibration that altered<br />
the <strong>water</strong> (García Flórez, n.d.). This is the<br />
principle used in homeopathic medicine,<br />
where <strong>water</strong> maintains its healing effects<br />
without the need <strong>of</strong> having the specific<br />
substance.<br />
It is from these hexagonal formations and<br />
their variations that <strong>water</strong> has the capacity<br />
to retain a large amount <strong>of</strong> energy or<br />
vibrational waves (such as radio waves),<br />
which is why they are known as liquid<br />
crystals, since their molecules are so<br />
precisely organized that they can contain<br />
the energy. However, hydrogen bridges<br />
are weak enough to allow <strong>water</strong> to mold<br />
and accommodate both the vibration that<br />
affects it and the body that contains it.<br />
By the latter I refer to the ability <strong>of</strong> <strong>water</strong><br />
to become part <strong>of</strong> a river and then be<br />
transformed into the aloe vera <strong>of</strong> a plant or<br />
the blood <strong>of</strong> an animal.<br />
Photo recovered from: https://unsplash.com/photos/iyA6oTK6vig<br />
Image 22. shapes in the <strong>water</strong><br />
48 49
Masaru Emoto and the memory <strong>of</strong> <strong>water</strong><br />
The beginning <strong>of</strong> life as we know it has<br />
been said to be the product <strong>of</strong> <strong>water</strong>, that<br />
is why since the most ancient civilizations<br />
man has tried to understand this essential<br />
element for life and that constitutes<br />
everything in nature. For indigenous<br />
communities there has always been a<br />
connection with nature and a relationship<br />
<strong>of</strong> appreciation and respect, where she<br />
gives us what we need to live and we use<br />
it without breaking the balance, or at<br />
least that is how it is for ancient cultures,<br />
in modern science this has been lost.<br />
The Japanese physician Masaru Emoto,<br />
dedicated his research to the study <strong>of</strong><br />
<strong>water</strong> and its benefits for homeopathic<br />
medicine. His research focused on the<br />
study <strong>of</strong> the formation <strong>of</strong> crystals in<br />
<strong>water</strong>, where he demonstrated that, when<br />
<strong>water</strong> was exposed to different vibrations,<br />
it generated crystals with different<br />
characteristics that communicated the<br />
state <strong>of</strong> the <strong>water</strong> and its alterations.<br />
Para sus experimentos Emoto tomó aguas<br />
de distintos lugares tanto naturales,<br />
como agua procesada o de los grifos en<br />
las ciudades. El objetivo era comprobar si<br />
existía alguna diferencia entre estas aguas<br />
cuando se congelaban, lo que descubrió<br />
fue que en la mayoría de los casos el<br />
agua proveniente de los grifos o lugares<br />
contaminados no generaba cristales<br />
completos o en general no formaba<br />
ningún cristal. Sin embargo, al someter<br />
un mismo tipo de agua a la influencia de<br />
distintas vibraciones se formaban cristales<br />
que recordaban la vibración a la que se<br />
había expuesto, es así que las palabras<br />
o sonidos positivos generaban cristales<br />
bellos, armónicos y completos (imagen<br />
23), mientras que aquellos expuestos a<br />
palabras o sonidos negativos y agresivos<br />
no generaban cristales o eran incompletos<br />
(imagen 24).<br />
These experiments led Emoto to propose<br />
that, on the one hand, <strong>water</strong> is an element<br />
that can be influenced, that changes<br />
according to the vibrations to which it<br />
has been exposed, so it can be said that<br />
<strong>water</strong> is capable <strong>of</strong> remembering and<br />
reflecting that which has influenced<br />
it. In addition, by performing the same<br />
experiment with <strong>water</strong> in its free and<br />
natural state and comparing it with treated<br />
or purified <strong>water</strong> from cities, he concluded<br />
that for <strong>water</strong> to form crystals it must be<br />
in equilibrium and pure, so the principles<br />
<strong>of</strong> life in <strong>water</strong> are the ability to move,<br />
change and flow, since it is through these<br />
movements or states <strong>of</strong> chaos that <strong>water</strong><br />
is revitalized by being charged with energy.<br />
During his studies Emoto managed to<br />
determine that the most beautiful crystal <strong>of</strong><br />
all is formed under the influence <strong>of</strong> the words<br />
“love and gratitude” (image 23), two words<br />
that reflect positive energies, but are also<br />
opposites. The first one, love, means to give<br />
or deliver a positive feeling towards another<br />
being, while gratitude means to receive a<br />
feeling with the same value as the previous<br />
one, but in the opposite sense, which leads<br />
50 51
us to understand that these two words<br />
generate a balance between them or what<br />
Emoto calls is the capacity to resonate.<br />
The capacity to resonate indicates that the<br />
<strong>water</strong> <strong>of</strong> a being or a place is influenced by<br />
vibrations that reach it and that it is capable<br />
<strong>of</strong> transmitting these vibrations to another<br />
being or place that contains <strong>water</strong>, it is due<br />
to this capacity that it is said that <strong>water</strong> has<br />
memory. For Emoto this is fundamental for<br />
medicine, because we are beings mainly<br />
composed <strong>of</strong> <strong>water</strong> and therefore when we<br />
receive positive and balanced vibrations,<br />
these will generate the same effect in the<br />
<strong>water</strong> <strong>of</strong> our body (Emoto, 2020).<br />
Photo recovered from: http://www.menteyexito.org/wp-content/<br />
uploads/2017/04/c092ae655d768526719c0eadcb38374e.jpg<br />
Image 23. Crystal formed by the vibration <strong>of</strong> the words “love and gratitude”.<br />
Photo retrieved from: the book The miracle <strong>of</strong> <strong>water</strong><br />
Image 24. Crystal formed by the vibration <strong>of</strong> the words “I can’t”.<br />
52 53
Analysis / conclusions<br />
As stated at the beginning <strong>of</strong> this research<br />
on <strong>water</strong>, the objective is to be able to<br />
understand the different scales <strong>of</strong> this<br />
element and thus identify the fractal<br />
character that constitutes it. If we return<br />
to the theories and mathematical models<br />
used to explain fractals, we find the Koch<br />
curve, which resembles the shape <strong>of</strong> a<br />
snowflake (Poizat, Sauter and Spodarev,<br />
2014). It is from this approach that we<br />
can understand the relationship between<br />
fractals and <strong>water</strong> in its different states.<br />
As we have already seen, Masaru Emoto’s<br />
crystals show the property <strong>of</strong> <strong>water</strong> to be<br />
influenced by the different vibrations that<br />
constitute the world and life itself and<br />
then reflect, from hexagonal formations,<br />
different crystals that contain the intrinsic<br />
properties <strong>of</strong> fractals; self-similar,<br />
irregular and harmonic formations that<br />
come from a dynamic and chaotic system.<br />
Subsequently, it became evident that the<br />
hexagonal formations that are generated in<br />
the formation <strong>of</strong> crystals, which is in turn<br />
the solid state <strong>of</strong> <strong>water</strong>, is directly related<br />
to molecular formations that are generated<br />
between different <strong>water</strong> molecules in the<br />
liquid state, which are called clusters.<br />
These formations are <strong>of</strong> vital importance,<br />
as they are responsible for retaining the<br />
energy and incorporating the vibrations<br />
that travel through the <strong>water</strong>; it is for<br />
this reason that it is said that <strong>water</strong> has<br />
memory and reflects those vibrations that<br />
it remembers in its molecular structure;<br />
additionally, it is this molecular composition<br />
that generates characteristics in <strong>water</strong><br />
that classify it as a liquid crystal. One <strong>of</strong><br />
these is the malleability <strong>of</strong> <strong>water</strong>, which<br />
allows it to adapt not only to the vibrations<br />
that influence it, but also to the body that<br />
contains it, and as mentioned, <strong>water</strong> is the<br />
fundamental element for life and therefore<br />
all living bodies possess it even when it<br />
manifests itself in different compositions.<br />
On the other hand, another characteristic<br />
that is evident from the hexagonal structure<br />
<strong>of</strong> the cluster, is that <strong>water</strong> manifests itself<br />
as a compact and apparently stable element,<br />
when observed on a larger scale, such as a<br />
river, the sea or a lake; in these cases <strong>water</strong><br />
takes on the appearance <strong>of</strong> being a complete<br />
surface.<br />
Photo recovered from: https://unsplash.com/photos/VuBzplNNi0k<br />
54 55
Propuesta<br />
Value<br />
proposal<br />
de valor<br />
Foto recuperada de: https://unsplash.com/photos/3Ik7xWYJv3U
Initial proposal<br />
moodboards analysis<br />
Understanding the way in which <strong>water</strong> is<br />
conformed at the molecular level, from<br />
clusters and the properties <strong>of</strong> rigidity and<br />
malleability that this structure gives to<br />
<strong>water</strong> in liquid state, I consider that it is from<br />
the geometric structures (pentagonal and<br />
hexagonal), that the fractal character <strong>of</strong> this<br />
element is manifested, taking into account<br />
that from these formations it is possible<br />
to observe the aspects <strong>of</strong> self-similarity,<br />
being an element composed <strong>of</strong> parts that<br />
at different scales look similar, and <strong>of</strong> the<br />
proportionality that exists between the<br />
different molecules and that allows the<br />
generation <strong>of</strong> stable structures. Additionally,<br />
it is from these formations that <strong>water</strong><br />
acquires the capacity to adapt and transform<br />
itself to become an essential part <strong>of</strong> the body<br />
that contains it.<br />
Based on geometric patterns, the proposal<br />
is to develop surfaces that, like <strong>water</strong>,<br />
can adapt to different spaces where<br />
they interact, reflecting on the one hand<br />
the malleability <strong>of</strong> this element to be<br />
influenced by external bodies, but also the<br />
rigidity that the patterns give it to be an<br />
element <strong>of</strong> containment. These surfaces<br />
will show multiple proportional patterns,<br />
based on geometric shapes, reflecting<br />
the compositions <strong>of</strong> the clusters and how<br />
these are a reflection <strong>of</strong> the elements that<br />
surround or influence them.<br />
To understand the forms <strong>of</strong> <strong>water</strong> I made<br />
three moodboars from which I could analyze<br />
different characteristics <strong>of</strong> the forms that<br />
are expressed in it. It should be noted that<br />
for this inquiry I made an approach to images<br />
<strong>of</strong> <strong>water</strong>, without entering into a chemical<br />
study to analyze its molecular structures.<br />
However, based on the discoveries <strong>of</strong> Masaru<br />
Emoto, where it is possible to recognize<br />
the pentagonal or hexagonal shapes <strong>of</strong> the<br />
molecular composition <strong>of</strong> <strong>water</strong>, from the<br />
physical manifestation in the form <strong>of</strong> crystals,<br />
it was sought that the <strong>water</strong> in the images<br />
used reflected geometric shapes from which<br />
the forms <strong>of</strong> its molecular structure are<br />
sought to be recognized.<br />
Photo retrieved from: https://unsplash.com/photos/lUPHw5bM7HM<br />
58 59
Frozen fluidity<br />
The first moodboar reflects <strong>water</strong> in<br />
solid state or more specifically in frozen<br />
formations, so the images used correspond<br />
to crystals and ice formations in <strong>water</strong><br />
currents. This moodboard has the concept <strong>of</strong><br />
Frozen Fluidity, which is understood as each<br />
drop is a frozen instant, which in its deep<br />
interior remains a flowing vitality ready<br />
to follow its path. The concept describes<br />
on the one hand the movement <strong>of</strong> <strong>water</strong><br />
that stops and is trapped in the crystals,<br />
but also the fluidity that lasts in the <strong>water</strong><br />
currents thanks to the frozen formations<br />
<strong>of</strong> its surface, in this case it highlights the<br />
importance <strong>of</strong> movement in <strong>water</strong> because<br />
despite being contained in a solid state is<br />
<strong>water</strong> that has traveled the world, which is<br />
alive and contains a lot <strong>of</strong> energy.<br />
In this case, two important aspects are<br />
highlighted in relation to fractals, the first<br />
one is that, as Emoto mentions, only good<br />
quality <strong>water</strong> or, taking into account Mu<br />
Shik Jhon’s research, <strong>water</strong> with hexagonal<br />
structure has the property <strong>of</strong> forming frozen<br />
crystals and each one is unique, since, as<br />
it happens with fractals, the <strong>water</strong> present<br />
there corresponds to only one part seen at a<br />
given moment in time.<br />
60 61
The second moodboar shows <strong>water</strong> in its<br />
liquid state, which is found free in nature<br />
and is affected by the sun’s rays, since it<br />
is from this mixture that geometric shapes<br />
are reflected both on the surface as seen<br />
in the images, and at the bottom <strong>of</strong> the sea.<br />
The concept in this case is that <strong>of</strong> Sinuous<br />
Malleability, understood as infinite and<br />
changing <strong>water</strong> that in its warm mixture<br />
with the sun’s rays permeates and moves<br />
those who enter into its being, and speaks<br />
<strong>of</strong> the appearance that <strong>water</strong> takes on when<br />
influenced by the sun’s rays that intermingle<br />
to generate a surface that, although liquid,<br />
appears to be compact but malleable.<br />
are appreciated at the same time, which<br />
allows me to infer that the <strong>water</strong> present<br />
in these images has a very organized and<br />
optimal molecular structure as is the case<br />
<strong>of</strong> hexagonal formations and that is why it<br />
manifests itself in this way.<br />
This appearance can be related to the<br />
characteristic <strong>of</strong> <strong>water</strong> as a liquid crystal, in<br />
which the properties <strong>of</strong> a solid and a liquid<br />
Sinuous malleability<br />
62 63
Fickle resistance<br />
Finally, the third moodboard represents<br />
the delicacy observed in the <strong>water</strong>, for<br />
this reason images <strong>of</strong> <strong>water</strong> bubbles<br />
and crystalline <strong>water</strong> currents are taken<br />
that allow observing the depth and its<br />
harmony. For this case the concept used<br />
was that <strong>of</strong> Flickle Resistance, which is<br />
understood as bubbles, s<strong>of</strong>t windows, that<br />
come and go on the surface and reveal<br />
the vital immensity that vibrates under<br />
the flowing mantle, and that speaks <strong>of</strong> the<br />
characteristic contrast that this element<br />
has, which can be as delicate and volatile<br />
as a bubble that explodes and evaporates<br />
in a strong breeze, but as strong and<br />
resistant as a drop <strong>of</strong> <strong>water</strong> that collects<br />
and transports the energy that keeps<br />
alive the different systems <strong>of</strong> the planet.<br />
In this case the purity <strong>of</strong> the <strong>water</strong> and<br />
the fact that it comes from a natural<br />
stream, which also has a large amount<br />
<strong>of</strong> rocks in its soil, allows us to infer that<br />
the minerals <strong>of</strong> the soil, the constant<br />
movement oxygenates and nourishes it<br />
and therefore we can infer that it is <strong>water</strong><br />
with a hexagonal structure. Additionally,<br />
we can appreciate the geometric shapes<br />
that generate the edges <strong>of</strong> the bubbles and<br />
that can be interpreted as a smaller scale<br />
observation <strong>of</strong> a quantity <strong>of</strong> <strong>water</strong>.<br />
64 65
Moodboards and references<br />
After analyzing and determining the forms<br />
<strong>of</strong> <strong>water</strong> in the three cases defined above,<br />
I carried out an exploration <strong>of</strong> references<br />
that would allow me to understand<br />
different representations <strong>of</strong> the forms I<br />
was looking for based on materials, colors<br />
and compositions. After selecting the<br />
references that responded best with the<br />
moodboards, I made a second composition<br />
<strong>of</strong> each one integrating images <strong>of</strong> the<br />
moodboard and images <strong>of</strong> the references<br />
in order to better analyze the relationships<br />
between the two and define their particular<br />
forms.<br />
Photo retrieved from:: https://i.pinimg.<br />
com/564x/3c/2e/63/3c2e6361e1294560aae30241fe9004df.jpg<br />
Image 26. Mathieu Lehanneur, Liquid marble<br />
Photo retrieved from: https://www.elisastrozyk.com./wooden-rugs<br />
Image27. Elisa Strozyk, Fading<br />
Photo retrieved from: https://dazedimg-dazedgroup.<br />
netdna-ssl.com/467/azure/dazed-prod/1210/6/1216181.jpg<br />
Image 28.Iris Van Herpen, AW17 couture Show<br />
Photo retrieved from: https://design-milk.com/scale-flexible-modularacoustic-partition-system/scale-layerxwovenimage-partition-1/<br />
Image 29. Bejamin Hubert, Scale<br />
Image 25. Karen LaMonte, Dreamscape Drapery Study<br />
Photo retrieved from: https://www.karenlamonte.com/Contemporary-<br />
Scultpures-Prints/Drapery-Sculptures-Bas-Relief/i-7dwCKCF/A<br />
66 67
For the first moodboard, Frozen Fluidity, I<br />
looked for references where from solid<br />
and static forms I could transmit fluidity<br />
and movement, this with the objective <strong>of</strong><br />
understanding in a better way the way in<br />
which I could transmit two opposite states<br />
such as fluidity and movement with the<br />
frozen and static. One <strong>of</strong> the main references<br />
is the work <strong>of</strong> Karen LaMonte and especially<br />
the work called Drapery sculpture (LaMonte,<br />
2008) (image 25), which as its name suggests<br />
are sculptures that resemble draped fabrics,<br />
this reference allowed me to understand how<br />
from static forms can generate movement<br />
from folds or superimpositions that in turn<br />
generate harmonic pieces.<br />
In the composition that links the moodboard<br />
with the references I could find geometric<br />
shapes that generated the movement and<br />
fluidity <strong>of</strong> <strong>water</strong> without losing the rigidity<br />
<strong>of</strong> the materials and <strong>of</strong> a static piece. In this<br />
composition you can see a close-up <strong>of</strong> a<br />
table by French designer Mathieu Lehanneur,<br />
more specifically from his Liquid Marble<br />
collection (“Mathieu Lehanneur ‘Liquid<br />
Marble’ installation at the Musée des Arts<br />
décoratifs, Paris - urdesignmag”, 2017) (image<br />
26), this piece is very interesting for the<br />
contrast generated between the rigidity <strong>of</strong> a<br />
material such as marble and the movement<br />
generated by the carved forms, additionally<br />
when viewed in conjunction with the images<br />
<strong>of</strong> frozen <strong>water</strong> similarities are appreciated<br />
as the color changes that are generated by<br />
the shadows and brightness product <strong>of</strong> the<br />
solid structures. Likewise, other textures<br />
and structures that show the geometric<br />
formations that are generated in the frozen<br />
<strong>water</strong> and that by their arrangement in space<br />
and difference in sizes and shapes transmit<br />
fluidity despite being in a solid state are also<br />
appreciated.<br />
68 69
For the second moodboard, Sinuous<br />
malleability, I looked for the references to<br />
reflect opposite states in its materiality,<br />
from something that seems rigid and<br />
compact, but is formed by parts that give<br />
it movement or something that I know<br />
has a lot <strong>of</strong> movement but is rigid to the<br />
touch. In this case the main referent was<br />
the work <strong>of</strong> the designer Elisa Strosyk who<br />
uses pieces <strong>of</strong> wood on fabric to generate<br />
rigid structures, but with great movement<br />
(“Elisa Strozyk | Poligom”, 2011)(image 27).<br />
became evident from this composition is<br />
the mixture <strong>of</strong> colors between the different<br />
modules or forms <strong>of</strong> <strong>water</strong>, to give the<br />
feeling <strong>of</strong> fluidity and movement in the piece<br />
without making it look fragmented, but on<br />
the contrary, the colors give unity to the<br />
totality <strong>of</strong> modules.<br />
In the composition between the referents<br />
and the moodboard, it is evident the need<br />
to use modules that generate structure in<br />
the piece due to their geometry, but also<br />
that represent a contrast that generates<br />
movement. Another important aspect that<br />
70 71
Finally, for the moodboard <strong>of</strong> Voluble<br />
Resistance, I looked for references that<br />
allowed me to understand the delicacy <strong>of</strong><br />
the forms from pieces with transparencies<br />
and defined edges, which generate the<br />
sensation <strong>of</strong> unity and resistance within the<br />
pieces. Among the references, two stand<br />
out that use transparencies and edges as<br />
constituent elements <strong>of</strong> the pieces, but<br />
using materials that are contrary in terms<br />
<strong>of</strong> their rigidity. The first referent are the<br />
dresses <strong>of</strong> the designer Iris Van Herpen,<br />
these are characterized by the movement<br />
and fluidity they generate, product <strong>of</strong><br />
the combination <strong>of</strong> materials and the<br />
arrangement <strong>of</strong> the fabrics, also handles<br />
transparencies as seen in the fabric <strong>of</strong><br />
one <strong>of</strong> the dresses <strong>of</strong> the AW17 couture<br />
show (Hope Allwood, 2017) (image 28), in<br />
which the use <strong>of</strong> transparencies and the<br />
combination <strong>of</strong> colors evoke me the colors<br />
and shapes <strong>of</strong> <strong>water</strong>.<br />
On the other hand, there is the reference<br />
<strong>of</strong> the designer Benjamin Hubert, who<br />
created Scale (image 29), which is a flexible<br />
modular system that has sustainability<br />
as a fundamental element (Williamson,<br />
2015). In this piece different modules are<br />
generated from a hexagonal structure,<br />
where both solid spaces and empty spaces<br />
that maintain this same structure are<br />
appreciated and I relate it to the geometric<br />
shapes that are generated with the bubbles<br />
in the <strong>water</strong>, which can show an internal<br />
surface or on the contrary may seem empty,<br />
but maintain the structure <strong>of</strong> its edges.<br />
72 73
geometries<br />
After making the compositions with<br />
the referents and understanding the<br />
shapes that were present in each one<br />
and in the moodboards, I carried out<br />
two exercises <strong>of</strong> geometrization <strong>of</strong><br />
the moodboards to better understand<br />
the shapes <strong>of</strong> each one and the<br />
relationship <strong>of</strong> these shapes with<br />
the concepts. It should be noted that,<br />
during the previous exercise, the<br />
presence <strong>of</strong> other geometric shapes<br />
different from the hexagons and<br />
pentagons that were being studied<br />
during the research became evident.<br />
The first geometrization exercise was<br />
carried out digitally, which allowed<br />
me to relate the shapes I found in the<br />
moodboards with the colors present<br />
in the images and from this to find the<br />
rhythms generated from the size and<br />
color <strong>of</strong> the modules, which is linked to<br />
the property <strong>of</strong> self-similarity seen in<br />
natural fractals.<br />
With the second geometrization<br />
exercise, which I performed manually<br />
on the moodboards, I found more clearly<br />
the irregular patterns that are generated<br />
from the modules and that together<br />
show the proportionality that exists in<br />
the chaotic geometric shapes <strong>of</strong> nature.<br />
In the case <strong>of</strong> the first moodboard I could<br />
observe how the composition is made up<br />
<strong>of</strong> several small modules that by their<br />
different colors and locations within<br />
the set form volumes <strong>of</strong> different sizes<br />
that can be associated with the frozen<br />
formations that make up the moodboard.<br />
Additionally, it is observed that within the<br />
composition there are accumulations<br />
<strong>of</strong> small modules in some parts that<br />
contrast with other larger ones that<br />
balance the composition, which I<br />
interpret as the vibrations present in the<br />
<strong>water</strong> within its particular context and<br />
that are manifested in the shapes, sizes<br />
and colors <strong>of</strong> the different modules<br />
transmitting the properties <strong>of</strong> cold and<br />
volume <strong>of</strong> frozen <strong>water</strong>.<br />
74 75
In the second moodboard the first<br />
geometrization is very similar to the<br />
previous moodboard, however, in this<br />
case the modules have a larger size<br />
and less contrasting shades <strong>of</strong> blue and<br />
green, which generates the sensation <strong>of</strong><br />
being flat modules with a large surface.<br />
As for the distribution <strong>of</strong> the modules,<br />
there are also some concentrations <strong>of</strong><br />
small ones, but for the most part there is<br />
a proportional mixture <strong>of</strong> large, medium<br />
and small modules, giving a sense <strong>of</strong><br />
unity and order in the composition,<br />
which is what, in terms <strong>of</strong> the molecular<br />
composition <strong>of</strong> <strong>water</strong>, allows it to take<br />
on the characteristics <strong>of</strong> a liquid crystal.<br />
76 77
For the third moodboard, the first<br />
geometrization was not very helpful,<br />
because I focused more on generating<br />
transparencies than on the shapes<br />
present in the images, although the<br />
importance <strong>of</strong> the edges <strong>of</strong> the figures<br />
to achieve this sensation was evident.<br />
However, with the second geometrization<br />
I focused on the shapes, highlighting the<br />
presence <strong>of</strong> almost regular polygons,<br />
with different number <strong>of</strong> sides, which<br />
overlap each other giving depth to the<br />
composition from flat surfaces. In this<br />
case the edges <strong>of</strong> the figures take<br />
a relevant role, since they evidence<br />
the transparencies, and also delimit<br />
the different geometric modules that<br />
conform it.<br />
78 79
analysis for design<br />
By understanding the characteristics<br />
<strong>of</strong> each moodboard, I was able to<br />
establish the relationship between<br />
these and the concepts worked<br />
previously, to establish the way in<br />
which <strong>water</strong> as a fractal element was<br />
studied for the development <strong>of</strong> the<br />
design proposal.<br />
The first concept that I worked on<br />
was that <strong>of</strong> self-similarity, taking<br />
into account that in the moodboards<br />
as well as in the geometrizations the<br />
presence <strong>of</strong> figures in different sizes<br />
that when put together form a whole,<br />
so for the development <strong>of</strong> the pieces<br />
I will work on the development <strong>of</strong><br />
geometric modules in different sizes,<br />
that when put together form the total<br />
piece and reflect the irregularity <strong>of</strong><br />
the fractals. Likewise, these modules<br />
arise from the clusters present in the<br />
<strong>water</strong> and that as we have already<br />
seen are the elements that generate<br />
the characteristic properties <strong>of</strong><br />
this element, so we will look for the<br />
conformation and characterization <strong>of</strong><br />
the pieces from the characteristics<br />
<strong>of</strong> its modules and the relationship <strong>of</strong><br />
these with their respective moodboard.<br />
From the geometries, the<br />
proportionality <strong>of</strong> the moodboards<br />
was highlighted, reflected both in the<br />
size <strong>of</strong> the modules already mentioned<br />
and the relationship between them,<br />
as well as in the distribution <strong>of</strong><br />
colors within the compositions. For<br />
this reason, proportionality will be<br />
a characteristic that will be worked<br />
on in the development <strong>of</strong> the pieces,<br />
seeking that the sizes <strong>of</strong> the modules<br />
are proportional to each other,<br />
making use <strong>of</strong> the Fibonacci chain to<br />
determine the sizes <strong>of</strong> the modules. In<br />
the case <strong>of</strong> the proportions between<br />
the color zones, it will be sought that<br />
these correspond to those found<br />
from the moodboards in terms <strong>of</strong> the<br />
colors used, but also that they look<br />
harmonious within the compositions,<br />
reflecting movement and rhythm<br />
within them.<br />
Finally, the last property to be<br />
reflected in the pieces corresponds<br />
to the dynamic systems and their<br />
characteristics. In this case two<br />
elements will be taken for its<br />
representation, the first corresponds<br />
to the distribution <strong>of</strong> the modules,<br />
which is subject to the person who<br />
makes the composition <strong>of</strong> the piece,<br />
since, as explained with the chaos<br />
theory, any change however small will<br />
result in a different conformation,<br />
which indicates that even if the same<br />
composition is taken as a basis, the<br />
distribution <strong>of</strong> the modules will always<br />
be different. On the other hand, it<br />
has been established that <strong>water</strong> as<br />
a fractal element and as a dynamic<br />
system is in constant change, and<br />
that the changes are subject to the<br />
context in which it is located, which<br />
is why the pieces can be adapted to<br />
80 81
different contexts, so that the same<br />
surface acquires a particular use or<br />
arrangement according to the place<br />
where it is located.<br />
Foto recuperada de: https://i.pinimg.com/originals/<br />
eb/c5/ba/ebc5badb7c3675d2846ac96b02e9fd44.jpg<br />
Imagen 20. Cataratas, el agua en caida genera un movimiento fractal<br />
82
final<br />
piezas<br />
pieces<br />
finales<br />
Foto recuperada de: https://www.nature-p0rn.com/wp-content/uploads/2019/01/steve-huntington-374991-unsplash-min-1024x683.jpg
Prototypes<br />
After understanding the forms<br />
present in the three moodboards I<br />
started the prototyping stage, which<br />
aims to explore different methods to<br />
reach the definition <strong>of</strong> the modules<br />
that represent each moodboard<br />
and find the optimal material for its<br />
realization. Additionally, there were<br />
three principles for the development<br />
<strong>of</strong> the pieces, these correspond to<br />
the findings <strong>of</strong> the research and<br />
are: self-similarity, translated into<br />
modules that are equal in shape but<br />
with different sizes; proportionality,<br />
reflected in the irregular patterns<br />
that are formed from the regular<br />
modules and that, although their<br />
arrangement seems random, maintain<br />
the balance in the piece, as well as<br />
the relationship with the colors <strong>of</strong><br />
the modules and the formation <strong>of</strong><br />
color zones; the pieces function as<br />
dynamic systems that is evident in the<br />
arrangement <strong>of</strong> the modules, which is<br />
the result <strong>of</strong> a subjective decision, so<br />
that each composition will be similar<br />
but different from the others, besides<br />
being pieces that relate directly to<br />
their environment adapting its shape<br />
to it; the last element are the clusters<br />
as the pattern present in the <strong>water</strong> and<br />
will be represented by the geometric<br />
modules that make up each <strong>of</strong> the<br />
prototypes and final pieces.<br />
The exploration for the moodboard<br />
<strong>of</strong> Fluidez congelada focused on<br />
exploring modules from origami that<br />
would allow to generate volumes<br />
within the piece. The first prototype<br />
consists <strong>of</strong> triangular modules called<br />
Trinity box (AxensWorkshops, 2012),<br />
in this case although the aim was to<br />
generate a volume, the shapes and the<br />
paper used made the piece look very<br />
strong, which did not correspond with<br />
the harmonic forms <strong>of</strong> frozen <strong>water</strong>.<br />
Then I made other explorations with<br />
origami modules, which did not work<br />
because they did not give the desired<br />
volume. Finally, I found the book A<br />
constellation <strong>of</strong> origami polyhedra<br />
(Montroll, 2004) in which different<br />
modules <strong>of</strong> regular and irregular<br />
polygons are presented, from these<br />
modules I made the second prototype<br />
that is made <strong>of</strong> modules <strong>of</strong> a triangular<br />
bipyramid cut in half and modules <strong>of</strong> a<br />
pentagonal bipyramid, each <strong>of</strong> these<br />
figures have three different sizes and<br />
with different colored papers that give<br />
dynamism and proportion to the piece.<br />
From this last prototype, the<br />
development <strong>of</strong> the final proposal was<br />
started, so the next step was to make<br />
the modules in fabric, for which we used<br />
sheet fabric, which due to its crumpled<br />
texture facilitates the generation <strong>of</strong><br />
folds that form the module. For the<br />
realization <strong>of</strong> the first fabric modules,<br />
two straw cardboard templates were<br />
made, taking the shape <strong>of</strong> the paper<br />
86 87
modules <strong>of</strong> the prototype and another<br />
figure was added that corresponds to<br />
the base <strong>of</strong> the module, this base is<br />
necessary to fill the module and thus<br />
give consistency to the figure. Starting<br />
from the initial templates, three other<br />
templates were made for each figure in<br />
three larger sizes. These modules were<br />
made with cotton fabric that preserves<br />
the cardboard texture; however, as the<br />
size increased, it became necessary to<br />
add a layer <strong>of</strong> interlining at the base<br />
<strong>of</strong> the modules to preserve the shape<br />
<strong>of</strong> the module and prevent it from<br />
distorting when filled.<br />
Imagen 30. Trinity box<br />
Foto tomada por: Laura Zamudio<br />
Foto tomada por: Laura Zamudio<br />
Image 31. Composition <strong>of</strong> regular polyhedra in origami on paper.<br />
Picture taken by: Laura Zamudio<br />
Imagen 31. Moldes en tela de los poliedros regulares<br />
Picture taken by: Laura Zamudio<br />
Image 32. Modules in regular polyhedron fabric in three sizes.<br />
88 89
For the sinuous malleability<br />
moodboard I started with some<br />
prototypes that generated a<br />
structure very similar to the one<br />
worked in the referent from painted<br />
fabric with the hydrographic<br />
technique, to represent the fluidity<br />
<strong>of</strong> <strong>water</strong> in the and triangular<br />
modules in straw cardboard that<br />
will generate a rigid structure with<br />
empty spaces that give movement.<br />
This prototype is interesting for the<br />
shapes that are generated as well<br />
as the appearance achieved with the<br />
painted fabric, however, it does not<br />
show the geometric shapes that are<br />
visualized in the geometrizations and<br />
that represent the cluster. For the<br />
following prototype lency cloth was<br />
used to make the modules, which are<br />
regular hexagons <strong>of</strong> different colors<br />
and two sizes, these hexagons are<br />
composed <strong>of</strong> an edge that gives the<br />
shape and strips <strong>of</strong> 0.5 or 1 centimeter<br />
attached to the edge. Each module<br />
consists <strong>of</strong> two hexagons <strong>of</strong> different<br />
colors intermingled strips to reveal<br />
both colors and give movement and<br />
fluidity to the piece.<br />
This last prototype was very<br />
fractioned by the hexagonal borders,<br />
so we looked for a way to eliminate<br />
this border and that the internal<br />
movement <strong>of</strong> each module would<br />
highlight and give rise to the whole<br />
composition. The last prototype<br />
is again made in lency cloth and<br />
consists <strong>of</strong> hexagonal modules <strong>of</strong><br />
different colors and sizes, where a<br />
single integrated edge is maintained<br />
to the color strips, which as in the<br />
previous case are folded to make<br />
visible the two colors <strong>of</strong> each module<br />
and generate movement and fluidity<br />
in the total piece. In this case the<br />
modules are placed mixing colors,<br />
sizes and direction to give harmony to<br />
the piece and generate the feeling <strong>of</strong><br />
unity present in the moodboard <strong>water</strong>.<br />
Picture taken by: Laura Zamudio<br />
Image 33. Straw cardboard modules on fabric painted with hydrography.<br />
90 91
Picture taken by: Laura Zamudio<br />
Image 34. Hexagonal modules woven with edges<br />
Picture taken by: Laura Zamudio<br />
Image 35. Hexagonal modules woven without border and in three sizes.<br />
The prototyping process <strong>of</strong> the third<br />
moodboard, Voluble resistance,<br />
differs in the shapes, especially in<br />
the first prototypes, since the aim<br />
was to generate transparencies and<br />
highlight the edges <strong>of</strong> the shapes. The<br />
first prototypes were made from cuts<br />
in an elastic fabric, however, they did<br />
not give the expected sensation so it<br />
was discarded and I do not consider<br />
necessary to show it. For the first<br />
prototype we tried to generate fluid<br />
shapes and transparencies from<br />
strips <strong>of</strong> fabric <strong>of</strong> different blues<br />
and some reds, which when joined<br />
at different points generated waves<br />
that allowed us to see the mixed<br />
colors when viewing the entire<br />
piece. However, this prototype<br />
handled a completely different<br />
language to the explorations <strong>of</strong> the<br />
previous moodboards, so I made<br />
a completely different prototype,<br />
in which the transparencies were<br />
preserved by using chiffon and<br />
tulle, but the shapes <strong>of</strong> the modules<br />
did not seek to generate waves<br />
but on the contrary correspond to<br />
geometric shapes that are evident<br />
in the second geometrization for<br />
this moodboard.<br />
This prototype not only handles a<br />
language according to the previous<br />
proposals, but also transmits<br />
the moodboard sensations such<br />
as the delicacy <strong>of</strong> <strong>water</strong> and the<br />
harmony between all the elements.<br />
92 93
In this prototype were handled<br />
non-regular geometric modules<br />
<strong>of</strong> different sizes, for the final<br />
prototype it was decided to use<br />
three regular figures such as the<br />
pentagon, hexagon and octagon<br />
taking into account that in the<br />
previous prototype the figures with<br />
more sides were distorted or even<br />
took an almost round appearance,<br />
each figure is used in three sizes<br />
and in different colors.<br />
Picture taken by Laura Zamudio<br />
Picture taken by: Laura Zamudio<br />
Image 37. Irregular modules with transparencies.<br />
Imagen 36. Tejido<br />
94 95
Design proposal<br />
After making the different prototypes<br />
for each piece, the decision was made to<br />
develop as the final piece the proposal<br />
corresponding to the Frozen Fluidity<br />
moodboard, this decision was made<br />
taking into account the time available<br />
for the completion <strong>of</strong> the project and<br />
the development <strong>of</strong> the proposal. It was<br />
also taken into consideration which<br />
proposal best reflected the principles<br />
<strong>of</strong> self-similarity, proportionality and<br />
that it would work as a dynamic system.<br />
For the development <strong>of</strong> the final piece,<br />
a photographic montage (image 38)<br />
was first made on a double bed, in order<br />
to begin to understand the relationship<br />
between the modules and the bed,<br />
understanding it as the environment<br />
that influences the <strong>water</strong> or in this<br />
case the piece. This montage allowed<br />
me to analyze the sizes, distribution<br />
and materials <strong>of</strong> the modules, in order<br />
to make the necessary corrections.<br />
First, it became evident that the size<br />
<strong>of</strong> the modules was not proportional<br />
and harmonious with the surface <strong>of</strong><br />
the bed, and the relationship between<br />
them also needed to be reviewed. As<br />
for the material, it became evident the<br />
need to integrate the surface <strong>of</strong> the<br />
piece as an integral part <strong>of</strong> the design,<br />
and upon seeing the modules on the<br />
bed, the decision was made to change<br />
the fabric used in the prototypes to<br />
lency cloth. This decision was made<br />
because when reviewing the prototype<br />
on paper the shapes were solid and<br />
defined, which was not being achieved<br />
with the fabric, while the lency fabric,<br />
being a more rigid material, allowed to<br />
better define the faces <strong>of</strong> the modules<br />
and by leaving the seams on the outside<br />
it was possible to define the vertices<br />
<strong>of</strong> the figures, giving cleanliness and<br />
structure to the figures (image 39).<br />
Likewise, 3 sizes were defined<br />
following the Fibonacci chain for their<br />
diameters, so the smallest ones have<br />
a radius <strong>of</strong> 3 cm, the medium ones <strong>of</strong> 5<br />
cm and the large ones <strong>of</strong> 7 cm, which,<br />
although not belonging to the chain,<br />
looks proportional in relation to the<br />
others and to the surface <strong>of</strong> the bed. On<br />
the other hand, the decision was made<br />
Picture taken by: Laura Zamudio<br />
Image 38. Photomontage <strong>of</strong> first volume modules<br />
Picture taken by: Laura Zamudio<br />
Image 39. Modules in lency cloth, final sizes<br />
96 97
to combine modules with volume and<br />
flat modules, which at first sight seem<br />
to have volume due to the combination<br />
<strong>of</strong> colors, this was done to generate<br />
greater harmony in the piece.<br />
The second important element<br />
that had to be reviewed was the<br />
composition <strong>of</strong> the piece, since up to<br />
that moment the distribution <strong>of</strong> the<br />
modules was randomly made from<br />
the available modules. However, when<br />
testing on the bed, the arrangement <strong>of</strong><br />
the modules did not show harmony or<br />
proportionality.<br />
With this in mind, I carried out a<br />
digital exploration that allowed me<br />
to develop a composition that would<br />
bring together both the previously<br />
mentioned concepts and their<br />
design manifestations, as well as<br />
the characteristics and sensations<br />
reflected in the moodboard. For this<br />
exploration I took as inspiration three<br />
photographs <strong>of</strong> frozen landscapes in the<br />
world, in which I found a way to express<br />
the essence <strong>of</strong> the concept.<br />
The first image corresponds to an<br />
ice formation through which light<br />
is reflected on the <strong>water</strong>, resulting<br />
in different shades <strong>of</strong> blue that<br />
complement the cold environment <strong>of</strong><br />
the image (image 40). For this case I<br />
made a delimitation <strong>of</strong> the color zones<br />
taking three representative colors, in<br />
order to make a composition in which<br />
the modules were arranged according<br />
to the desired shades (image 41). For<br />
the composition with the modules, a<br />
digital recreation <strong>of</strong> both the bed and<br />
the modules was made, taking the real<br />
measurements, this allowed me to<br />
check the proportionality between all<br />
the elements. Additionally, with this<br />
composition we determined the colors<br />
that make up each color zone and the<br />
distribution <strong>of</strong> volume modules that<br />
delimit the zones and flat modules<br />
that make them up (image 42).<br />
98 99
Photo retrieved from: https://i.pinimg.com/564x/38/83/<br />
bb/3883bbc54af2dceec167051fa03f2dbd.jpg<br />
Image 40.<br />
Image 41. Color silhouettes<br />
Image 42. Composition with modules 1<br />
100 101
The second image corresponds to<br />
the ceiling <strong>of</strong> the Marble Caverns<br />
in Patagonia (image 43), which is<br />
made up <strong>of</strong> frozen formations that<br />
assimilate waves, thus allowing to<br />
show movement and fluidity from<br />
static structures. In this case I made<br />
again the process <strong>of</strong> delimitation <strong>of</strong><br />
color zones highlighting the different<br />
waves <strong>of</strong> the cavern to take them as<br />
a reference for the distribution <strong>of</strong><br />
the modules from the distribution<br />
<strong>of</strong> colors and volumes (image 44).<br />
However, in this case the volumes<br />
were used to generate the two darkest<br />
and the lightest zones, while the<br />
others were formed by flat modules<br />
with different color combinations<br />
(image 45). In this composition the<br />
proportionality generated by the color<br />
zones is more evident, since it is from<br />
these that the composition transmits<br />
fluidity despite being static, also in<br />
this case the volume <strong>of</strong> the modules<br />
helps to generate more movement<br />
within the piece.<br />
Image 43. Marble caverns<br />
Photo retrieved from: de: https://i.pinimg.<br />
com/564x/85/20/4e/85204e7df0990cd00aef10b158ccace0.jpg<br />
Image 44. Color silhouettes<br />
102 103
The third image corresponds to a photo<br />
<strong>of</strong> a natural phenomenon in which<br />
frozen methane bubbles accumulate<br />
under the surface <strong>of</strong> the frozen <strong>water</strong><br />
<strong>of</strong> Abraham Lake (image 46), in this<br />
case I took the image as a reference<br />
for the composition, but without<br />
determining color zones, since as<br />
can be seen in the image the colors<br />
are mixed, although organized leaving<br />
the lighter ones over the darker<br />
ones, which generates depth. In the<br />
composition with the modules I tried<br />
to generate the mixture <strong>of</strong> colors seen<br />
in the image and from the mixture <strong>of</strong><br />
volumes, planes and empty spaces to<br />
give movement to the piece (image 47).<br />
Photo retrieved from: https://i.pinimg.com/564x/9e/<br />
ea/3b/9eea3b1b4948b9d83b44bf9adc0929ad.jpg<br />
Image 46. Frozen methane bubbles<br />
Image 45. Composition with modules 2<br />
104 105
After analyzing the three composition<br />
proposals, I decided to opt for the<br />
second option, as I consider it to<br />
be the one that best integrates the<br />
relationship between the volume<br />
modules and planes with the color<br />
zones, which are the reflection<br />
<strong>of</strong> proportionality in the piece.<br />
Additionally, the undulating forms <strong>of</strong><br />
the zones generate movement and<br />
fluidity from static structures, thus<br />
representing the meaning <strong>of</strong> the<br />
concept.<br />
Image 45. Composition with modules 3<br />
106 107
Final pieces<br />
The final process consisted <strong>of</strong> the<br />
material development <strong>of</strong> the chosen<br />
composition, for which the different<br />
triangles needed to form both the flat<br />
modules and the volume modules were<br />
cut in the different colors <strong>of</strong> lency<br />
cloth available. The colors <strong>of</strong> the cloth<br />
correspond to the range <strong>of</strong> blues, sea<br />
<strong>water</strong>, grays, mint green and beige to<br />
highlight the lighter areas.<br />
For the base <strong>of</strong> the piece it was decided<br />
to use the lightest blue cloth, which<br />
allows the piece to look cool despite<br />
being made <strong>of</strong> a warm material. This<br />
base has a size <strong>of</strong> 1 meter wide by 2<br />
meters long, the measures correspond<br />
to the size <strong>of</strong> a double bed being a little<br />
longer to overcome the width <strong>of</strong> the<br />
bed. On the other hand, Peptel paper<br />
was used to adhere the flat modules<br />
to the base fabric, this paper works as<br />
a layer between the two fabrics and<br />
sticks to them when exposed to the<br />
heat <strong>of</strong> the iron. The volume modules<br />
were sewn by machine to speed up the<br />
manufacturing process and were sewn<br />
by hand on the base, using an invisible<br />
stitch that is resistant and preserves<br />
the neatness <strong>of</strong> the product.<br />
Finally, a photographic session <strong>of</strong> the<br />
piece was carried out, in which the<br />
possibilities <strong>of</strong> use were explored.<br />
In this session the dynamism <strong>of</strong> the<br />
blanket became evident, since when<br />
placed on different surfaces such as<br />
the bed or the s<strong>of</strong>a, it adapted to its<br />
context. In the same way, by involving<br />
people to interact with the blanket, it<br />
took on a new narrative articulating<br />
the context, use and body <strong>of</strong> the<br />
person, which evidenced the property<br />
<strong>of</strong> <strong>water</strong> to adapt and remember the<br />
vibrations that influence it, to reflect<br />
them in its appearance.<br />
The latter is <strong>of</strong> great importance, since<br />
it is the reason why <strong>water</strong> manifests<br />
itself as a dynamic system that under<br />
the influence <strong>of</strong> the context in which<br />
it is, it transforms itself by taking the<br />
vibrations and manifesting them in its<br />
appearance. In the photos you can see<br />
how the piece adapts to each situation,<br />
but also how the environment or the<br />
person who is interacting with it is<br />
infected by the sensation that the<br />
piece reflects. This is related to the<br />
property <strong>of</strong> <strong>water</strong> to resonate with<br />
other systems composed <strong>of</strong> <strong>water</strong>,<br />
so that a calm <strong>water</strong> can spread that<br />
feeling to plants, animals, people or<br />
in general to the living beings that<br />
surround it. In the case <strong>of</strong> the piece,<br />
it can be seen how the two people are<br />
infected by the feeling <strong>of</strong> warmth and<br />
comfort that the blanket transmits<br />
and in the case <strong>of</strong> the bed and the<br />
s<strong>of</strong>a, the surfaces attract and invite to<br />
interact with them.<br />
108 109
Picture taken by: Laura Zamudio<br />
Image 46. Blanket on the bed Picture taken by: Laura Zamudio Image 47. Blanket on the bed<br />
110 111
Picture taken by: Laura Zamudio<br />
Picture taken by: Laura Zamudio<br />
Image 48. Blanket on the bed - volumes<br />
Image 49. Blanket on the s<strong>of</strong>a<br />
112 113
Picture taken by: Laura Zamudio<br />
Picture taken by: Laura Zamudio<br />
Image 50. Blanket and person on the bed<br />
Image 51. Blanket person on a chair<br />
114 115
and<br />
Another important element for the<br />
final piece is the graphic development<br />
<strong>of</strong> the brand, logo, packaging and<br />
other elements necessary for the<br />
completion <strong>of</strong> the product as such.<br />
For the logo (image 52), we sought to<br />
generate a composition in which the<br />
use <strong>of</strong> geometries stood out, combining<br />
different shades <strong>of</strong> blue to give the<br />
appearance <strong>of</strong> <strong>water</strong>. Regarding the<br />
distribution <strong>of</strong> the modules, the idea<br />
was that the composition should have<br />
movement and that, although it gives<br />
the appearance <strong>of</strong> being a drop <strong>of</strong><br />
<strong>water</strong>, it also shows that it transforms<br />
and evolves, making reference to the<br />
capacity <strong>of</strong> <strong>water</strong> to adapt and transform<br />
itself according to the environment.<br />
Likewise, two versions were proposed,<br />
one for a white background and the other<br />
with a dark blue background, as part <strong>of</strong><br />
the brand’s graphic identity.<br />
Among the complementary elements for<br />
the product, we developed the packaging<br />
design, the label that accompanies it and<br />
another style <strong>of</strong> label that highlights the<br />
brand. As for the packaging, we decided<br />
on a blue fabric lining (image 53), with<br />
a transparent part through which the<br />
blanket can be seen; the lining works<br />
both for transportation and storage. The<br />
blanket is accompanied by 3 cushions<br />
with the shapes <strong>of</strong> the modules and a<br />
cover for washing, with which it can be<br />
washed in the washing machine, when it<br />
does not have a roller or it can be washed<br />
in the laundry.<br />
As for the labels, the first one (image<br />
54) corresponds to the packaging<br />
label, which contains the product<br />
specifications such as measurements,<br />
elements it contains and the care that<br />
must be taken to ensure its durability.<br />
While the second (image 55) is an<br />
advertising element <strong>of</strong> the brand, which<br />
contains the logo and the brand. These<br />
elements accompany the piece and<br />
characterize it as a finished product.<br />
116 117
LA GEOMETRÍA DEL AGUA<br />
LA GEOMETRÍA DEL AGUA<br />
Image 52. Logos on white and blue backgrounds<br />
Imagen 53. Packaging<br />
118 119
LA GEOMETRÍA DEL AGUA<br />
Manta memoria del agua<br />
Cama doble<br />
Paño lency<br />
Manta<br />
1 mt x 2mts<br />
3 cojines<br />
34 cm<br />
1 funda para lavado<br />
50 cm x 65 cm<br />
Lavar en la funda de lavado<br />
Lavado ropa delicada<br />
Lavar con agua fría<br />
Secar en secadora<br />
Planchar a vapor<br />
No presionar al planchar<br />
No retorcer<br />
No estirar<br />
No usar clorox<br />
Image 53. Packaging label<br />
Image 55. Advertising label<br />
120
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