Geometry of water

geometry La geometría 

of del water agua

The geometry of water 

Laura Zamudio Robles 

Degree project 

2020-1

index 

INTRODUCTION 

1. Introduction 

2. Why fractals? 

3. Understanding fractals 

a. The father of fractals 

b. What are the fractals? 

c. Supporting theories 

d. Types of fractals 

4. Application of fractals 

a. Fractals in medicine 

b. Fractals in art 

c. Fractals in nature 

d. Analysis / conclusions 

5. Searching of the fractal for the proyect 

a. Water as a research path 

b. Dr. Mu Shik Jhon and the geomety of water 

c. Cluster - the water alpabeth 

d. Masaru Emoto and the memory water 

e. Analysis / conclusions 

4 

5 

7 

27 

39 

6. Value proposal 

a. Initial approach 

b. Moodboards analysis 

c. Moodboards and references 

d. Geometries 

e. Analysis for design 

7. Final pieces 

a. Prototypes 

b. Design proposal 

c. Pieces 

d. Brand 

8. Bibliography 

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83 

121 

This paper reports the conceptual and material 

development of a design project inspired by 

the fractals of water. The project begins with 

a research on fractals, in order to understand 

them from their mathematical approach and 

their representations in different fields of 

knowledge. From this research, two main 

categories of fractal classification are 

determined: mathematical and natural fractals. 

For the development of the project, natural 

fractals are taken as the object of research and 

it is focused on the study of water as an element 

analyzed from fractality. From this, Dr. Mu Shik 

Jhon’s research on the molecular structure of 

water is used, which results in the formation of 

geometric structures called clusters. 

From these structures, the development of the 

proposal begins, in which the product design 

seeks to represent the geometry generated 

by the cluster in the water, making use of four 

principles that arise from research on fractals 

and water and that guide the development of 

prototypes and final pieces. These principles 

are self-similarity, proportionality, dynamic 

systems that reflect the vibrations that modify 

the structure of water and the cluster as its 

constitutive pattern. 

4 5

why fractals? 

Nature, architecture and art are just some 

of the multiple contexts in which fractals 

can be found and although we often do 

not recognize them as such, they manifest 

themselves in organized forms that produce 

an inexplicable beauty. In my case, these forms 

have attracted my attention since I was very 

young and involuntarily I have approached 

them throughout my life through drawings 

of mandalas, observing the rosettes or the 

characteristic forms of Islamic art, and more 

recently in the contemplation of the forms of 

nature. I do not remember the moment when I 

discovered that all these forms were fractal 

representations, but my interest in these forms 

has remained and has been complemented and 

developed throughout my career as a designer, 

so I decided to go into this research for the 

development of my degree project. 

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Entendiendo 

understanding 

los fracals fractales 

Foto recuperada de: https://flic.kr/p/pvgfEQ

The theory of fractals is relatively recent, if 

compared to other scientific theories that have 

generated important changes in the way of 

understanding the world. Fractal geometry has 

become the tool for studying irregular shapes, 

especially the apparently random shapes of 

nature. Due to the change that fractals represent 

for the study of the world, it has awakened a great 

interest in studying them from different branches 

of knowledge, not only from the sciences. 

the father of fractals 

The first to use the term “fractal” was the 

Polish mathematician Benoit Mandelbrot. 

However, before understanding fractals, it 

is important to know a little more about his 

life and the questions that led him to the 

development of the fractal theory and for 

which he is called the father of fractals, even 

though he was not the first to talk about them. 

Benoit Mandelbrot was born in Warsaw in 1924 

into a Jewish family, which forced him to migrate 

to France during the First World War and to go into 

Picture 1. Benoit Mandelbrot 

Photo recovered from : https://flic.kr/p/5Z5v4U 

hiding during the Second World War. Mandelbrot 

migrated with his family in 1936 to France where his 

uncle took charge of his education. Subsequently, 

he entered Paris to study, but had to withdraw due 

to the outbreak of the Second World War, which 

motivated him to become self-taught. Due to his 

unconventional education, he develops a great 

interest in geometry and the observation of nature 

(Sanz, 2019). 

For Mandelbrot, nature was a complex system that 

could not be understood from the traditional laws 

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10 11

of Euclidean geometry. In nature there were 

irregular and chaotic forms, which apparently 

had no order or way to be understood. However, 

for Mandelbrot there were patterns, so he 

dedicated his studies to analyze the irregular 

forms of nature and to find a mathematical 

formula that could explain what he observed. 

During his observations he was able to 

determine that shapes in nature such as 

clouds, coastlines, rivers, trees and so on, 

had self-similar shapes, meaning that when 

looking at a small portion of the shapes, 

it is similar to the whole of what is being 

observed. An example of this is tree branches, 

which when taking a single one gives the 

impression of looking at the tree on a smaller 

scale, as seen in image 2 (Ventura, 2019). 

This principle of self-similarity would 

allow Mandelbrot to establish that his 

observations corresponded to a new 

type of geometry, which explained the 

irregular and chaotic forms of nature. 

The first studies on this new geometry were 

presented in his first article published in 

Science magazine in 1967, entitled “How long is 

the coast of Great Britain?” (Gaussianos, 2010). 

In this article he posed the situation in which 

someone would want to measure the coast of 

Great Britain, in which case he would encounter 

the problem that, being an irregular surface, the 

final measurement will depend on the unit of 

measurement used, so that the forms of nature 

cannot be studied from classical mathematics. 

Image 2. Branching of a tree showing its fractal formation. 

Photo recovered from : https://flic.kr/p/dNY6uW 

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Image 3. Simulation of the Mandelbrot ensemble 

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what are fractals? 

From the above and in the same article 

Mandelbrot uses for the first time the term 

fractal, which comes from the Latin Fractus 

and refers to the self-similar property that 

these possess, i.e. it is a whole that is formed 

of several similar parts at different scales. It 

can also be understood from the example of 

the measurement of the coast, in which it is 

stated that while normally the measures used 

are whole numbers such as 1 or 2, for the coast 

of Great Britain it will be more or less 1.25. 

Finally, it is necessary to mention that 

Mandelbrot relies on previous studies that 

dealt with the subject, such as the studies 

carried out by the French mathematician 

Gaston Julia. However, none of them 

was able to make a demonstration that 

would support the approaches of fractal 

mathematics. Mandelbrot was able to 

make the first simulation of fractal theory, 

because he worked for IBM and there he had 

total freedom to develop his theory and the 

technological tools (computers) from which 

he was able to arrive at the equation Zn=Z2+C, 

with which he generated the “Mandelbrot 

Set” (see image 3). Because it was he who 

gave it its name and managed to make the 

first simulation of a fractal, he is known as 

the father of fractals and is credited with 

the discovery of a new geometry capable 

of mathematically explaining nature. 

As already mentioned, Mandelbrot was not 

the first to study the irregularity of nature, 

but by proposing fractal geometry he opened 

the door for new researchers from different 

branches of knowledge to become interested 

in studying and understanding this new world 

of mathematics. Therefore, one would think 

that the first thing necessary to enter into such 

research would be to ask oneself what are 

fractals, which is paradoxical is that there is no 

precise definition, not even Benoit Mandelbrot 

managed to propose a satisfactory definition. 

However, it is possible to reach an 

understanding of them by understanding the 

characteristics that they possess and that 

differentiate them from traditional geometry. 

To begin with, fractals are geometric figures 

fractioned in different scales and self-similar, 

this means that when observing a fractal and 

regardless of the scale in which it is seen, it 

will always be similar since it is composed of 

several copies similar to the original figure, 

but in smaller scales. This property is of great 

importance, since in nature there are no 

perfect shapes, understood from Euclidean 

geometry, since being shapes composed of 

other smaller ones generates the irregularity of 

nature (What are fractals? - Silicon News, n.d.). 

Another characteristic found in fractals is 

that they have finite areas but their length 

or perimeter is infinite, to better understand 

this property we will take as an example the 

article about the coast of Great Britain. In 

this example we can say that Great Britain 

has a defined land area, however, when trying 

to measure the perimeter of the coast, the 

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esulting measurement will be different 

depending on the unit of measurement used 

given the irregularity of the surface, therefore, 

it is assumed that by taking a smaller scale 

the measurement will increase, but you can 

always take a smaller scale making the 

length increase to infinity (see image 4). 

Another way in which fractals can be understood 

is from the mathematical principles that 

support them. 

“In mathematical terms a fractal is a shape that 

begins with an object that is constantly altered 

by infinite application of a certain rule. This 

can be described by means of a mathematical 

formula or by means of words.” (Cruz Gomez, 

Perez Abad and Gomez Garcia, n.d.) 

The previous quote can be understood from 

the Mandelbrot Fractal (image 3), which starts 

from the complex equation Zn=Z2+C, where C 

corresponds to the points of the plane that are 

part of the set from the sequence 0, f(0), f(f(f(0)), 

f(f(f(f(0))), ... taking into account that this 

sequence must be performed for each of the 

points of the plane and thus determine whether 

or not they belong to the set. It is because of the 

latter that these equations are called dynamical 

systems, which is also the reason why before 

Mandelbrot and more specifically before having 

access to computers it was impossible to 

perform a simulation that would prove fractal 

geometry from a mathematical explanation 

(What is the Mandelbrot fractal?, 2010). 

It is important to emphasize that the 

contribution of computers corresponds to their 

Photo recovered from: https://vonneumannmachine.files.wordpress.com/2012/11/costagranbretac3b1a.jpg 

Simulation of the measurement of the coast of Great Britain with different scales. 

16 17

ability to perform and process a large number of 

equations, which allows generating figures such 

as fractals that can be observed at different 

scales, making these forms infinite. Given the 

infinite characteristic of fractals, it can be said 

that they are figures in movement, in constant 

development and that they are observed at a 

specific time and scale but not in their totality. 

Although the mathematical explanation of 

fractals is complex to understand, there are 

examples such as the Koch curve (image 5), 

Sierpinski’s triangle (image 6), which show 

the principle of self-similarity from Euclidean 

geometric figures, as for example in the 

Sierpinski triangle, in which we start from an 

initial equilateral triangle and create three new 

triangles from the corners of the first one, this 

process can be repeated infinitely maintaining 

always the principles of self-similarity and 

infinity of the fractal. In the case of the Koch 

curve it can be observed how the total figure 

generates a shape similar to that of snowflakes, 

which allows us to understand the relationship 

between fractal geometry and the shapes of 

nature (Poizat, Sauter and Spodarev, 2014). 

It should be clarified that the review of the 

mathematical principles of fractal theory is 

part of the research, as an element to reach a 

theoretical understanding of fractals and their 

characteristics, but the formulas and theories 

exposed above will not be taken as guidelines for 

the development of the project. 

Photo recovered from: http://images.treccani.it/enc/media/share/images/orig/ 

system/galleries/Enciclopedia_della_Matematica/fig_lettk_00530_001.jpg 

Image 5. Representation of Koch’s curve 

Photo recovered from: http://scpdptomatematicas.blogspot. 

com/2017/11/construccion-de-nuestro-fractal.html 

Image 6. Representation of the Sierpimski triangle. 

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supporting theories 

As mentioned above, Benoit Mandelbrot 

studied and complemented his studies for 

the approach of fractal geometry, based 

on several theories previously proposed, 

some of these approaches were close to 

fractal theory directly and others were not. 

The first theory of which it is necessary to 

speak is the Julia set, proposed by the French 

mathematician Gaston Julia. This theory 

corresponds in general terms to the theory 

proposed by Mandelbrot, since the latter based 

his investigations on those carried out by Julia, 

with the difference that in this theory the term 

C of the equation corresponds to a complex 

number, which may or may not be part of the set, 

making the required calculations even greater. 

Another difference, and perhaps the most 

important, is that Gaston Julia was never able 

to see a simulation of a fractal, so his theory 

was left aside in mathematical research and 

only taken up again by Benoit Mandelbrot. 

Another of the theories that greatly 

influenced the approach of fractal geometry 

was what is known as chaos theory or the 

butterfly effect, since, as mentioned, fractals 

are the geometry of the chaotic forms of 

nature. Chaos theory was proposed by the 

meteorologist Edward Lorenz in 1960, while he 

was conducting mathematical experiments 

to predict the weather, using 12 equations. 

According to the knowledge of the time, it was 

believed that the results obtained from the 

same origin (a number with at least 2 known 

decimal places) would result in a pattern 

similar to the previous one, however, being 

dynamic systems, even the smallest change 

in the beginning would generate a change in 

the subsequent development. To explain the 

above, we take as an example the movement of 

a butterfly flapping its wings, which represents 

a small disturbance in the air but can generate 

a tsunami on the other side of the world. 

From this discovery Lorenz focused his studies 

on the development of 3 equations that were 

derived from the 12 previously mentioned, 

but that had the same effect, thus arriving at 

the equations that would explain the chaos 

theory and that later would be proven to 

describe the movement of a water whirlpool 

(image 7), being a fractal form of nature (Cruz 

Gómez, Pérez Abad and Gómez García, s.f.). 

One of the main characteristics of fractals is 

their self-similarity, however, it is worth noting 

Photo recovered from: https://www.elcompositorhabla.com/corps/ 

elcompositorhabla/data/resources/image/Ruth/Lorentz%20Attractor1.png 

Image 7. Simulation of the Lorenz equations 

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that each of the parts that compose it complies 

with the ideal or harmonic proportions. This idea 

of harmonic proportions comes from Pythagoras’ 

theory of numbers and proportions, so it can be 

thought that there is a relationship between 

Pythagorean theories and fractal theory. 

For the fractal theory the proportions are of 

great importance and are appreciated between 

the different elements that compose a fractal, 

so a relationship with the theory of numbers 

and proportions of Pythagoras is appreciated. 

This theory proposes a relationship between 

music and mathematics, since musical tones 

and intervals can be expressed in numerical 

ratios, being the comparison of one quantity 

with another, making the intervals concordant 

or discordant (harmonic or disharmonic) with 

each other. 

To better understand the relationship between 

music and mathematics, an instrument called 

Monochord (image 8) was used, which consists 

of a single string supported on a wooden 

base (like a guitar), which is stretched until 

it reaches a fundamental sound determined 

as a tone. Subsequently, it was divided 

into 12 equal parts, in such a way that the 

proportion between the string fragments 

was maintained, producing different sounds 

depending on where the string was stepped 

on. Using this instrument, it was possible to 

define the octave, the fourth and the fifth, 

which are the concordant sounds. From these 

sounds the general property of the harmonic 

arithmetic mean ab=mh was generated, where 

there is a proportionality between ab equal 

to that between mh. (Correa Pabón, 2006). 

Proportions and harmony are fundamental 

factors in the world of fractals, since they are 

intrinsic characteristics of these, making them 

beautiful to the eye and being the manifestation 

of an order and a reason for being in the apparent 

chaos of nature. It is worth noting that these 

harmonic patterns of fractals and their beauty 

are one of the main reasons why they arouse so 

much interest in different disciplines. 

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docs-images/70/63756304/images/166-0.jpg 

Image 8. Pythagorean Monochord 

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types of fractals 

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org/wikipedia/commons/4/4b/Fractal_fern_explained.png 

Image 9. Accurate self-similarity 

Since fractal theory is very recent and under 

constant study, it is not completely defined 

and classified, so different forms and 

parameters have been proposed to classify 

the known fractals, highlighting mainly their 

properties of self-similarity and linearity. 

Regarding the classification by self-similarity, 

fractals are divided into three groups; the 

first are those with exact self-similarity 

(image 9), which means that they appear 

identical in their different scales, these 

are fractals generated by iterated function 

systems, which is a program specialized 

in the generation of fractals. The second 

group corresponds to fractals with quasisimilarity 

(image 10), which means that in 

their different scales they are spread out but 

not identical. Finally, there are fractals with 

statistical self-similarity (image 11), which 

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culturacolectiva.com/content/2013/03/fractales.jpg 

Image 10. Quasi-likelihood 

Photo recovered from: https://flic.kr/p/75Rior 

is the weakest because the only condition 

is that it complies with the numerical or 

statistical measures in its different scales. 

For the classification by linearity the division 

is given in two groups; the first are the linear 

fractals (image 12) that are constructed 

from a change in the variation of its scales, 

but without losing the equality in all its 

scales. On the other hand, the nonlinear 

fractals (image 13) are created from 

complex distortions generating a similar 

but not identical structure for the different 

scales of the fractal (Fractal Classes, 2015). 

Finally, we can speak of a third classification 

that corresponds to the mathematical 

fractals or fractals created on computer 

(image 14) and natural fractals (image 15). 

Photo recovered from: https://flic.kr/p/7LF3Wb 

Image 12. Linear fractal 

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Image 13. Non-linear fractal 

Statistical self-similarity 

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Photo recovered from: https://flic.kr/p/5bMnJa 

Image 14. Mathematical fractal 

Mathematical fractals are called fractal 

sets and are defined from mathematical 

formulations and simulated in computers, 

some examples are the Mandelbrot set, or the 

Koch curve. On the other hand, natural fractals 

are those found in nature so they are not so 

precise in their different scales, however, 

these are the result of evolution processes 

making them more complex (Gómez Cumaco, 

2009). Additionally, natural fractals do not 

have infinite scales, but they do comply with 

the self-similar characteristic. 

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originals/95/28/3f/95283fdc7c216f0fc7e90bc5bade968c.jpg 

Image 15. Natural fractal - Aloe polyphylla 

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aplicaciones applicationde 

of fractals 

los fractales 

Foto recuperada de: https://flic.kr/p/25PSiWK

Fractals in medicine 

Fractals, being a relatively new theory, but 

mainly for having the ability to explain a large 

number of aspects of nature, which so far 

seemed not to follow any kind of principle, 

have aroused an interest and fascination 

for different branches of knowledge. In 

the following, we will present some of the 

applications that have been given to fractals, 

which in my opinion are contrasting because 

they are seen from apparently very different 

disciplines, but which allow us to see the 

versatility of fractal theory. 

In the human body a great variety of fractal 

structures can be found, so the fractal 

theory is very useful in the field of medicine 

especially to understand the functioning 

of these systems, some examples are the 

nervous system (image 16), circulatory 

or pulmonary system. In the case of the 

circulatory system, fractal formation is what 

ensures that the blood pressure is constant 

throughout the body, so that all cells can 

receive the necessary supplies carried by 

the blood. Another case is the prediction of 

diseases such as osteoporosis from the first 

changes in the bones. (Cruz Gomez, Perez 

Abad and Gomez Garcia, n.d.) 

Image 16. Nervous system 

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AAAAAAAAUiY/Rx46lYJsD6E/s1600/Sistema-nervioso-neuronas-biologia.jpg 

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Fractals in art 

Fractals arouse great interest and attraction 

for those who see them, due to the beauty 

produced by the creation of harmonic and 

proportional patterns. The first fractals 

proposed in the world of art emerged since 

the appearance of fractal theory in 1980 from 

different computer experiments, however, 

these were initially very similar to each 

other, so they were not considered pieces 

of art. As the study in programming and the 

advances in computers became possible to 

create new fractal patterns by experimenting 

with new mathematical formulas or by 

making variations in the existing ones. 

From 1995 onwards, the development of fractal 

art began to focus on color patterns, since 

the experimentations based on mathematical 

formulas had been almost entirely developed. 

The experimentations from color consisted of 

developing algorithms that allowed coloring 

the different points of the plane in different 

ways, depending on given conditions, such as 

whether or not they belonged to the fractal 

set, or the oldest one which is by escape 

time, which consists of determining whether 

the point of the plane tends to infinity or not, 

when subjected to the fractal algorithm. The 

most important thing in the development 

of color algorithms is that it allows to 

generate very different compositions from 

the same fractal base. (Calonge, 2013). 

One of the main discussions about fractal art 

has to do with whether the images generated 

by computers can be considered works of art 

or not and where is the creativity of the artist 

during the simulation of an algorithm. Faced 

with these questions, some fractal artists such 

as Jean-Paul Agosti (image 17) or Susan Conde 

decided to dedicate themselves solely to 

making works based on their manual skills with 

painting, highlighting the fractality of nature. 

However, it should be noted that the use of 

fractal patterns as inspiration for art has 

been known long before fractal art. Cultures 

such as the Arabian, used patterns of nature, 

to make the paintings of their mosques, as 

seen in the Asfahan Mosque or in the Indian 

culture with the development of mandalas 

known as Kolam (Poizat, Sauter and Spodarev, 

2014). Another example that has been found 

between fractal theory and art is the work of 

the painter Jackson Pollock, who belonged to 

abstract expressionism and who developed the 

techniques of Action-painting and Dripping. It 

Photo recovered from: https://biblioteca.acropolis.org/wpcontent/uploads/2014/10/Jean_Paul_Agosti-Harmonie.jpg 

Image 17. Jean-Paul Agosti, Garden of Metamorphosis. 

32 33

should be noted that Pollock did not create 

his works thinking in the fractality of nature, 

but inspired by the rhythms of nature and it 

was the physicist Richard Taylor, who during 

an artistic experiment in the Yorkshire Park, 

in which they mounted a structure and let 

the paint spread without any control during 

a storm, obtained as a result a canvas that 

reminded him of Pollock’s work. After the 

experiment Taylor decided to investigate 

Pollock’s work and found that his painting 

techniques consisted of spilling paint and 

other materials using his whole body, while 

following the rhythms of nature or the chants 

of the Navajo Indians. During the analysis of 

the works Taylor used the technique of Boxcounting 

in which he divides the work into 

boxes and to study each part until completing 

the work, with this technique he discovered 

the fractal character that Pollock’s works 

have (image 18) and that was perfected 

throughout his work. From this discovery it 

was found that the rhythms of nature, being 

chaotic, are fractal. (Paissan, 2015). 

Imagen 18. Jacson Pollock, Number 1 

Photo recovered from: https://flic.kr/p/otURgt 

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Fractals in nature 

Although it has been said from the beginning 

that the fractal theory comes from observation 

and the desire to understand nature, the most 

explored examples are usually mathematical 

and those made on computer, so I consider it 

important to explore the fractals that exist in 

nature and that as Pollock said are the result 

of life itself. Since the beginning of humanity, 

nature has 1 always been attributed a divine 

character, given its capacity to surprise by 

being chaotic and incomprehensible on many 

occasions, this divine character of nature is due 

to the harmonic and proportional forms that 

are created and which Mandelbrot describes as 

fractals. 

1. Throughout human history, within diverse cultures around the world, 

one of the main characteristics attributed to divinity is aesthetics. With 

geometry, rhythm and chromatics as three of the favorite resources of 

this divine discourse, nature reaches the most spectacular and at the 

same time the most discreet divine manifestation as a hyper-aesthetic 

paradox”, (Villar, J., 2010). 

Nature is characterized by evolving and 

developing while maintaining order, harmony 

and balance among all beings that exist and in 

themselves, so harmony is a feature that stands 

out. One of the differences that exist between 

natural and digital fractals has to do with the 

infinite characteristic, since in nature they do 

not have this property and generally they come 

to present a specific amount of self-similar 

scales. However, natural fractals retain the 

property of being changing or dynamic systems, 

as they are constantly evolving which in turn 

makes them more complex, from this principle 

Mandelbrot presents two fundamental variables 

for their understanding; irregularity at the level 

of form and pattern at the level of rhythm. 

(Villar, 2010). 

Finalmente, cabe resaltar que los fractales 

se encuentran en una gran cantidad de 

fenómenos naturales tales como, hongos, 

plantas, truenos e incluso en el universo, lo 

que nos indica que desde el elemento más 

pequeño hasta lo más grande se rige por los 

principios fractales. 

Photo recovered from: https://flic.kr/p/aUFgL 

36 37

Analysis / conclusions 

After understanding the fractal theory and the 

different examples, both from simulations, 

as well as in art and in nature in general, it is 

possible to determine that fractals are divided 

into two large groups, mathematical and 

natural, for the development of this project I 

will take as a point of interest the natural ones. 

Fractals are dynamic systems with 

figures in constant motion, which can be 

observed in time. In the case of natural 

fractals we can determine that they are 

systems with two main characteristics, 

proportionality understood as irregular 

patterns that determine the shape of the 

fractal, and self-similarity, which generates 

rhythms, resulting in fractal composition. 

Proportionality is a characteristic that 

is understood from the irregularity that 

characterizes nature and that is governed 

under the chaos theory, or more specifically 

it is called chaos geometry. This theory 

states that a change, no matter how 

small, in a dynamic system will generate 

a major reaction in the development of 

the system, due to the fact that these 

systems are highly sensitive to change and 

are in constant development. In addition, 

it should be noted that these variations 

preserve the proportional relationships 

between all the elements of the system, 

since the changes occur progressively. 

In the case of natural fractals, changes come 

from the context that surrounds them and 

from the fortuitous events that affect them. 

An example of this can be the growth of a 

tree branch that, when exposed to a great 

force produced by the wind, can bend or even 

break and give rise to a new branch, which 

integrates harmoniously with the whole. 

On the other hand, self-similarity is the 

main characteristic of fractals and is 

directly related to the chaotic development 

of dynamic systems, since, as I mentioned 

before, changes in the system occur 

progressively, which indicates that there are 

rhythms in the development of the system 

that give rise to fractal compositions. These 

compositions are the result of the union 

of multiple parts similar to each other, but 

which are in different scales, or in different 

degrees of development, taking into account 

that fractals are observed in a certain time 

and not in their totality. 

38 39

Searching of the fractal 

En busca del fractal 

for the proyect 

para el proyecto 

Foto recuperada de: https://flic.kr/p/aasMrK

Si bien la naturaleza y los seres que 

pertenecemos a ella somos sistemas 

dinámicos y en constante cambio, estos 

tienen un ritmo o una secuencia en la que 

se van dando. Estos ritmos son diferentes y 

esenciales para el desarrollo de cada sistema, 

son los responsables de que la naturaleza 

fluya y en el caso de los fractales, que tengan 

un orden interno perceptible y repetible, a 

esto se le conoce como ritmo dinámico. (“El 

Ritmo de la naturaleza en nosotros – Mindalia 

Noticias y Artículos”, s.f.) 

leaves following the theory of the golden 

ratio; the roots of some plants such as 

mangroves or the Guacari tree, which have 

adapted to their contexts by making their 

roots grow out of the ground, evidencing their 

fractal growth present both in the branches 

and in the roots and among themselves; and 

the third case corresponds to the movements 

or forms generated by water in nature, rivers, 

waterfalls, sea, etc. , in which their fractal 

character is not always completely evident, 

but which when observed in detail reveal a 

large number of fractal patterns. 

Taking into account the characteristics 

defined by proportionality and self-similarity 

in fractals, select three possible avenues 

of investigation; plants such as succulents, 

which have a fractal arrangement of their 

Photo recovered from: https://flic.kr/p/a3fD1E 

42 43

water as a research path 

Dr. Mu Shik Jhon and the geometry of water 

As previously mentioned, natural fractals are 

dynamic systems, which must be understood 

from their capacity to flow and evolve. For 

this reason I decided to investigate the 

movements of water, since it is an element 

where these characteristics are evident 

and constant. However, this makes it appear 

to be a stable system when viewed on a 

large scale, which only shows its fractal 

character when observed on smaller scales, 

so I find it intriguing to delve into its different 

scales both from the contexts in which 

it is found, as well as from its states and 

the characteristics it has in each of them. 

Additionally, during the investigation of the 

three possible paths I found that water, 

although it is mentioned on several occasions 

as an element where the principles of 

fractality can be observed, has not been 

studied or at least there is not as much 

information as in the case of plants. Due to the 

lack of information that I could find, I found it 

intriguing to go into the investigation of water 

to understand it under the fractal principles. 

In the investigation of the fractal character 

of water I found two scientists who have 

focused on the study of this element, 

finding two points of view from which water 

is studied at different scales and states, 

but resulting in a fractal understanding of 

water from its molecular composition to its 

different manifestations in nature. 

Water is an element that can be in three 

states, solid, liquid and gaseous, so it can be 

transformed and become part of everything 

that exists in the world from living beings 

to the atmosphere, but more importantly 

it is constantly transforming to transport 

energy. It is precisely these changes and 

the characteristics it has in each of its 

states, which arouses the interest of the 

Korean chemist and physicist Mu Shik Jhon, 

who states that water can have a hexagonal 

or pentagonal molecular structure, with 

important differences between the two. 

Water is the fundamental element for the 

life of all living beings, it is responsible for 

transporting vital energy, however, to fulfill 

this function it must have a good quality. The 

good quality of water is due to its molecular 

structure, which is explained by Mu Shik Jhon. 

Dr. Mu Shik Jhon states that water needs 

to move and flow to be charged with 

energy, this conclusion is obtained from 

studying water in its natural state, where 

movements persist in the form of vortices, 

helicoids, whirlpools and in general chaotic 

movements of water. Mu Shik Jhon explains 

that these movements are fundamental for 

the purification of water, since they allow 

water molecules (H2O) to attract other 

disorganized water molecules and structure 

them into organized molecules with a 

large amount of trapped energy. From the 

crystalline water approach and its relation 

to the structural order of water molecules 

Mu Shik Jhon proposes that in nature one 

can find molecules with hexagonal or 

44 45

pentagonal formations. The former are those 

found in natural, crystalline water, especially 

in what is known as super-frozen water such 

as that of glaciers. This structure provides the 

greatest benefits for living organisms, since 

it generates strong bonds that allow large 

amounts of energy to be contained in addition 

to containing more oxygen atoms. In terms of 

its behavior, this structure allows water to 

flow naturally and facilitates the formation 

of harmonic crystals; these hexagonal 

formations are known as clusters (image 21). 

On the other hand, pentagonal structures are 

commonly found in water when it has been 

exposed to ions of elements such as fluorine 

or magnesium or others that are used to 

decontaminate water for human consumption, 

in other words, the water that comes out of 

the taps in cities. This structure breaks 

down easily, so it contains low amounts of 

energy and oxygen. (“The Geometry of Water 

(I). Biological aspects.”, 2015) 

Photo recovered from: https://www.researchgate.net/profile/Pei_Zhong_Feng/publication/301933759/figure/fig7/ 

AS:360516078850052@1462965139883/a-molecular-structure-of-water-b-molecular-structure-of-ice-crystals.png 

Cluster, hexagonal formation of water molecules. 

46 47

Cluster - the water alphabet 

The cluster is the hexagonal structure 

that is formed from the union of multiple 

water molecules (image 21), these 

unions occur through hydrogen bonds 

which are weaker than the bonds of the 

water molecule (H2O), so they allow it to 

be malleable and take multiple forms. 

It should be noted that this cluster 

composition occurs in the liquid state of 

water, and it is also important to bear in 

mind that this structure is changeable 

and adapts to the vibrations that pass 

through the water and influence it. 

This last characteristic of the clusters 

is the one that allows us to understand 

the idea of the memory of water, 

since as it happens with the crystal 

formations (solid state of water), where 

each crystal starts from a hexagonal 

structure, none is equal to another since 

the hexagonal formations that they reflect, 

vary depending on the vibration to which 

the water has been exposed. The same 

happens with the clusters, where although 

they are all formations between tetrahedra 

(hydrogen bridge junctions), which together 

form hexagons, these are not the same in 

all cases, since they depend on the vibration 

that has altered them, so if we could 

understand these hexagonal formations 

we could know the vibration that altered 

the water (García Flórez, n.d.). This is the 

principle used in homeopathic medicine, 

where water maintains its healing effects 

without the need of having the specific 

substance. 

It is from these hexagonal formations and 

their variations that water has the capacity 

to retain a large amount of energy or 

vibrational waves (such as radio waves), 

which is why they are known as liquid 

crystals, since their molecules are so 

precisely organized that they can contain 

the energy. However, hydrogen bridges 

are weak enough to allow water to mold 

and accommodate both the vibration that 

affects it and the body that contains it. 

By the latter I refer to the ability of water 

to become part of a river and then be 

transformed into the aloe vera of a plant or 

the blood of an animal. 

Photo recovered from: https://unsplash.com/photos/iyA6oTK6vig 

Image 22. shapes in the water 

48 49

Masaru Emoto and the memory of water 

The beginning of life as we know it has 

been said to be the product of water, that 

is why since the most ancient civilizations 

man has tried to understand this essential 

element for life and that constitutes 

everything in nature. For indigenous 

communities there has always been a 

connection with nature and a relationship 

of appreciation and respect, where she 

gives us what we need to live and we use 

it without breaking the balance, or at 

least that is how it is for ancient cultures, 

in modern science this has been lost. 

The Japanese physician Masaru Emoto, 

dedicated his research to the study of 

water and its benefits for homeopathic 

medicine. His research focused on the 

study of the formation of crystals in 

water, where he demonstrated that, when 

water was exposed to different vibrations, 

it generated crystals with different 

characteristics that communicated the 

state of the water and its alterations. 

Para sus experimentos Emoto tomó aguas 

de distintos lugares tanto naturales, 

como agua procesada o de los grifos en 

las ciudades. El objetivo era comprobar si 

existía alguna diferencia entre estas aguas 

cuando se congelaban, lo que descubrió 

fue que en la mayoría de los casos el 

agua proveniente de los grifos o lugares 

contaminados no generaba cristales 

completos o en general no formaba 

ningún cristal. Sin embargo, al someter 

un mismo tipo de agua a la influencia de 

distintas vibraciones se formaban cristales 

que recordaban la vibración a la que se 

había expuesto, es así que las palabras 

o sonidos positivos generaban cristales 

bellos, armónicos y completos (imagen 

23), mientras que aquellos expuestos a 

palabras o sonidos negativos y agresivos 

no generaban cristales o eran incompletos 

(imagen 24). 

These experiments led Emoto to propose 

that, on the one hand, water is an element 

that can be influenced, that changes 

according to the vibrations to which it 

has been exposed, so it can be said that 

water is capable of remembering and 

reflecting that which has influenced 

it. In addition, by performing the same 

experiment with water in its free and 

natural state and comparing it with treated 

or purified water from cities, he concluded 

that for water to form crystals it must be 

in equilibrium and pure, so the principles 

of life in water are the ability to move, 

change and flow, since it is through these 

movements or states of chaos that water 

is revitalized by being charged with energy. 

During his studies Emoto managed to 

determine that the most beautiful crystal of 

all is formed under the influence of the words 

“love and gratitude” (image 23), two words 

that reflect positive energies, but are also 

opposites. The first one, love, means to give 

or deliver a positive feeling towards another 

being, while gratitude means to receive a 

feeling with the same value as the previous 

one, but in the opposite sense, which leads 

50 51

us to understand that these two words 

generate a balance between them or what 

Emoto calls is the capacity to resonate. 

The capacity to resonate indicates that the 

water of a being or a place is influenced by 

vibrations that reach it and that it is capable 

of transmitting these vibrations to another 

being or place that contains water, it is due 

to this capacity that it is said that water has 

memory. For Emoto this is fundamental for 

medicine, because we are beings mainly 

composed of water and therefore when we 

receive positive and balanced vibrations, 

these will generate the same effect in the 

water of our body (Emoto, 2020). 

Photo recovered from: http://www.menteyexito.org/wp-content/ 

uploads/2017/04/c092ae655d768526719c0eadcb38374e.jpg 

Image 23. Crystal formed by the vibration of the words “love and gratitude”. 

Photo retrieved from: the book The miracle of water 

Image 24. Crystal formed by the vibration of the words “I can’t”. 

52 53

Analysis / conclusions 

As stated at the beginning of this research 

on water, the objective is to be able to 

understand the different scales of this 

element and thus identify the fractal 

character that constitutes it. If we return 

to the theories and mathematical models 

used to explain fractals, we find the Koch 

curve, which resembles the shape of a 

snowflake (Poizat, Sauter and Spodarev, 

2014). It is from this approach that we 

can understand the relationship between 

fractals and water in its different states. 

As we have already seen, Masaru Emoto’s 

crystals show the property of water to be 

influenced by the different vibrations that 

constitute the world and life itself and 

then reflect, from hexagonal formations, 

different crystals that contain the intrinsic 

properties of fractals; self-similar, 

irregular and harmonic formations that 

come from a dynamic and chaotic system. 

Subsequently, it became evident that the 

hexagonal formations that are generated in 

the formation of crystals, which is in turn 

the solid state of water, is directly related 

to molecular formations that are generated 

between different water molecules in the 

liquid state, which are called clusters. 

These formations are of vital importance, 

as they are responsible for retaining the 

energy and incorporating the vibrations 

that travel through the water; it is for 

this reason that it is said that water has 

memory and reflects those vibrations that 

it remembers in its molecular structure; 

additionally, it is this molecular composition 

that generates characteristics in water 

that classify it as a liquid crystal. One of 

these is the malleability of water, which 

allows it to adapt not only to the vibrations 

that influence it, but also to the body that 

contains it, and as mentioned, water is the 

fundamental element for life and therefore 

all living bodies possess it even when it 

manifests itself in different compositions. 

On the other hand, another characteristic 

that is evident from the hexagonal structure 

of the cluster, is that water manifests itself 

as a compact and apparently stable element, 

when observed on a larger scale, such as a 

river, the sea or a lake; in these cases water 

takes on the appearance of being a complete 

surface. 

Photo recovered from: https://unsplash.com/photos/VuBzplNNi0k 

54 55

Propuesta 

Value 

proposal 

de valor 

Foto recuperada de: https://unsplash.com/photos/3Ik7xWYJv3U

Initial proposal 

moodboards analysis 

Understanding the way in which water is 

conformed at the molecular level, from 

clusters and the properties of rigidity and 

malleability that this structure gives to 

water in liquid state, I consider that it is from 

the geometric structures (pentagonal and 

hexagonal), that the fractal character of this 

element is manifested, taking into account 

that from these formations it is possible 

to observe the aspects of self-similarity, 

being an element composed of parts that 

at different scales look similar, and of the 

proportionality that exists between the 

different molecules and that allows the 

generation of stable structures. Additionally, 

it is from these formations that water 

acquires the capacity to adapt and transform 

itself to become an essential part of the body 

that contains it. 

Based on geometric patterns, the proposal 

is to develop surfaces that, like water, 

can adapt to different spaces where 

they interact, reflecting on the one hand 

the malleability of this element to be 

influenced by external bodies, but also the 

rigidity that the patterns give it to be an 

element of containment. These surfaces 

will show multiple proportional patterns, 

based on geometric shapes, reflecting 

the compositions of the clusters and how 

these are a reflection of the elements that 

surround or influence them. 

To understand the forms of water I made 

three moodboars from which I could analyze 

different characteristics of the forms that 

are expressed in it. It should be noted that 

for this inquiry I made an approach to images 

of water, without entering into a chemical 

study to analyze its molecular structures. 

However, based on the discoveries of Masaru 

Emoto, where it is possible to recognize 

the pentagonal or hexagonal shapes of the 

molecular composition of water, from the 

physical manifestation in the form of crystals, 

it was sought that the water in the images 

used reflected geometric shapes from which 

the forms of its molecular structure are 

sought to be recognized. 

Photo retrieved from: https://unsplash.com/photos/lUPHw5bM7HM 

58 59

Frozen fluidity 

The first moodboar reflects water in 

solid state or more specifically in frozen 

formations, so the images used correspond 

to crystals and ice formations in water 

currents. This moodboard has the concept of 

Frozen Fluidity, which is understood as each 

drop is a frozen instant, which in its deep 

interior remains a flowing vitality ready 

to follow its path. The concept describes 

on the one hand the movement of water 

that stops and is trapped in the crystals, 

but also the fluidity that lasts in the water 

currents thanks to the frozen formations 

of its surface, in this case it highlights the 

importance of movement in water because 

despite being contained in a solid state is 

water that has traveled the world, which is 

alive and contains a lot of energy. 

In this case, two important aspects are 

highlighted in relation to fractals, the first 

one is that, as Emoto mentions, only good 

quality water or, taking into account Mu 

Shik Jhon’s research, water with hexagonal 

structure has the property of forming frozen 

crystals and each one is unique, since, as 

it happens with fractals, the water present 

there corresponds to only one part seen at a 

given moment in time. 

60 61

The second moodboar shows water in its 

liquid state, which is found free in nature 

and is affected by the sun’s rays, since it 

is from this mixture that geometric shapes 

are reflected both on the surface as seen 

in the images, and at the bottom of the sea. 

The concept in this case is that of Sinuous 

Malleability, understood as infinite and 

changing water that in its warm mixture 

with the sun’s rays permeates and moves 

those who enter into its being, and speaks 

of the appearance that water takes on when 

influenced by the sun’s rays that intermingle 

to generate a surface that, although liquid, 

appears to be compact but malleable. 

are appreciated at the same time, which 

allows me to infer that the water present 

in these images has a very organized and 

optimal molecular structure as is the case 

of hexagonal formations and that is why it 

manifests itself in this way. 

This appearance can be related to the 

characteristic of water as a liquid crystal, in 

which the properties of a solid and a liquid 

Sinuous malleability 

62 63

Fickle resistance 

Finally, the third moodboard represents 

the delicacy observed in the water, for 

this reason images of water bubbles 

and crystalline water currents are taken 

that allow observing the depth and its 

harmony. For this case the concept used 

was that of Flickle Resistance, which is 

understood as bubbles, soft windows, that 

come and go on the surface and reveal 

the vital immensity that vibrates under 

the flowing mantle, and that speaks of the 

characteristic contrast that this element 

has, which can be as delicate and volatile 

as a bubble that explodes and evaporates 

in a strong breeze, but as strong and 

resistant as a drop of water that collects 

and transports the energy that keeps 

alive the different systems of the planet. 

In this case the purity of the water and 

the fact that it comes from a natural 

stream, which also has a large amount 

of rocks in its soil, allows us to infer that 

the minerals of the soil, the constant 

movement oxygenates and nourishes it 

and therefore we can infer that it is water 

with a hexagonal structure. Additionally, 

we can appreciate the geometric shapes 

that generate the edges of the bubbles and 

that can be interpreted as a smaller scale 

observation of a quantity of water. 

64 65

Moodboards and references 

After analyzing and determining the forms 

of water in the three cases defined above, 

I carried out an exploration of references 

that would allow me to understand 

different representations of the forms I 

was looking for based on materials, colors 

and compositions. After selecting the 

references that responded best with the 

moodboards, I made a second composition 

of each one integrating images of the 

moodboard and images of the references 

in order to better analyze the relationships 

between the two and define their particular 

forms. 

Photo retrieved from:: https://i.pinimg. 

com/564x/3c/2e/63/3c2e6361e1294560aae30241fe9004df.jpg 

Image 26. Mathieu Lehanneur, Liquid marble 

Photo retrieved from: https://www.elisastrozyk.com./wooden-rugs 

Image27. Elisa Strozyk, Fading 

Photo retrieved from: https://dazedimg-dazedgroup. 

netdna-ssl.com/467/azure/dazed-prod/1210/6/1216181.jpg 

Image 28.Iris Van Herpen, AW17 couture Show 

Photo retrieved from: https://design-milk.com/scale-flexible-modularacoustic-partition-system/scale-layerxwovenimage-partition-1/ 

Image 29. Bejamin Hubert, Scale 

Image 25. Karen LaMonte, Dreamscape Drapery Study 

Photo retrieved from: https://www.karenlamonte.com/Contemporary- 

Scultpures-Prints/Drapery-Sculptures-Bas-Relief/i-7dwCKCF/A 

66 67

For the first moodboard, Frozen Fluidity, I 

looked for references where from solid 

and static forms I could transmit fluidity 

and movement, this with the objective of 

understanding in a better way the way in 

which I could transmit two opposite states 

such as fluidity and movement with the 

frozen and static. One of the main references 

is the work of Karen LaMonte and especially 

the work called Drapery sculpture (LaMonte, 

2008) (image 25), which as its name suggests 

are sculptures that resemble draped fabrics, 

this reference allowed me to understand how 

from static forms can generate movement 

from folds or superimpositions that in turn 

generate harmonic pieces. 

In the composition that links the moodboard 

with the references I could find geometric 

shapes that generated the movement and 

fluidity of water without losing the rigidity 

of the materials and of a static piece. In this 

composition you can see a close-up of a 

table by French designer Mathieu Lehanneur, 

more specifically from his Liquid Marble 

collection (“Mathieu Lehanneur ‘Liquid 

Marble’ installation at the Musée des Arts 

décoratifs, Paris - urdesignmag”, 2017) (image 

26), this piece is very interesting for the 

contrast generated between the rigidity of a 

material such as marble and the movement 

generated by the carved forms, additionally 

when viewed in conjunction with the images 

of frozen water similarities are appreciated 

as the color changes that are generated by 

the shadows and brightness product of the 

solid structures. Likewise, other textures 

and structures that show the geometric 

formations that are generated in the frozen 

water and that by their arrangement in space 

and difference in sizes and shapes transmit 

fluidity despite being in a solid state are also 

appreciated. 

68 69

For the second moodboard, Sinuous 

malleability, I looked for the references to 

reflect opposite states in its materiality, 

from something that seems rigid and 

compact, but is formed by parts that give 

it movement or something that I know 

has a lot of movement but is rigid to the 

touch. In this case the main referent was 

the work of the designer Elisa Strosyk who 

uses pieces of wood on fabric to generate 

rigid structures, but with great movement 

(“Elisa Strozyk | Poligom”, 2011)(image 27). 

became evident from this composition is 

the mixture of colors between the different 

modules or forms of water, to give the 

feeling of fluidity and movement in the piece 

without making it look fragmented, but on 

the contrary, the colors give unity to the 

totality of modules. 

In the composition between the referents 

and the moodboard, it is evident the need 

to use modules that generate structure in 

the piece due to their geometry, but also 

that represent a contrast that generates 

movement. Another important aspect that 

70 71

Finally, for the moodboard of Voluble 

Resistance, I looked for references that 

allowed me to understand the delicacy of 

the forms from pieces with transparencies 

and defined edges, which generate the 

sensation of unity and resistance within the 

pieces. Among the references, two stand 

out that use transparencies and edges as 

constituent elements of the pieces, but 

using materials that are contrary in terms 

of their rigidity. The first referent are the 

dresses of the designer Iris Van Herpen, 

these are characterized by the movement 

and fluidity they generate, product of 

the combination of materials and the 

arrangement of the fabrics, also handles 

transparencies as seen in the fabric of 

one of the dresses of the AW17 couture 

show (Hope Allwood, 2017) (image 28), in 

which the use of transparencies and the 

combination of colors evoke me the colors 

and shapes of water. 

On the other hand, there is the reference 

of the designer Benjamin Hubert, who 

created Scale (image 29), which is a flexible 

modular system that has sustainability 

as a fundamental element (Williamson, 

2015). In this piece different modules are 

generated from a hexagonal structure, 

where both solid spaces and empty spaces 

that maintain this same structure are 

appreciated and I relate it to the geometric 

shapes that are generated with the bubbles 

in the water, which can show an internal 

surface or on the contrary may seem empty, 

but maintain the structure of its edges. 

72 73

geometries 

After making the compositions with 

the referents and understanding the 

shapes that were present in each one 

and in the moodboards, I carried out 

two exercises of geometrization of 

the moodboards to better understand 

the shapes of each one and the 

relationship of these shapes with 

the concepts. It should be noted that, 

during the previous exercise, the 

presence of other geometric shapes 

different from the hexagons and 

pentagons that were being studied 

during the research became evident. 

The first geometrization exercise was 

carried out digitally, which allowed 

me to relate the shapes I found in the 

moodboards with the colors present 

in the images and from this to find the 

rhythms generated from the size and 

color of the modules, which is linked to 

the property of self-similarity seen in 

natural fractals. 

With the second geometrization 

exercise, which I performed manually 

on the moodboards, I found more clearly 

the irregular patterns that are generated 

from the modules and that together 

show the proportionality that exists in 

the chaotic geometric shapes of nature. 

In the case of the first moodboard I could 

observe how the composition is made up 

of several small modules that by their 

different colors and locations within 

the set form volumes of different sizes 

that can be associated with the frozen 

formations that make up the moodboard. 

Additionally, it is observed that within the 

composition there are accumulations 

of small modules in some parts that 

contrast with other larger ones that 

balance the composition, which I 

interpret as the vibrations present in the 

water within its particular context and 

that are manifested in the shapes, sizes 

and colors of the different modules 

transmitting the properties of cold and 

volume of frozen water. 

74 75

In the second moodboard the first 

geometrization is very similar to the 

previous moodboard, however, in this 

case the modules have a larger size 

and less contrasting shades of blue and 

green, which generates the sensation of 

being flat modules with a large surface. 

As for the distribution of the modules, 

there are also some concentrations of 

small ones, but for the most part there is 

a proportional mixture of large, medium 

and small modules, giving a sense of 

unity and order in the composition, 

which is what, in terms of the molecular 

composition of water, allows it to take 

on the characteristics of a liquid crystal. 

76 77

For the third moodboard, the first 

geometrization was not very helpful, 

because I focused more on generating 

transparencies than on the shapes 

present in the images, although the 

importance of the edges of the figures 

to achieve this sensation was evident. 

However, with the second geometrization 

I focused on the shapes, highlighting the 

presence of almost regular polygons, 

with different number of sides, which 

overlap each other giving depth to the 

composition from flat surfaces. In this 

case the edges of the figures take 

a relevant role, since they evidence 

the transparencies, and also delimit 

the different geometric modules that 

conform it. 

78 79

analysis for design 

By understanding the characteristics 

of each moodboard, I was able to 

establish the relationship between 

these and the concepts worked 

previously, to establish the way in 

which water as a fractal element was 

studied for the development of the 

design proposal. 

The first concept that I worked on 

was that of self-similarity, taking 

into account that in the moodboards 

as well as in the geometrizations the 

presence of figures in different sizes 

that when put together form a whole, 

so for the development of the pieces 

I will work on the development of 

geometric modules in different sizes, 

that when put together form the total 

piece and reflect the irregularity of 

the fractals. Likewise, these modules 

arise from the clusters present in the 

water and that as we have already 

seen are the elements that generate 

the characteristic properties of 

this element, so we will look for the 

conformation and characterization of 

the pieces from the characteristics 

of its modules and the relationship of 

these with their respective moodboard. 

From the geometries, the 

proportionality of the moodboards 

was highlighted, reflected both in the 

size of the modules already mentioned 

and the relationship between them, 

as well as in the distribution of 

colors within the compositions. For 

this reason, proportionality will be 

a characteristic that will be worked 

on in the development of the pieces, 

seeking that the sizes of the modules 

are proportional to each other, 

making use of the Fibonacci chain to 

determine the sizes of the modules. In 

the case of the proportions between 

the color zones, it will be sought that 

these correspond to those found 

from the moodboards in terms of the 

colors used, but also that they look 

harmonious within the compositions, 

reflecting movement and rhythm 

within them. 

Finally, the last property to be 

reflected in the pieces corresponds 

to the dynamic systems and their 

characteristics. In this case two 

elements will be taken for its 

representation, the first corresponds 

to the distribution of the modules, 

which is subject to the person who 

makes the composition of the piece, 

since, as explained with the chaos 

theory, any change however small will 

result in a different conformation, 

which indicates that even if the same 

composition is taken as a basis, the 

distribution of the modules will always 

be different. On the other hand, it 

has been established that water as 

a fractal element and as a dynamic 

system is in constant change, and 

that the changes are subject to the 

context in which it is located, which 

is why the pieces can be adapted to 

80 81

different contexts, so that the same 

surface acquires a particular use or 

arrangement according to the place 

where it is located. 

Foto recuperada de: https://i.pinimg.com/originals/ 

eb/c5/ba/ebc5badb7c3675d2846ac96b02e9fd44.jpg 

Imagen 20. Cataratas, el agua en caida genera un movimiento fractal 

82

final 

piezas 

pieces 

finales 

Foto recuperada de: https://www.nature-p0rn.com/wp-content/uploads/2019/01/steve-huntington-374991-unsplash-min-1024x683.jpg

Prototypes 

After understanding the forms 

present in the three moodboards I 

started the prototyping stage, which 

aims to explore different methods to 

reach the definition of the modules 

that represent each moodboard 

and find the optimal material for its 

realization. Additionally, there were 

three principles for the development 

of the pieces, these correspond to 

the findings of the research and 

are: self-similarity, translated into 

modules that are equal in shape but 

with different sizes; proportionality, 

reflected in the irregular patterns 

that are formed from the regular 

modules and that, although their 

arrangement seems random, maintain 

the balance in the piece, as well as 

the relationship with the colors of 

the modules and the formation of 

color zones; the pieces function as 

dynamic systems that is evident in the 

arrangement of the modules, which is 

the result of a subjective decision, so 

that each composition will be similar 

but different from the others, besides 

being pieces that relate directly to 

their environment adapting its shape 

to it; the last element are the clusters 

as the pattern present in the water and 

will be represented by the geometric 

modules that make up each of the 

prototypes and final pieces. 

The exploration for the moodboard 

of Fluidez congelada focused on 

exploring modules from origami that 

would allow to generate volumes 

within the piece. The first prototype 

consists of triangular modules called 

Trinity box (AxensWorkshops, 2012), 

in this case although the aim was to 

generate a volume, the shapes and the 

paper used made the piece look very 

strong, which did not correspond with 

the harmonic forms of frozen water. 

Then I made other explorations with 

origami modules, which did not work 

because they did not give the desired 

volume. Finally, I found the book A 

constellation of origami polyhedra 

(Montroll, 2004) in which different 

modules of regular and irregular 

polygons are presented, from these 

modules I made the second prototype 

that is made of modules of a triangular 

bipyramid cut in half and modules of a 

pentagonal bipyramid, each of these 

figures have three different sizes and 

with different colored papers that give 

dynamism and proportion to the piece. 

From this last prototype, the 

development of the final proposal was 

started, so the next step was to make 

the modules in fabric, for which we used 

sheet fabric, which due to its crumpled 

texture facilitates the generation of 

folds that form the module. For the 

realization of the first fabric modules, 

two straw cardboard templates were 

made, taking the shape of the paper 

86 87

modules of the prototype and another 

figure was added that corresponds to 

the base of the module, this base is 

necessary to fill the module and thus 

give consistency to the figure. Starting 

from the initial templates, three other 

templates were made for each figure in 

three larger sizes. These modules were 

made with cotton fabric that preserves 

the cardboard texture; however, as the 

size increased, it became necessary to 

add a layer of interlining at the base 

of the modules to preserve the shape 

of the module and prevent it from 

distorting when filled. 

Imagen 30. Trinity box 

Foto tomada por: Laura Zamudio 

Foto tomada por: Laura Zamudio 

Image 31. Composition of regular polyhedra in origami on paper. 

Picture taken by: Laura Zamudio 

Imagen 31. Moldes en tela de los poliedros regulares 


Image 32. Modules in regular polyhedron fabric in three sizes. 

88 89

For the sinuous malleability 

moodboard I started with some 

prototypes that generated a 

structure very similar to the one 

worked in the referent from painted 

fabric with the hydrographic 

technique, to represent the fluidity 

of water in the and triangular 

modules in straw cardboard that 

will generate a rigid structure with 

empty spaces that give movement. 

This prototype is interesting for the 

shapes that are generated as well 

as the appearance achieved with the 

painted fabric, however, it does not 

show the geometric shapes that are 

visualized in the geometrizations and 

that represent the cluster. For the 

following prototype lency cloth was 

used to make the modules, which are 

regular hexagons of different colors 

and two sizes, these hexagons are 

composed of an edge that gives the 

shape and strips of 0.5 or 1 centimeter 

attached to the edge. Each module 

consists of two hexagons of different 

colors intermingled strips to reveal 

both colors and give movement and 

fluidity to the piece. 

This last prototype was very 

fractioned by the hexagonal borders, 

so we looked for a way to eliminate 

this border and that the internal 

movement of each module would 

highlight and give rise to the whole 

composition. The last prototype 

is again made in lency cloth and 

consists of hexagonal modules of 

different colors and sizes, where a 

single integrated edge is maintained 

to the color strips, which as in the 

previous case are folded to make 

visible the two colors of each module 

and generate movement and fluidity 

in the total piece. In this case the 

modules are placed mixing colors, 

sizes and direction to give harmony to 

the piece and generate the feeling of 

unity present in the moodboard water. 


Image 33. Straw cardboard modules on fabric painted with hydrography. 

90 91


Image 34. Hexagonal modules woven with edges 


Image 35. Hexagonal modules woven without border and in three sizes. 

The prototyping process of the third 

moodboard, Voluble resistance, 

differs in the shapes, especially in 

the first prototypes, since the aim 

was to generate transparencies and 

highlight the edges of the shapes. The 

first prototypes were made from cuts 

in an elastic fabric, however, they did 

not give the expected sensation so it 

was discarded and I do not consider 

necessary to show it. For the first 

prototype we tried to generate fluid 

shapes and transparencies from 

strips of fabric of different blues 

and some reds, which when joined 

at different points generated waves 

that allowed us to see the mixed 

colors when viewing the entire 

piece. However, this prototype 

handled a completely different 

language to the explorations of the 

previous moodboards, so I made 

a completely different prototype, 

in which the transparencies were 

preserved by using chiffon and 

tulle, but the shapes of the modules 

did not seek to generate waves 

but on the contrary correspond to 

geometric shapes that are evident 

in the second geometrization for 

this moodboard. 

This prototype not only handles a 

language according to the previous 

proposals, but also transmits 

the moodboard sensations such 

as the delicacy of water and the 

harmony between all the elements. 

92 93

In this prototype were handled 

non-regular geometric modules 

of different sizes, for the final 

prototype it was decided to use 

three regular figures such as the 

pentagon, hexagon and octagon 

taking into account that in the 

previous prototype the figures with 

more sides were distorted or even 

took an almost round appearance, 

each figure is used in three sizes 

and in different colors. 

Picture taken by Laura Zamudio 


Image 37. Irregular modules with transparencies. 

Imagen 36. Tejido 

94 95

Design proposal 

After making the different prototypes 

for each piece, the decision was made to 

develop as the final piece the proposal 

corresponding to the Frozen Fluidity 

moodboard, this decision was made 

taking into account the time available 

for the completion of the project and 

the development of the proposal. It was 

also taken into consideration which 

proposal best reflected the principles 

of self-similarity, proportionality and 

that it would work as a dynamic system. 

For the development of the final piece, 

a photographic montage (image 38) 

was first made on a double bed, in order 

to begin to understand the relationship 

between the modules and the bed, 

understanding it as the environment 

that influences the water or in this 

case the piece. This montage allowed 

me to analyze the sizes, distribution 

and materials of the modules, in order 

to make the necessary corrections. 

First, it became evident that the size 

of the modules was not proportional 

and harmonious with the surface of 

the bed, and the relationship between 

them also needed to be reviewed. As 

for the material, it became evident the 

need to integrate the surface of the 

piece as an integral part of the design, 

and upon seeing the modules on the 

bed, the decision was made to change 

the fabric used in the prototypes to 

lency cloth. This decision was made 

because when reviewing the prototype 

on paper the shapes were solid and 

defined, which was not being achieved 

with the fabric, while the lency fabric, 

being a more rigid material, allowed to 

better define the faces of the modules 

and by leaving the seams on the outside 

it was possible to define the vertices 

of the figures, giving cleanliness and 

structure to the figures (image 39). 

Likewise, 3 sizes were defined 

following the Fibonacci chain for their 

diameters, so the smallest ones have 

a radius of 3 cm, the medium ones of 5 

cm and the large ones of 7 cm, which, 

although not belonging to the chain, 

looks proportional in relation to the 

others and to the surface of the bed. On 

the other hand, the decision was made 


Image 38. Photomontage of first volume modules 


Image 39. Modules in lency cloth, final sizes 

96 97

to combine modules with volume and 

flat modules, which at first sight seem 

to have volume due to the combination 

of colors, this was done to generate 

greater harmony in the piece. 

The second important element 

that had to be reviewed was the 

composition of the piece, since up to 

that moment the distribution of the 

modules was randomly made from 

the available modules. However, when 

testing on the bed, the arrangement of 

the modules did not show harmony or 

proportionality. 

With this in mind, I carried out a 

digital exploration that allowed me 

to develop a composition that would 

bring together both the previously 

mentioned concepts and their 

design manifestations, as well as 

the characteristics and sensations 

reflected in the moodboard. For this 

exploration I took as inspiration three 

photographs of frozen landscapes in the 

world, in which I found a way to express 

the essence of the concept. 

The first image corresponds to an 

ice formation through which light 

is reflected on the water, resulting 

in different shades of blue that 

complement the cold environment of 

the image (image 40). For this case I 

made a delimitation of the color zones 

taking three representative colors, in 

order to make a composition in which 

the modules were arranged according 

to the desired shades (image 41). For 

the composition with the modules, a 

digital recreation of both the bed and 

the modules was made, taking the real 

measurements, this allowed me to 

check the proportionality between all 

the elements. Additionally, with this 

composition we determined the colors 

that make up each color zone and the 

distribution of volume modules that 

delimit the zones and flat modules 

that make them up (image 42). 

98 99

Photo retrieved from: https://i.pinimg.com/564x/38/83/ 

bb/3883bbc54af2dceec167051fa03f2dbd.jpg 

Image 40. 

Image 41. Color silhouettes 

Image 42. Composition with modules 1 

100 101

The second image corresponds to 

the ceiling of the Marble Caverns 

in Patagonia (image 43), which is 

made up of frozen formations that 

assimilate waves, thus allowing to 

show movement and fluidity from 

static structures. In this case I made 

again the process of delimitation of 

color zones highlighting the different 

waves of the cavern to take them as 

a reference for the distribution of 

the modules from the distribution 

of colors and volumes (image 44). 

However, in this case the volumes 

were used to generate the two darkest 

and the lightest zones, while the 

others were formed by flat modules 

with different color combinations 

(image 45). In this composition the 

proportionality generated by the color 

zones is more evident, since it is from 

these that the composition transmits 

fluidity despite being static, also in 

this case the volume of the modules 

helps to generate more movement 

within the piece. 

Image 43. Marble caverns 

Photo retrieved from: de: https://i.pinimg. 

com/564x/85/20/4e/85204e7df0990cd00aef10b158ccace0.jpg 

Image 44. Color silhouettes 

102 103

The third image corresponds to a photo 

of a natural phenomenon in which 

frozen methane bubbles accumulate 

under the surface of the frozen water 

of Abraham Lake (image 46), in this 

case I took the image as a reference 

for the composition, but without 

determining color zones, since as 

can be seen in the image the colors 

are mixed, although organized leaving 

the lighter ones over the darker 

ones, which generates depth. In the 

composition with the modules I tried 

to generate the mixture of colors seen 

in the image and from the mixture of 

volumes, planes and empty spaces to 

give movement to the piece (image 47). 

Photo retrieved from: https://i.pinimg.com/564x/9e/ 

ea/3b/9eea3b1b4948b9d83b44bf9adc0929ad.jpg 

Image 46. Frozen methane bubbles 


104 105

After analyzing the three composition 

proposals, I decided to opt for the 

second option, as I consider it to 

be the one that best integrates the 

relationship between the volume 

modules and planes with the color 

zones, which are the reflection 

of proportionality in the piece. 

Additionally, the undulating forms of 

the zones generate movement and 

fluidity from static structures, thus 

representing the meaning of the 

concept. 


106 107

Final pieces 

The final process consisted of the 

material development of the chosen 

composition, for which the different 

triangles needed to form both the flat 

modules and the volume modules were 

cut in the different colors of lency 

cloth available. The colors of the cloth 

correspond to the range of blues, sea 

water, grays, mint green and beige to 

highlight the lighter areas. 

For the base of the piece it was decided 

to use the lightest blue cloth, which 

allows the piece to look cool despite 

being made of a warm material. This 

base has a size of 1 meter wide by 2 

meters long, the measures correspond 

to the size of a double bed being a little 

longer to overcome the width of the 

bed. On the other hand, Peptel paper 

was used to adhere the flat modules 

to the base fabric, this paper works as 

a layer between the two fabrics and 

sticks to them when exposed to the 

heat of the iron. The volume modules 

were sewn by machine to speed up the 

manufacturing process and were sewn 

by hand on the base, using an invisible 

stitch that is resistant and preserves 

the neatness of the product. 

Finally, a photographic session of the 

piece was carried out, in which the 

possibilities of use were explored. 

In this session the dynamism of the 

blanket became evident, since when 

placed on different surfaces such as 

the bed or the sofa, it adapted to its 

context. In the same way, by involving 

people to interact with the blanket, it 

took on a new narrative articulating 

the context, use and body of the 

person, which evidenced the property 

of water to adapt and remember the 

vibrations that influence it, to reflect 

them in its appearance. 

The latter is of great importance, since 

it is the reason why water manifests 

itself as a dynamic system that under 

the influence of the context in which 

it is, it transforms itself by taking the 

vibrations and manifesting them in its 

appearance. In the photos you can see 

how the piece adapts to each situation, 

but also how the environment or the 

person who is interacting with it is 

infected by the sensation that the 

piece reflects. This is related to the 

property of water to resonate with 

other systems composed of water, 

so that a calm water can spread that 

feeling to plants, animals, people or 

in general to the living beings that 

surround it. In the case of the piece, 

it can be seen how the two people are 

infected by the feeling of warmth and 

comfort that the blanket transmits 

and in the case of the bed and the 

sofa, the surfaces attract and invite to 

interact with them. 

108 109


Image 46. Blanket on the bed Picture taken by: Laura Zamudio Image 47. Blanket on the bed 

110 111



Image 48. Blanket on the bed - volumes 

Image 49. Blanket on the sofa 

112 113



Image 50. Blanket and person on the bed 

Image 51. Blanket person on a chair 

114 115

and 

Another important element for the 

final piece is the graphic development 

of the brand, logo, packaging and 

other elements necessary for the 

completion of the product as such. 

For the logo (image 52), we sought to 

generate a composition in which the 

use of geometries stood out, combining 

different shades of blue to give the 

appearance of water. Regarding the 

distribution of the modules, the idea 

was that the composition should have 

movement and that, although it gives 

the appearance of being a drop of 

water, it also shows that it transforms 

and evolves, making reference to the 

capacity of water to adapt and transform 

itself according to the environment. 

Likewise, two versions were proposed, 

one for a white background and the other 

with a dark blue background, as part of 

the brand’s graphic identity. 

Among the complementary elements for 

the product, we developed the packaging 

design, the label that accompanies it and 

another style of label that highlights the 

brand. As for the packaging, we decided 

on a blue fabric lining (image 53), with 

a transparent part through which the 

blanket can be seen; the lining works 

both for transportation and storage. The 

blanket is accompanied by 3 cushions 

with the shapes of the modules and a 

cover for washing, with which it can be 

washed in the washing machine, when it 

does not have a roller or it can be washed 

in the laundry. 

As for the labels, the first one (image 

54) corresponds to the packaging 

label, which contains the product 

specifications such as measurements, 

elements it contains and the care that 

must be taken to ensure its durability. 

While the second (image 55) is an 

advertising element of the brand, which 

contains the logo and the brand. These 

elements accompany the piece and 

characterize it as a finished product. 

116 117

LA GEOMETRÍA DEL AGUA 


Image 52. Logos on white and blue backgrounds 

Imagen 53. Packaging 

118 119


Manta memoria del agua 

Cama doble 

Paño lency 

Manta 

1 mt x 2mts 

3 cojines 

34 cm 

1 funda para lavado 

50 cm x 65 cm 

Lavar en la funda de lavado 

Lavado ropa delicada 

Lavar con agua fría 

Secar en secadora 

Planchar a vapor 

No presionar al planchar 

No retorcer 

No estirar 

No usar clorox 

Image 53. Packaging label 

Image 55. Advertising label 

120

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