strange attractors
STRUCTURE
complex
STRUCTURE
complex
complex
Complex, or non-linear, fractals are fractals that exhibit a self-similar structure,
but are not exactly self-similar. The overall appearance of a non-linear fractal
closely resembles some of its smaller parts but always with some variation. In
some cases smaller parts might look quite similar to the overall fractal and help
define the fractal’s overall shape, in other cases varying regions appear as twisted
or skewed scale copies of the original, while still other regions have shapes that
bear no resemblance to the original.
STRUCTURE
complex
complex number fractals
STRUCTURE
Strange Attractors are defined by an equation or system of
equations. The orbit points are generated by passing the
current orbit point through the equations to obtain
the next orbit point. This process is
repeated thousands (or millions) of times
This process
to produce the fractal data. Of course, most
equations will not produce a fractal and
the challenge is to find equations that do.
Quadratic Attractors and Cubic Attractors
are examples of these fractal types.
produce the fractal data.
Take an equation, solve it; take the result and fold it
back into the equation and then solve it again. Keep
doing this a million times. Each time Clifford Pickover
of IBM solved the equation he marked a point on
a graph and therefore he could follow the point as
it swept around the plotting space. It’s a little like
tracing the path of a fly as it whizzes around a room.
Pickover’s feedback sculpture is what scientists call
a strange attractor. All strange attractors are fractal.
THE MANDELBROT SET LOOKS VERY COMPLICATED, AND
YET IT IS GENERATED BY A VERY SIMPLE RULE:
REAL NUMBERS
fractal geometry:
2
3 4 5
3i
If one comes the real line and
the imaginary line at a right
angle a plane is created, a
complex plane.
2i
i
-5 -4 -3 -2 -1
1
2
An imaginary number is the
square root of a negative real
number. “Imaginary” comes from
imagining that -1 or any other
negative number has a square
root, a mathematician supposition
that is extremely useful. The unit
imaginary number is i=√-1. In a
similar geometric sense, there is a
line of imaginary numbers.
The arrow can be read as ‘goes to’ or
‘becomes’, for what this rule represents is a
transformation of two-dimensional space;
the letters z and c by convention indicate
generic points in this space, with z being
variable and c constant. In other words, the
rule transforms the point z to another point
in the space, while leaving c unchanged. This
two- dimensional space inhabited by z and
c, the home of the Mandelbrot Set and Julia
Sets, is central to mathematics, from quantum
mechanics to number theory.
3 4 5
-2i
-3i
-4i
12
The Mandelbrot set is the set of points on a
complex plane. The formula used to build the
Mandelbrot set, separates the points of the
complex plane into two categories: points
inside the Mandelbrot set, points outside the
Mandelbrot set. The Mandelbrot Set emerges
when we apply this rule over and over again,
taking the outcome of one transformation as
the input for the next.
The main body of the Mandelbrot Set consists
of a cardioid, or heart-shaped core, surrounded
by infinitely many circular buds. Each bud
is surrounded by a further infinity of smaller
buds, and, at the end of each of these chains of
buds, a spiral frond, sometimes lacy and floral,
sometimes straight and spiky.
The fronds, which comprise the boundary
of the Mandelbrot Set, actually consist of
infinitely many miniature copies of the whole
shape, joined together by bifurcating threads
of ever-smaller miniatures. In response to
this continual branching, these fronds are
also called dendrites, from the Greek for tree.
The name conjures up associations with the
straggly branching receptors of nerve cells in
our brains, which are also called dendrites. This
is no accident: evidently the functionality and
processing power of neurons derive from their
richly entwined fractal structure.
Pickover’s feedback sculpture is what scientists
call a “strange attractor,” which means “it has
some structure even though it’s very irregular”.
28
13
16
through visual media
29
17
STRUCTURE
PULL THE TABS TO SEE EACH
ITERATION OF THE FRACTALS.
linear
cantor set
Linear fractals or “classical” fractals are
exactly self-similar. If you look at a very small
part of a fractal’s overall shape, it looks exactly
like the original fractal, only smaller. We call
this size difference “the scalability factor” or
“scale”. These fractals begin with a “seed”, a
set of lines that form a basic structure. Next
you make duplicate copies of the original
seed and you use them to replace the lines
found in the original seed. You continue this
process at greater levels by replacing line
segments with seeds, whose lines in turn
get replaced by seeds, so on and so on,
forever. Since many of these fractals can be
easily drawn, they were the primary types
of fractals generated before computers.
STRUCTURE
linear
Karen Yee
The Cantor set, introduced by German
mathematician Georg Cantor in 1883, is a
remarkable construction involving only the real
numbers between zero and one. It was actually
discovered by Henry Smith, a professor of
geometry at Oxford in 1875, however without
explanation. Cantor was interested in what
happens when you apply these rules an
infinite number of times. Most people would
think, “Well, if I’ve thrown everything away,
eventually there’s nothing left.” Not the case;
there’s not just one point left, there’s not
just two points left. There’s infinitely many
points left. As you zoom in on the Cantor set,
the pattern stays the same, much like the
noise patterns that Mandelbrot had seen at IBM.
The Cantor set is the prototype of a fractal.
FIRST
ITERATION
SECOND
ITERATION
S. Gud d e r
31
FIFTH
ITERATION
LIFT EACH TAB TO REVEAL HOW THE CANTOR SET ITERATES BY ERASING THE MIDDLE THIRD.
It is self-similar, because
it is equal to two copies
of itself, if each copy is
shrunk by a factor of
1/3 and translated. The
Cantor set has no length
or interior. In technical
parlance, it has “zero
measure” A randomly
thrown dart is infinitely
unlikely to hit it. It is “nowhere dense’. Every part of it
consists almost completely of holes and yet despite being
nothing but totally disconnected points, it is uncountable.
In fact, it contains as many points as the whole line it is
carved from. Ever point is an “accumulation” or “limit” point,
meaning there are infinitely many other points from the set
in any neighborhood of it, no matter how small. Conversely,
the Cantor set contains all of its limit points.
There’s not just one
USE THIS OVERLAY TO COMPARE THE COMPUTER
GENERATED FRACTAL TRANSFORMATION FERN
TO A PHOTOGRAPH OF A REAL FERN.
point left, there’s not
just two points left.
CANTOR SET
STRAIGHT LINE
KOCH CURVE
There’s infinitely
many points left.
The essence of mathematics
30
FOURTH
ITERATION
A CROSS SECTION PERPENDICULAR TO THE PLANE
OF THE RINGS OF SATURN SEEMS TO HAVE THIS
STYLE OF FRACTAL STRUCTURE.
A Cantor set contains two one-third-sized copies of
itself. A straight line can be split into three on-third
sized copies of itself. A Koch curve consists of four onethird-sized copies of itself. A square is made up of nine
one-third-sized copies of itself. In some sense the Cantor
set and Koch curve lie on either side of the straight line.
The Koch curve is between the line and the square. It
takes up more space than the line, but less space than
the square. It lies somehow between the first and second
dimensions. This concept is called similarity dimension.
is not to make simple things complicated,
but to make complicated things simple.
THIRD
ITERATION
STRUCTURE
linear
One very simple way to understand fractals and
the meaning of “iteration” is to examine a simple
recursive operation that produces a fractal pattern
known as Cantor Set. The Cantor set is defined
by repeatedly removing the middle thirds of line
segments. This is the first step or iteration, and
then take the remaining two lines and repeat
the clipping procedure. Eventually after 5 or 10
iterations you have dozens of tiny lines which
take up only as much room as the two original
ones from the first step.
linear fractals
STRUCTURE
linear
A tactile exploration
-5 -4 -3 -2 -1 0 1
4i
COMPLEX NUMBERS
11
FOLLOW THE INSTRUCTIONS UNDERNEATH TO BUILD
A KALEIDOCYCLE, WHICH SIMULATES THE ZOOMING
CHARACTERISTIC OF A MANDELBROT FRACTAL.
IMAGINARY NUMBERS
Geometrically a real number
is any position on a line which
extends infinitely.
10
is repeated
thousands of times to
TWO COPIES
OF ITSELF
1/3 FULL SIZE
THREE COPIES
OF ITSELF
1/3 FULL SIZE
FOUR COPIES
OF ITSELF
1/3 FULL SIZE
LESS THAN 1-D
ONE DIMENSIONAL
(1-D)
MORE THAN 1-D
The old-fashioned way to store an image digitally is to impose
a grid on it and record the average gray level of each grid cell
(pixel). The finer the detail needed, the finer the grid must be
and the larger the record. Fineness of detail is a matter of
persistence in iteration. Left, sketched on the photograph, is an
outline of the fern and collaged upon it are four trans-formed
copies of it. This is the result of iterating the transformations.
This process is known as recursion:
the repeated application of a rule to
successive results will be explained in
further detail on the following page. 33
32
48
38
39
Not surprisingly, fractals occur in other geographical
features. The land surface of countries is an obvious
example, which can display fractality from scales of
hundreds of miles down to a few feet. Mountain
ranges contain many peaks, each with subsidiary
summits, with hillsides made up of large and small
undulations resulting from local geology or erosion by
streams, and smaller hillocks and tufts formed by soil
irregularities. Mathematical fractal constructions have
been used very effectively to simulate realistic looking
landscapes. Fractal landscapes have been utilized widely
in art and movies, with fractal
planet and landscape simulations
pioneered in the early 1980s films
Whether we are conscious
Star Trek II: The Wrath of Khan
of it or not, fractal shapes
and Star Wars: Return of the Jedi
and used in many later films.
fractal simulations
Not only does fractal geometry describe natural phenomena, it can
also be used to simulate, or “fake” or “forge,” them. In other words,
fractal principles can be used to create exceedingly realistic images
or models of these natural phenomena. For instance, topological
representations created fractally are indistinguishable from the real
thing. Furthermore, these fractal simulations are useful; petroleum
engineers routinely use them because they can accurately model the
distribution of oil in sedimentary rock. This characteristic of fractals
explains why school children find it so easy to create plausible, but
fictitious, maps of rivers and coastlines; one just draws a squiggly
line that approximates the contours of the original.
fractals in nature
coastlines are not circles, and bark is not smooth,
nor does lightning travel in a straight line.
Be no it Mand e l br o t
The Frac t al Geo met ry o f Nat ure
This stained cross section of
cells in a cucumber bears a
curious resemblance to the
purely mathematical fractal
pattern generated on a computer.
have become part of our
Mathematical fractal constructions have been
used very effectively to simulate realistic looking
landscapes. Methods used to generate random
curves such as Brownian motion may be extended
to produce random surfaces.
RELEVANCE
nature
RELEVANCE
Clouds are not spheres, mountains are not cones,
nature
thought patterns.
Usually it is the overall
appearance of the
landscape that is
important rather than its
precise form, and random fractal constructions can
produce and vary such scenery very efficiently and at
a much lower cost than creating elaborate film sets.
The mountains on the left are created
with a Brownian fractal function,
in comparison to a photograph of
mountains on the right.
RELEVANCE
nature
abstract
Most of the fractals encountered so far in this book lie in the
idealized world of the mathematician, where it is theoretically
possible to repeat a construction step forever, or view an object
at arbitrarily fine scales. Of course, the real world is not like
that—in reality, we only encounter approximate fractals. If we
zoom in too closely on a real object any self-similarity will be
lost, and eventually we encounter molecular or atomic structure.
Nevertheless, it can be very useful to regard natural objects as
fractals if they exhibit irregularities or self-similarity when viewed
over a significant range of scales. But all scientific descriptions
or `models’ of reality are approximate, and this is a further
instance. This section will take a look at fractal geometry in
natural occurring existence.
Intelligence can be applied to conventional storage to reduce
the space required. A simple thing to do if there is a run of pixels
with the same value is to encode them with two numbers—a
count of the pixels in the run, and the value common to them
all—rather than recording all the values in the run. And there are
much more elaborate methods. What they all have in common is
that they somehow find and remove redundancies in the image.
Fractals are in a way the ultimate in redundancy, so this method
is potentially the most effective compression of all.
Whether we are conscious of it or not, fractal shapes
have become part of our thought patterns. Perception
scientists have carried out psychology experiments
that shown we are most comfortable with objects
whose dimensions range between 1.2 - 1.4 and 2.2
- 2.4, these coincidentally are the dimensions most
commonly found in trees, mountains and clouds.
Perhaps as we coexist harmoniously with the fractal
patterns in nature, our visual systems process
thoughts that themselves have become fractalized,
subconsciously in tune with the world around us.
The ocean is a large drop;
a drop is a small ocean.
Ral ph Wal d o E me r s o n
49
fractals in the universe
Fractals will maybe revolutionize the way that
the universe is seen. Cosmologists usually assume
that matter is spread uniformly across space. But
observation shows that this is not true. Astronomers
agree with that assumption on “small” scales, but
most of them think that the universe is smooth at
very large scales. However, a dissident group of
scientists claims that the structure of the universe
is fractal at all scales. If this new theory is proved
to be correct, even the big bang models should
be adapted. Some years ago we proposed a new
approach for the analysis of galaxy and cluster
correlations based on the concepts and methods
of modern Statistical Physics. This led to the
surprising result that galaxy correlations are
fractal and not homogeneous up to the limits
of the available catalogs.
Many more redshifts have been measured and we
have extended our methods also to the analysis
of number counts and angular catalogs. The result
is that galaxy structures are highly irregular and
self-similar. The usual statistical methods, based
on the assumption of homogeneity, are therefore
inconsistent for all the length scales probed until
now. A new, more general, conceptual framework
is necessary to identify the real physical properties
of these structures. But at present, cosmologists
need more data about the matter distribution in
the universe to prove (or not) that we are living
in a fractal universe.
EXPLORE INTO MORE
FRACTAL IMAGERY FROM
NATURE PHOTOGRAPHY.
By using the iterative features of “recursive programming” Dawkins created a biomorph program that simulations evolution
and iterates genes which resemble the trilobites that swam in the oceans of the Cambrian era 570 million years ago.
complex
Mathematics and art are considered by some to be in opposition.
Math is perceived to be intimidating to artistically creative people.
There are many visual elements of math most creative people are
unaware of.
52
53
Canyonlands from
Island-in-the-Sky, Utah
54
55
60
61
62
63
This algorithmically designed piece of jewelry is a metaphor for fractals’ history for the multiplicity
of different contacts fractals create between fields. Here is it compared to an Apollony Gasket.
fractals in our bodies
nature
RELEVANCE
applications
If people do not believe that
applications
Jo h n Lo uis v o n Ne umann
Already, technological and commercial advances have stemmed from
such questions – for example, a compact antenna for mobile phones,
new ways to analyze the movements of the stock market, and efficient
methods to compress the data in computer images, squeezing more
pictures onto a CD. Once our eyes have been opened to the fact that
fractal objects possess a distinctive character and structure, and are
not just irregular or random, it becomes obvious that the universe is full
of fractals. Indeed, it may even be one. Fractals teach us not to confuse
complexity with irregularity, and they open our eyes to new possibilities.
Fractals represent an entire new regime of mathematical modeling,
which science is just beginning to explore.
The nervous system exhibits fractal patterns
seen at both visible and microscopic levels.
Looking at the cerebellum, a structure located
at the base of the brain, one can see a series
of continually smaller nerve branches forming
a network that sends sensory information
throughout the body. This branching structure
is called the arbor vitae which literally translated
means the “tree of life”.
Spiral lens structure in the compound eye of
a firefly (left) and computer simulation of a
spiraling DNA ladder from the top.
64
65
66
67
Another system which demonstrates
fractal structure, is the cardiovascular
system, which is composed of the heart,
arteries, and veins. Blood vessels which
transport blood through the body form
another branching network. Blood is
carried away from the heart by arteries
leading to all parts of the body, and
these divide into narrower arterioles
and end in fine capillaries of about
0.01mm in diameter. Oxygen and
nutrients pass out through the thin
capillary walls to body tissue and
waste products are absorbed into
the capillaries. For this to function,
every cell in the body must be within
around 0.1mm of a blood vessel.
Moreover, for the circulatory system
to function efficiently, the distance
between the heart and the capillaries
should be no longer than necessary.
This requires an intricate branching
fractal network, comprising a total
length of blood vessels of around
60,000 miles. An understanding of
these new principles derived from
fractal geometry has provided us with
greater insight into the human body.
What’s absolutely amazing is
that you can translate
what you see in the
natural world in the
LEARN ABOUT FRACTALS AND
THEIR APPLICATIONS IN ART.
language of mathematics.
68
69
FRACTALS ARE
CONSTANTLY BEING
APPLIED TO NEW
APPLICATIONS.
HERE IS A SHORT
LIST OF PLACES
WHERE FRACTALS
ARE BEING USED:
•
•
•
•
•
•
•
•
•
•
•
aggregation growths
art
cardiovascular system analysis
computer graphics
cellular automata modeling
chemistry
city planning
crystallography
data compression
dielectric processes
electrochemistry
•
•
•
•
•
•
•
•
•
•
•
epidemics
geology
image rendering
image processing
lung analysis
kidney structures analysis
material science
mammography
meteorology
metallurgy
music
•
•
•
•
•
•
•
•
•
network structure
peace and conflict studies
psychology
plant structures
population biology
semiconductors
stellar formation
structural engineering
traffic flow
72
73
SYSTEM MAP: Opening spread | First spread | Interior spread | Final spread with interactive pocket
The way in which a medium can convey research affects how
wide of an audience it can reach. Through paper-based interactive
elements housed in the book, including overlays, tabbed reveals,
and paper folding, math concepts become a hands-on experience
that facilitates learning and makes math a visually interesting and
interactive experience for anyone, regardless of their knowledge of
mathematics research affects how wide of an audience it can reach.
This project was for completed for my senior thesis in Graphic
Design, over a course of ten weeks.
Full size: 10.5"x9.5", 80 pages
RELEVANCE
mathematics is simple, it is only because
they do not realize how complicated life is.
In the breathing or respiratory system, the
windpipe, or trachea, splits into two bronchial
tubes leading into the two lungs. These tubes
split into narrower tubes, which continue to
split repeatedly until, after about 11 levels of
branching, they reach numerous very fine tubes
called bronchioles which end in microscopic
thin-walled sacs called alveoli. A lung contains
around 400 million very closely spaced alveoli.
Air breathed through the mouth or nose passes
down the trachea and into the lung to reach
the alveoli from whence oxygen is passed
into the bloodstream and carbon dioxide is
absorbed from the blood to be exhaled. Adult
human lungs are about 12 inches long and 5
inches wide, but because of their branching
fractal structure have an enormous surface
area of about 100 square yards. It is the fractal
structure that achieves this large area within
a confined space and thus enables oxygen
to be supplied to the blood efficiently and
in adequate quantity for the whole body.
within a confined space.
On a cellular level, these nerve branches
are comprised of cells with similar
structure, neurons and astrocytes,
that also display these fractal patterns.
Neurons, which are the cells primarily
responsible for the transfer and
processing of information within the
nervous system, contain cell extensions
called dendrites. Dendrites receive
information in the form of electrical
impulses from a variety of sources.
This network of dendrites, which
is often called the “dendritic tree”
can be seen in Purkinje cells of the
cerebellum, which are organized into
short, asymmetric branches that break
off into smaller branches and so on.
RELEVANCE
applications
fractals
Are they important? Undoubtedly. Turbulence in the atmosphere makes
it difficult for Earth-based telescopes to produce accurate images of
stars; a turbulent atmosphere is well modeled by a fractal distribution
of the refractive index. Light bouncing off the ocean, with its myriad
waves on many scales, closely resembles reflection from a fractal mirror.
And the way trees absorb energy from the wind is closely related to
the ‘vibrational modes’ of a fractal – and it is such modes that create
the sound of a drum. The natural world provides an inexhaustible supply
of important problems in fractal physics.
RELEVANCE
applications
linear
Fractals are mathematical patterns that infinitely repeat and have
strong connections to nature and applications in daily life. My
presentation will be on the design of an interactive print book,
with an electronic tablet companion, to demonstrate how these
tangible components can make learning mathematical concepts an
accessible and rewarding and enjoyable experience.
the importance of fractals
that achieves this large area
RELEVANCE
applications
STRUCTURE
It is the fractal structure
Some of the most extensive examples
of branching fractal networks are to be
found within the bodies of humans or other
mammals, in particular in the breathing, blood
circulation, and nervous systems. Whilst
these remarkable structures lead to efficient
physiological operation of the body, it is still
not clear what evolutionary or biophysical
mechanisms underlie this branching growth.
system map
color system
The book is divided into two sections, Structure and Relevance,
overview
that further break down into two subsections.
- Each chapter opens up with large imagery in the color
the corresponds to the specific chapter.
STRUCTURE
- Discusses concepts for both Complex and Linear fractals
- Explains what fractals are, how they were discovered, and
how they are created
- Bulk of the interactive elements are
s t housed
r u c t u r ein othese
v e r vchapters
iew
- Green is used through out the book as an interactivity indicator.
- Each chapter ends with a green pocket that holds an
interactive element related to the content of the chapter.
- Color palette is inspired by psychedelic poster art from the 70s.
This time period is relevant to the rise of computers, which led
to large strides in research of fractal geometry.
RELEVANCE
- Subsections: Nature and Application
S E L F S I M I L(less
ARITY
- This half of the book is a more visual experience
interactivity than the first half)
chapter relate back
O n w h a t e v e r s c a l e , a n d w i t h i n - Interactive elements at the end of each
PEANO CURVE
a g how
i v e n r athey
n g e y oare
u examine a
- Explains where fractals exist in nature and
to the color of the chapter they (live
I N A in.
KOCH SNOWFLAKE)
fractal it will always appear
important in other elements of our lives t o h a v e t h e s a m e s h a p e o r
interactive elements
same degree of irregularity.
These are the interactive elements that are included in the book to heighten the learning
experience for the user. By using interactivity in a print form it creates a tangible and
visual experience while learning new concepts.
overview
END OF CHAPTER POCKET INTERACTIVE ELEMENTS
overview
icon system
ITERATION
1
o rreiterate
e p e a t a n o p econcepts
ration,
Icons seen here are used throughout the book Tto
that show
2
generally using the last result
up in
multiple
places.
There
is
also
a
green,
the
color
used
for
interactivity,
icon
structure overview
of that operation as the input.
to indicate where you can use an iPad to digitally experience fractals.
structure overview
FRACTAL CONCEPT: Self-similarity
A pull tab shows how the Mandelbrot
set is self-similar using frosted vellum
as an overlay.
3
4
KOCH CURVE
SELF SIMILARITY
SOEnL F
w hS aI M
t eIvLeA
r Rs Ic T
aY
le, and within
a given range you examine a
c thaal t ietv w
yn
s da pwpi et h
a irn
Of rnaw
e ri lsl caallw
e ,a a
t
o
h
a
v
e
t
h
e
s
a
m
e
s
h
a
p
e
o
r
a given range you examine a
r eygs ual a
fsraam
c tea ld ei tg rweiel l oaf l iwr a
p rpietayr.
to have the same shape or
same degree of irregularity.
This kaleidocycle is a visual representation of the “zooming in” you experience with the self-similar
qualities of fractals. The user is given instructions and a pre-scored template to glue together.
PEANO CURVE
(IN A KOCH SNOWFLAKE)
PEANO CURVE
The dimension of a fractal (IN A KOCH SNOWFLAKE)
in general is not a whole
number, not an integer.
FRACTAL DIMENSION
PASCAL TRIANGLE
SIERP
PASCAL TRIANGLE
SIERP
CANTOR SET
BOUN
FRACTAL CONCEPT: Recursion
ITERATION
Multiple pull tabs that show
recursion.
r eApTeIaOt Na n o p e r a t i o n ,
ITToE R
generally using the last result
a et aotp ae rnaot p
i oenr aatsi otnh ,e i n p u t .
Toof rt he p
generally using the last result
of that operation as the input.
1
3
1
50 2
3
4
2
REVIEW MORE FRACTALS NOT
INCLUDED ON THIS PAGE
4
KOCH CURVE
KOCH CURVE
FRACTAL CONCEPT: Iteration
This overlay lets the user discover the similarities between a computer generated fern using a
fractal equation, and a photograph of a real fern.
LEVY C CURVE
LEVY C CURVE
FRACTAL DIMENSION
FTRhAe CdTi A
mLe nDs Ii M
o nE oNfS aI Of N
ractal
in general is not a whole
Tnhuemdbiemr e, nnsoi to na no fi nat ef rg ae cr t. a l
in general is not a whole
number, not an integer.
A visual representation using acetate
overlays of the Cantor Set. This
fractal is created by eliminating the
middle third of a line.
50
REVIEW MORE FRACTALS NOT
INCLUDED ON THIS PAGE
REVIEW MORE FRACTALS NOT
INCLUDED ON THIS PAGE
CANTOR SET
CANTOR SET
BOUNDARY OF A DRAGON CURVE
BOUNDARY OF A DRAGON CURVE
50
FRACTAL CONCEPT: Iteration
This circle explains the process of
iteration. Passing colors through a
blue overlay shows that the output
of the first iteration becomes the
input of the next.
This small booklet has a further exploration of fractals in nature. It only uses black and white
photography to showcase form and pattern that naturally exist.
FRACTAL CONCEPT: Form
This element is a study of threedimensional form that fractals can
take. This element is white to let
the user focus on form.
These oversized postcards give more information on fractals in art, and can be used as miniposters if the user desired.
Full book with pocket interactive elements
H
HI