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Where Art and Maths
I'm a bit of a "late bloomer", having "hit the Maths Wall" just
after "O" level. It took a very enthusiastic teacher to reawaken my
early love for the subject and, as with many converts, I've been
thoroughly smitten with the love of maths I had when very young. As
I am also doing an art degree, for fun, I wondered what would
happen when you mix the two.
Plenty of artists and mathematicians have mixed both subjects; most
notable of these being, of course, Leonard da Vinci. Choosing one
topic, perspective, I constructed a timeline to see how much
cross-over there actually was. Apparently, many early artists were
mathematicians and/or scientists. More than a few were also
Having drawn a spider diagram to organise my thoughts, there were
so many areas I could explore, but didn't have the time. It was
time to narrow down the field of enquiry. One area of mathematics
stood out for me - fractals. Ever since I saw my first
computer-created fractal image I have been fascinated by their
beauty. The question was how I could create my own, original
Looking first at the original Mandelbrot equations, I tried to
calculate variables to make a Mandelbrot plot. This is not easy
maths! At one glance it is obvious to see why fractals didn't
really come into their own until computers grew up a bit. I lost
and found and lost myself again in the complicated calculations. As
my programming skills only reach to BASIC, and I didn't have Visual
Basic up and running on my computer, I thought there had to be a
more artistic solution.
Here is a traditional Mandlebrot fractal. It has a very
distinctive appearance and shape, which is reiterated (repeated) in
varying sizes, throughout. If you zoom in on any fractal, you see
the same level of detail and complexity as you do with the original
image. For a more detailed explanation of what fractals are and
their possibilities, follow this link:
There are a number of fractals that may be hand drawn, the first of
these being the Cauliflower.
Another fractal which can be drawn is the Van Koch Snowflake. Here
are a few images of my early attempts.
If you would like to explore the Snowflake to create you own images
then this previous
challenge shows you how the Snowflake begins. It gives you
step-by-step instructions to create your own Van Koch Snowflake. It
would be a good idea to look at this before proceeding further.
uses the program LOGO to draw similar fractals. You
would need access to that program to be able to complete the
My tutor challenged me to create my own, original fractals, using
IT-based methods, rather than paint and brush. I suppose I'm a bit
old-fashioned, but it was quite difficult to accept that digital
art is as valid a medium as using acrylics and a brush! Finding the
right software that would enable me to do this was the next step.
There are a number of programs, other than LOGO, out there. Some
are simply Mandelbrot viewers, which enable you to zoom in on
portions of the Mandelbrot set. You can choose the area to look at
and even change the colour of the image. However, there is little
or no real mathematical calculation involved. These can give you
some interesting images, but they are hardly original. The
limitation of this particular program is that it only works to
about six levels of magnifications, then degrades into a series of
colourful parallel lines. Still, the results are quite
Further manipulation of such images, using a program such as
Paint.net, can improve the interest of the image. Photographs can
also be 'fractalised' in this way. Paint.net has a tool bar which
allows manipulation of fractal images. This has a 'render as
fractal' facility, which takes a regular picture and creates a
Julia or Mandelbrot fractal from the digital information stored
within the photo.
However, in the images below I have used sections from fractals
cropped from the Mandelbrot viewer program.
The image on the left has been 'embossed' and has 'show outline'
added. The image on the right has had a portion of the image
'twisted'. This facility has provision for you to input the degree
of rotation. The images below these show how a photo can be
rendered into a Mandelbrot fractal.
||Adding a twist to a Mandelbrot
|A Palm at Kew Gardens
||... given the Mandelbrot treatment
The Mandelbrot Generator used in the first four images can be found
at the following web address:
For a more in-depth, technical look at how Fractals can become
artwork, there is a great step-by-step tutorial by Damien Jones, at
the following address:
My next stage involved finding software with which I could interact
and input my own variables. If you would like to try this program,
it's free for a month, if you can put up with the watermark across
your images. I cropped around it, before I decided that I had to
have the real thing. The website is:
The fractals all start from the basic Mandelbrot, Julia or Orbit
Trap equations. These are already in place. From that point, you
have a huge variety of variables you may manipulate, from basic
issues such as enlarging or adjusting the centre point, to adding
transformations, such as rotation or shear. This link
you to several examples, showing you the screenshots of the
mathematical equations and how you can adjust them.
Here are a few examples of the fractals I have been growing with
it. The first two were made using the trial version.
These were generated once I bought the program. Their complexity
comes from growing familiarity with what changes can be made to
My current challenge is to make a Mathematical model of a fractal.
Working initially on a SMART board, I created a pattern, which I
repeated to form the surface of the blocks from which I would build
the fractal model, but also to form the template of how the blocks
would join together. It is very much still a work-in-progress. My
formula is Zn = Zo/2. The variable 'Z' refers to the dimensions of
the cubes; 'n' means 'new' and 'o' means 'old'. So the formula
reads: The dimensions of the new level of cubes are half the
dimensions of the previous level.
|This is the initial design
||This is the cover for one face of each cube
This handily copied to form a net to wrap around a cube. This led
to the mass construction of far more cubes than I had
Take one corner, for instance: it contains one $12$cm$^3$, seven
$6$cm$^3$ and forty-nine $3$cm$^3$.
Given that there are four corners at the front of the model; can
you work out how many cubes are needed to complete the top two
I also intend to add three projecting corners from the four
$6$cm$^3$ digitally produced on the background. These will consist
only of the $6$cm$^3$ and the $3$cm$^3$ cubes. That being the case;
how many cubes would I need in total, in order to complete the
I hope that maybe some of you reading this will start to be
inspired to venture further in your own directions. It has been a
wonderful journey for me, using my particular interests and skills.
Whatever your interests and skills are there may be a place for
them in this area of Mathematics and Art. It may simply open a new
door for you that you did not know existed. Enjoy.