Toward a
Fractal Spline.
By Mark Graham
Many cultures, traditions and disciplines employ geometry
to artistic effect. From the ancient Moorish decorations of the Alhambra Palace to the
tiling of a texture map for the hull of a CG spaceship – everywhere you look 
geometry is a bridge between mathematics and art.
Patterns, in particular, have fascinated man over centuries
and perhaps this is due to the rhythm of repetition and the pleasure of having our
expectations confounded by unexpected variation.
Brian Eno has said that repetition is a form of change and
this is certainly true of the mathematics of fractals.
This tutorial will take you, step by step, through the
creation of a complex geometric object (a spline) from simple mathematical principles to
finished artwork suitable for output for any application (in this case, an original
mandala design for a CD sleeve) using Satori (PhotoXL or FilmFX) 2.5.
There are an infinite number of patterns we could use. I am
indebted to the Penguin Dictionary of Curious and Interesting Geometry by David Wells for
my introduction to the remarkable "Hilbert’s SpaceFilling Curve" which I
have used as the basis for my design. It is so simple and so perfect and works like
this...
We are going to construct a curve using, for the moment at
least, only straight lines.
We could describe the curve has having the following
property...
…that it goes round the centre of a square in such a
way that, were we to subdivide the square into 4 smaller squares, it would pass through
the centre of each of the smaller squares.
And so on...and so on...ad infinitum. It becomes a sort of
fractal curve  always replicating on a smaller scale the pattern of the path in the
previous stage.
This is a job that suits Satori’s resolution
independence perfectly. RI mimics the mathematics of fractals in that no matter how
closely you look at something you always see the same amount of detail.
Open Satori and create a canvas 1000 pixels wide by 1000
pixels high as a nominal starting point although size/resolution is irrelevant at this
stage  being vectorbased anything can be scaled or sized to any dimension at any time.
I’ve used a Zoom View with a
magnification of x32 by clicking the In button on the Zoom Controls
Palette 5 times.
From the Geometry, Shapes tab select Open
Bezier Spline.
From the Geometry, Properties tab set an Outline
Width of 2.
Use the Grids\Constraints toolbar to set an Interval
of x10 and y10 and apply Grid Snap.
Remember, we are going to create a curve using straight
lines. This helps us understand, at its simplest, the beautiful structure underlying the
curve.
These "lines" are vectors (directions between
coordinates) which plot the shortest path between any two clicks we make. The clicks
themselves form a path that defines the spine of the curve. Later we will edit the
properties at these control points to alter the shape of the curve.
Satori’s UI permits this way of working. In normal
practice when creating a spline one might expect to employ the "click and drag"
method setting values for the handles (curvyness) as you proceed.
In this case it important not to get the control points out
of position so just click the mouse button ONCE ONLY to define each "corner".
We’ll do the frilly stuff in a minute.
So you just draw the shape leftclicking for each vertex
remembering that you can move any which you find to be misplaced.
Click on the thumbnail to see the full image
The path of the curve at its simplest.
"The first continuation of the curve"
"By the time we get to level three it’s probably time to put the kettle
on"
But realistically it’s only 52 clicks (count ‘em)
and takes two minutes at most.
"hmmm...can anyone think of a shortcut?"
Now, once the path of the curve is set proceed to Edit
Top Object and click on any vertex and drag out the control handles.
There are three for each vertex  two control the tension
into and out of the curve and are "locked" together while the third sets the
vertex coordinates (its position).
Experiment with different settings noticing how moving the
handles changes the shape of the curve by varying the tension at the vertex. A mouseright
click will launch the Edit Spline menu where you can access further
selections such as Break Handles.
"Using Handles"
"Theme and Variations"
It soon becomes clear that there is an infinite number of
possibilities. Once the shape of the curve is defined we can proceed to thinking about its
properties  color, thickness etc.
For this design for a CD sleeve I’ve chosen a white
line with a black border over a blue background. Edit the object, set the
properties and Finish Edit.
"Edit the OBJECT PROPERTIES at any time"
Next I’ll crop the artwork with an Invert Cut,
Circle then I’ll add a border, with Outline, Circles.
"Add an INVERT CUT CIRCLE"
The design requires a line of text. I’ll place this,
choose the size and font, render the result and then move the layer up to centre the
design as a whole.
I can save out many different versions and can alter every
detail of the design without compromise.
"The final result"
