“A fractal is a way of seeing infinity.” - Benoît B. Mandelbrot
1. Intention
To work with form in our art in a more sophisticated way we will peer into the nature of structure itself. In this lesson we will explore surface and light by analyzing their fractal properties. By the end of this lesson, we will apply these properties to render a “crystal mountain”.
2. Fractals
A pattern of similar parts to the whole is what we will refer to as a fractal. Just think - a cloud is made of smaller clouds which are made of smaller clouds, etc. Applying this to our rendering processes, we can understand rendering as drawing surfaces made of surfaces. To acheive depth as we render surfaces fractally, we gradually alter the look of the fractal, according to a desired effect (in fact, this usually facilitates a stronger sense of depth than atmospheric gradients).
The image below depicts how pioneer Johfra alters his fractals according to perceptual distance. The closer formations feel distinct and further formations fade into obscure ridges. As a brief exercise, observe how Johfra emphasizes occlusion and tonality in his work.
Even in writing this very document itself, I am working fractally by breaking concepts into paragraphs into sentences into words into letters...
3. Planes
A flat surface is what we will refer to as a plane. Sculpting form as planes is inherently a binary modality because each decision to alter the surface is performed abruptly - either you alter the surface or you do not (less strict decisions, i.e. curves, are to come in the next lesson).
Mountains represent one of the most commonly understood planar fractal phenomenon. Notice in the image below how Bierstadt emphasizes planes throughout his work below to cohesively render a scene with depth. In particular, he captures both the distinct quality of near formations and abstract quality of far formations. In fact, the majority of Bierstadt’s work stands out from nearly all landscape painters, both old and new, in his sincerity towards working with planes.
Effectively, working with planes allows the artist convey a strong sense of form with minimal effort. In fact, pre-constructing objects as planes is common place for imaginary artists. The image below depicts one way to construct the head via planes. There are infinitely many ways for condensing the musculoskeletal anatomical information each with their own respective tradeoffs.
4. Rays
For our purposes in artistic rendering, we consider light to travel linearly through space in what we will refer to as a ray. As it turns out, light rays split infinitely along a recursive, or fractal, structure. Physically speaking, this is also a binary modality because when a light ray hits a surface it “divides” into two.
Refraction: a ray division changes direction as it enters
Reflection: a ray division changes direction as it bounces
Surfaces of specific materials can exhibit varying degrees of behavior, like glass (more refractive) or metal (more reflective) or clay (neither). In the case of glass, we see a warped view through the object because light rays enter according to their angle with respect to the surface. In the case of metal, we see a warped view in front of the object because light rays bounce according to their angle with respect to the surface. In the case of clay, the material exhibits physical properties which prevent light from being focused directly through or away from the surface (we typically refer to this as a diffuse material).
For our rendering purposes we will not be concerned with reflection because simulating it requires a mathematical understanding (projection mapping) which extends beyond the scope of the course.
To formalize this a bit more, when looking at the surface of an object we see an accumulation of light from every possible path reflecting back to our eye. The distribution of light along those directions depends on the physical properties of the surface material. If this point seems confusing, consider it as not completely relevant for our practical purposes and revisit it later.
The image above depicts objects rendered in a “crystalline” material (essentially, rough glass). Note specifically how the light tends to aggregate opposite the light source, except for the nearest brightest area. Thus, this model essentially is an inverted occlusion model coupled with the the highlight portion of the tonal model.
Summary
The fundamental nature of surface and light is fractal
Fractals are infinitely composed self-similar patterns
Planes fractally model abrupt changes in surface direction
Refraction fractally models abrupt changes in light direction