A spiral is a three-dimensional curve rotating around a fixed position that is receding or approaching the viewer—think of a spiral staircase or the spiral arms of a galaxy. Spirals range in size from the tiny spiral in a flower to the vast reaches of intergalactic space; the spiral is a universal shape that occurs over and over again.
A fractal image is an image where small areas of the image are replicated across the larger image in its entirety.
There a number of different kinds of fractal images, ranging from those generated mathematically (take a look at the note on page 126 about fractal art) to photographs of fractals in nature. With natural fractals, examples include the pattern of an ice crystal that mimics the pattern of a frozen pond (page 121), branches in a complex tree that resemble the tree itself (page 123), and portions of a landscape that look like the landscape as a whole (page 174–175).
Keep in mind that visual uses of spirals and fractals are not intended to be precisely mathematical in many images. Most representations of spirals and fractals are not exact: a spiral staircase is often irregular to fit the needs of the architecture, and a portion of a fractal image can be reminiscent of the image as a whole without precisely duplicating the image. Outside of the world of pure mathematics, precise fractal replication is aspirational rather than exact.
A spiral is a shape—think of a chambered nautilus shell such as the one on page 108—and also a template for a pattern within an image. For more about working with pattern templates, turn to “Design Patterns” on page 104.
A fractal is not a shape in and of itself, but represents a family of shapes. Since each fractal contains information about how the fractal was created and about the fractal as a whole, a fractal can also be thought of as a kind of blueprint for life and a way of creation.
Both fractals and spirals are often found in nature. A spiral can be a fractal, and a fractal image often contains spirals; the two shapes are related.
Fractals are “cousins” of the spiral, often accompany spirals, and are a design pattern that is invaluable to learn to work with. The fractal way of life, which is closely related to our sense of the infinite, is always significant.
Spirals are, themselves, powerful shapes with emotional and spiritual connotations.
In this chapter, we’ll explore how to use spirals and fractals in your compositions and how to take advantage of their power.
Working with Spirals
When working photographically with spirals, there are three common points of view. These refer to the camera and focal plane position in relationship to the spiral subject matter. The spatial relationship of camera position to subject is discussed on page 61.
Perhaps the most common configuration is a head-on photo of a spiral that is in front of the camera; for example, the photo of the nautilus shown on page 108.
Another common camera-subject relationship involving spirals is to look down at the spiral. Examples include the mallow blossom shown on page 111 and the shell on page 114.
Looking up and through the spiral (for example, the staircase shown opposite) is somewhat less common. But gazing up a spiral is quite powerful because seeing to the end of the spiral brings a sense of completion and viewing the infinite.
It is important to identify which of these three scenarios your image will use before you undertake a detailed setup, as they are quite different in implication, meaning, and how you want to position the camera.
The first configuration, a frontal, head-on spiral, leads to a relatively flat composition that does not require much depth of field. The key point here is to position the focal plane of the camera as parallel as possible to the subject.
When you are looking down on a spiral, it’s important to think about how far down the tunnel of the spiral you are going to go to frame the image. There are always compositional choices and options, starting with changing your position.
Depending upon your equipment, you can also change the focal length of the lens. Particularly with spirals photographed close up (such as the shell, opposite), your point of focus and aperture can make a great deal of difference to the composition. Changing the point of focus when you are up close can actually alter the framing of the image. (If you don’t believe me, give it a try!)
As a result of these choices, you can frame the spiral at a distance or close up.
Peering through the camera down a spiral, there is always a center point. A compositional question becomes how much of a spiral around the center point will your image show?
The closer to the center point you are optically, the less your image will reveal of the spiral. However, if you pull back to show more of the spiral, then the spiral can become “tunnel-like” and you lose the sense of grandeur inherent in the spiral.
There’s a balance in this decision about whether to get close or capture more of the spiral from far away, and often an optimum point. This choice largely determines the kind of image you make.
The key thing to think about is symmetrical closure. In other words, the composition should be attractive at the edges of the spiral as well as in the center, regardless of where you choose to crop the composition.
When choosing to photograph a spiral looking up from underneath, the key thing to keep in mind is that you are looking up. Most upward spiral photos are captured with a very wide-angle lens, because there is no other way to capture the full extent of a widening spiral. Therefore, as a matter of geometry, you will need to be below your camera (unless you want a body part to appear in the image). This means lying or crouching on the ground under your tripod.
Whether you’re looking at your subject head on, up, or down, a spiral, like any other subject, needs to fit into the frame boundary of your image (framing is discussed in more detail starting on page 54). This means that as you previsualize a spiral composition, you should consider how the outer edges of the spiral interact with the frame to create an interesting composition.
Positioning the Center
In addition to the interaction between the spiral and the frame, the positioning of the spiral’s center is extremely significant. The most common compositional approach is to position the spiral’s center at or near the center of the photographic composition. You’ll find that roughly 99 percent of spiral photos take this approach.
The advantage of centering is that it leads to a sense of completeness and allows the viewer to see the whole spiral. A possible disadvantage is that placing the spiral center in the center of the composition can lead to an unexciting, static image.
If you decide to break the normal pattern either by omitting the spiral center—as in Caixa Forum Stairs on page 118—or by skewing the composition so the spiral center is close to one edge—as in Wright Stairs on page 119—do so carefully.
If you play visual games with the location of the center of the spiral, you need to know that you are doing so. You should provide clues so the viewer can clearly extrapolate the location of the spiral center, as well as visualize the completion of the spiral.
Spirals are everywhere and are a universally recognized shape and pattern. It’s great to internalize the use of the spiral as a design pattern in your work, and to understand that when you encounter a spiral there’s a good chance of making an exciting composition.
Fractals the Natural Way
The gist of the fractal—whether natural or mathematical—is to create complexity from simplicity. In other words, a simple transformation is used on a portion of the image to eventually create a highly complex overall image. In mathematics, this transformation is accomplished through recursion (repetition of a formula). Leaving math out of it, in the real world this transformation is accomplished primarily through a natural process that repeats itself.
In nature, fractals include all manner of branching patterns, such as trees, river deltas, lightning, and blood vessels, as well as spiral patterns like sea shells, hurricanes, and galaxies. In fact, the largest known fractal is a spiral galaxy.
Since all fractals are formed by simple repetition, if you combine repetition with rotation and resizing, it’s easy to create your own spiral-shaped fractals. You can also create your own universes and images using fractal repetition whether or not spirals are involved.
Many fractals can be found in the plant family, with examples including pine cones, sunflowers, aloes, and the agave cactus. Aloes and agaves are typical and interesting because they form a spiral by rotating a fixed amount before growing each new piece. The angle of rotation is determined by each plant’s best angle to the sun. So the angle of rotation of the fractal created by the plant’s spiral growth depends on the sun’s angle in the location where it is planted. Turn to page 117 to see a Gem of the Drakensberg aloe that grows in a spiral-fractal shape.
It is astounding and true that the incredible complexity of organic forms around us comes largely from simple repetition.
This is because most apparently complex fractal patterns start with a simple sub-element. The sub-element is modified slightly with rotations, resizing, mirroring, and so on. This modification occurs consistently and is repeated indefinitely, with the newly modified elements added to the original pattern.
So ultimately, a fractal is a larger pattern using repetition to create a complex and pleasing larger shape that is often symmetrical in part, and all based upon a simple element and method of growth.
Zabriskie View—One of the world’s great destinations for landscape photography is Death Valley National Park in California. Within Death Valley, no one should miss Zabriskie Point, a short distance from the parking lot and possessed of views that are always changing depending on the light. There is seldom a sunrise or sunset when photographers aren’t lined up with their tripods on the viewing platform above Zabriskie Point.
In making this image, I was mindful of the way the folds in the land repeated, with slight variations in the angles of the bluffs. It seemed to me that to make the most visual sense of this landscape meant emphasizing the fractal nature of the scene in front of me.
Nikon D850, 28-300mm Nikkor at 78mm, six exposures with shutter speeds ranging from 1/8 of a second to 3 seconds, each exposure at f/22 and ISO 64, tripod mounted.
Kinds of Fractal Compositions
In my work as an artist, I have created fractal images in a number of ways. Like some other visual artists with a background in mathematics (M.C. Escher comes to mind), besides intuitive visual approaches, I am always interested in artistic methods that involve computation. Of course, the proof of the art is in the pudding, as it were. Good art transcends whatever technique may have been used to make it.
Using computers and mathematical formulas to create fractal art is a significant trend in and of itself. I’ve played with these possibilities over the years. For more about mathematical fractal art, turn to page 126.
Back in the film days early in my career, I played with prismatic auxiliary lenses to create fractal-like effects in-camera. Maybe you’ve seen this effect in 1930s films such as those created by Busby Berkeley. Back in the day, I bought my prismatic lens in New York at a camera-gadget store called Spiratone. These days, this type of special-purpose, auxiliary lens is sometimes referred to as a “fractal lens.”
My favorite approach is to photograph fractals in nature or create them in the studio using specialized photographic techniques. Fractal Face 2, opposite, is an example of a studio fractal creation using in-camera multiple exposures.
Many of the photographs that illustrate this chapter are of fractal-like subjects that I have found in nature. To photograph fractals in nature requires identifying fractal structures when you see them, and understanding a coherent way to relate the fractal composition to the edge of the image.
Framing Fractals
Many, if not most, spirals are also fractal. Generally, the framing issue with fractal-spirals is straightforward. Whatever your position in relation to the spiral (see page 112 regarding the camera’s focal plane orientation to the spiral), the spiral should take up most of the space of the composition and have an apparent relationship to the boundary frame of the image. So this quickly becomes an issue of positioning the spiral.
Compositional challenges become more intriguing with a complex fractal such as California Live Oak, shown on pages 122–123. With a natural fractal, such as this tree, in some sense the composition has no beginning and no end. How do you put a border around a subject that is inherently borderless and constantly expanding?
I don’t have a simple answer to this conundrum other than to say it is a little bit like Edward Weston’s take on the study of composition and gravity. The response should be intuitive. Compositional framing of a large fractal can probably never be perfect, but it can be graceful. This kind of framing should not seem arbitrary. In other words, there should be a reason for drawing the boundary and border.
It’s a good idea to plan to include enough of the subject matter so that its fractal nature becomes clear, but not so much that the viewer becomes lost in the details of the composition. As with any composition, there needs to be a guiding, underlying principle even if this principle isn’t clear to the viewer. For example, the tree itself organizes the composition in California Live Oak.
With fractal compositions, considering how the viewer will enter the composition, the visual path the viewer will take, and how the viewer will exit the composition is particularly important. See “Entering & Exiting,” starting on page 136.
Creating Fractal Composites
Fractal compositions can be created in postproduction using compositing with surprisingly interesting results. Post-production fractals created in Photoshop (or with other comparable software) use the techniques for their creation that mimic the mathematical or natural approaches to fractal formation. As in mathematics and nature, this is a process that creates complexity from simplicity.
If you are going to invoke complexity from simplicity, this can be accomplished by creating some simple rules for processing the component parts of your image. There’s no magic formula for what these rules should be, and you most likely will need to find them using trial and error. Examples of the kinds of transformations that you can use to create an iterative rule include replication, rotation, inversion, and resizing.
You do want to consistently apply the rules once you have found them. Consistency is what gives a fractal-like composite the sense of being a regular pattern that is comprehensible.
To create the fractal-like image, you need to repeat the steps—meaning iterate your rules—many times. The way this works is that whatever the rule is, it gets applied and reapplied to the work that has already been altered. It’s this repeated reapplication that serves to create the fractal-like effect. The rule needs to be repeated regularly on the image that has already been modified with your rule.
For example, suppose I make a rule that a portion of the image will be resized smaller by 10 percent and rotated 90 degrees (most rules will be more complex). I apply this a first time, and now I have an image with a portion of itself that is smaller and rotated. Next, I could take the entire modified image with the changes and apply the rule once again to either the entire image or a portion of it.
Rinse and repeat, then rinse and repeat!
A number of kinds of transformations can work as rules, but you will need to find these by experimentation. Also, the image that the rule operates on must be chosen carefully. Not every kind of image will work for fractal replication in postproduction. Successful composite subject matter usually calls to mind architectural or natural fractal processes (for an example, take a look at Epic Stairs on pages 132–133).
Creative Use of Spirals and Fractals
If I only had two shapes or patterns in my artistic palette, I would likely choose the spiral and the fractal. For good reason, most folks love spirals. Fractals are best thought of as a pattern for creating patterns, rather than a pattern in and of themselves.
Images that make good use of spirals or fractals are always interesting, so I recommend that you look for these shapes and patterns in nature and in your work. The more you study spirals and fractals, the more likely you are to find creative approaches to working with them. The examples in this chapter show some of the ways I have worked with fractals. I hope you find them helpful.
Spirals echo many of the fundamental shapes of the universe around us and are great representations for our life voyage. Fractals demonstrate the building blocks of life and show how complex forms, shapes, and ways of being can come together using simple mechanisms.
Together, spirals and fractals are well worth study and experimentation. Artists and photographers who work creatively with spiral and fractal shapes and patterns will find their work enhanced and enriched.